Gibbs Measures and Phase Transitions, Second Edition

De Gruyter Studies in Mathematics 9 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Geszt...

Author: Hans-Otto Georgii

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De Gruyter Studies in Mathematics 9 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Hans-Otto Georgii

Gibbs Measures and Phase Transitions Second Edition

De Gruyter

Mathematics Subject Classification 2010: Primary: 60-02, 82-02; Secondary: 82B05, 60K35, 82B20, 82B26.

ISBN 978-3-11-025029-9 e-ISBN 978-3-11-025032-9 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data Georgii, Hans-Otto. Gibbs measures and phase transitions / by Hans-Otto Georgii. — [2nd ed.]. p. cm. — (De Gruyter studies in mathematics ; 9) Includes bibliographical references and index. ISBN 978-3-11-025029-9 1. Probabilities. 2. Phase transformations (Statistical physics) 3. Measure theory. I. Title. QC20.7.P7G46 2011 515'.42—dc22 2011006346

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ©2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ® Printed on acid-free paper Printed in Germany www.degruyter.com

To my family

Preface

This book deals with systems of infinitely many random variables attached to the vertices of a multi-dimensional lattice and depending on each other according to their positions. The theory of such "spatial random systems with interaction" is a rapidly growing branch of probability theory developed with the goal of understanding the cooperative effects in large random systems. The primary impetus comes from statistical physics. The range of applications also includes various other fields such as biology, medicine, chemistry, and economics, but this volume is only devoted to those concepts and results which are significant for physics. In the physicist's terminology, this subject is referred to as classical (i.e. non-quantum) equilibrium statistical mechanics of infinite lattice systems. As is well-known, statistical physics attempts to explain the macroscopic behaviour of matter on the basis of its microscopic structure. This effort also includes the analysis of simplified mathematical models. Consider, for example, the phenomenon of ferromagnetism. In a first approximation, a ferromagnetic metal (like iron) can be regarded as being composed of elementary magnetic moments, called spins, which are arranged on the vertices of a crystal lattice. The orientations of the spins are random but certainly not independent - they are subject to a spin-spin interaction which favours their alignment. It is plausible that this microscopic interaction is responsible for the macroscopic effect of spontaneous magnetization. What, though, are the essential features of the interaction giving rise to this phase transition? This sort of question is one of the motivations for the development and analysis of the stochastic models considered herein. Although the foundations of statistical mechanics were already laid in the nineteenth century, the study of infinite systems only began in the late 1960s with the work of R.L. Dobrushin, O.E. Lanford, and D. Ruelle who introduced the basic concept of a Gibbs measure. This concept combines two elements, namely (i) the well-known Maxwell-Boltzmann-Gibbs formula for the equilibrium distribution of a physical system with a given energy function, and (ii) the familiar probabilistic idea of specifying the interdependence structure of a family of random variables by means of a suitable class of conditional probabilities. One of the interesting features of this concept is the fact that (as a consequence of the implicit nature of the interdependence structure) a Gibbs measure for a given type of interaction may fail to be unique. In physical terms, this means that a physical system with this interaction can take several distinct equilibria. The phenomenon of non-uniqueness of a Gibbs measure

VIII

Preface

can thus be interpreted as a phase transition and is, as such, of particular physical significance. The main topics of this book are, therefore, the problem of non-uniqueness of Gibbs measures, the converse problem of uniqueness, and the question as to the structure of the set of all Gibbs measures. Due to its interdisciplinary nature, the theory of Gibbs measures can be viewed from different perspectives. The treatment here follows a probabilitistic, rather than a physical, approach. A prior knowledge of statistical mechanics is not required. The prerequisites for reading this book are a basic knowledge of measure theory at the level of a one-semester graduate course and, in particular, some familiarity with conditional expectations and probability kernels. The books by Bauer (1981) and Cohn (1980) contain much more than is needed. Some other tools which are used on occasion are a few standard results from probability theory and functional analysis such as the backward martingale convergence theorem and the separating hyperplane theorem. In all such cases a reference is given to help the uninitiated reader. My intention is that this monograph serve as an introductory text for a general mathematical audience including advanced graduate students, as a source of rigorous results for physicists, and as a reference work for the experts. It is my particular hope that this book might help to popularize its subject among probabilists, and thereby stimulate future research. There are four parts to the book. Part I (the largest part) contains the elements of the theory: basic concepts, conditions for the existence of Gibbs measures, the decomposition into extreme Gibbs measures, general uniqueness results, a few typical examples of phase transition, and a general discussion of symmetries. The other parts are largely independent of one another. Part II contains a collection of results closely related to some classical chapters of probability theory. The central objects of study are Markov fields and Markov chains on the integers and on trees, as well as Gaussian fields on Zd and other lattices. Part III is devoted to spatially homogeneous Gibbs measures on J.d. The topics include the ergodic decomposition, a variational characterization of shift-invariant Gibbs measures, the existence of phase transitions of prescribed types and a density theorem for ergodic Gibbs measures. Part IV deals with the existence of phase transitions in shiftinvariant models on Zd which satisfy a definiteness condition called reflection positivity. The non-uniqueness theorems provided here can be applied to various kinds of specific models having one of the following characteristic features: a stable degeneracy of ground states, a competition of several potential wells of different depths, or the existence of an SO (TV)-symmetry. Each part and each chapter begins with an introductory paragraph which may be consulted for further information on the contents and the interdependence of chapters. The Introduction is primarily addressed to readers who are not familiar with statistical physics. It provides some motivation for the definition of a Gibbs measure and indicates why the phenomenon of

Preface

IX

non-uniqueness can be interpreted as a phase transition. Many of the general notations of this book are already introduced in Section 1.1, but some standard mathematical notations are only explained in the List of Symbols at the end. With only a few exceptions, all historical and bibliographical comments are collected in a separate section, the "Bibliographical Notes". This section also includes a brief outline of numerous results which are not treated in the text, but this is by no means a full account of the vast literature. The bibliography contains only those papers which are referred to in the Bibliographical Notes or somewhere else in the book. A few words about the limitations of scope are appropriate. As stated above, this book is devoted to lattice models of classical statistical physics. It therefore neither contains a treatment of quantum-machanical models nor an analysis of interacting point particles in Euclidean space. Moreover, this book does not include a discussion of lattice systems with random interaction such as diluted ferromagnets and spin glass models, although this is a field of particular current interest. Even with these restrictions, the subject matter exceeds by far that which can be presented in detail in a single volume. There are two major omissions in this book: the Pirogov-Sinai theory of lowtemperature phase diagrams, and the immense field of ferromagnetic correlation inequalities and their applications. A sketch of these subjects can be found in the Bibliographical Notes (especially those on Chapters 2 and 19), and I urge the reader to investigate the literature given there. Also, since readability rather than generality has been my goal, systems of genuinely unbounded spins are treated here only sporadically rather than systematically, although references are given in the Bibliographical Notes. Some further topics which are not even touched upon are the field of "exactly solved" lattice models such as the eight- and six-vertex models (cf. Baxter (1982a)), the significance of unbounded spin systems for constructive quantum field theory (cf. Simon (1974), Guerra et al. (1975), and Glimm and Jaffe (1981)), lattice gauge theories (cf. Seiler (1982)), and stochastic time evolutions having Gibbs measures for their stationary measures (cf. Durrett (1981) and Liggett (1985) as well as Doss and Royer (1978) and Holley and Stroock (1981), e.g.). Finally, I take this opportunity to thank my academic teacher Konrad Jacobs for advice and encouragement during my first years as a probabilist, and in particular for guiding my interest towards the probabilistic problems of statistical mechanics. For many years, I had the good fortune of working in the groups of Hermann Rost in Heidelberg and Chris Preston in Bielefeld, and I gratefully acknowledge their influence on my work. I am particularly indebted to Paul Deuring, Aernout van Enter, Jôzsef Fritz, Andreas Greven, Harry Kesten, Hans-Rudolf Künsch, Reinhard Lang, Fredos Papangelou, Michael Röckner, and Herbert Spohn for reading various portions of the manuscript and making numerous valuable comments. Pol Mac Aonghusa looked over the English, and Mrs. Christine Hele typed the manuscript with

X

Preface

care and patience. Last but not least, I would like to express my gratitude to the editors of this series, in particular Heinz Bauer, for their stimulating interest in this project. Munich, May 1988

Hans-Otto Georgii

Preface to the Second Edition A second edition after 23 years of rapid development of the field? One may well say that this should give reason for a complete rewriting of the book. But, on the other hand, a selection of topics had to be made already in the first edition, and this particular selection has found its firm place in the literature. In view of this and some technical restrictions, the publisher and I decided to keep the book more or less in its previous state and to make only modest changes. Apart from the correction of a few minor errors which were already fixed in the Russian edition (Moscow: Mir 1992) and some small adjustments, these changes are - a new section, 15.5, on large deviations for Gibbs measures and the minimum free energy principle, and - a brief overview of the main progress since 1988, which is added to the Bibliographical Notes. In particular, the latter will show that many of the omissions here are filled by other texts. My thanks go to Anton Bovier, Franz Merkl, Herbert Spohn and especially Aernout van Enter for valuable comments on a first draft of the second addendum. I am also grateful to the publisher and the series editors for their constant interest in this work. Munich, January 2011

Hans-Otto Georgii

Contents

Introduction

1

Part I.

9

General theory and basic examples

Chapter 1 Specifications of random fields 1.1 1.2 1.3

Preliminaries Prescribing conditional probabilities ^-specifications

Chapter 2 2.1 2.2 2.3 2.4

Gibbsian specifications

Potentials Quasilocality Gibbs representation of pre-modifications Equivalence of potentials

Chapter 3

Finite state Markov chains as Gibbs measures

3.1 Markov specifications on the integers 3.2 The one-dimensional Ising model 3.A Appendix. Positive matrices Chapter 4 4.1 4.2 4.3 4.4 4.A

Local convergence of random fields Existence of cluster points Continuity results Existence and topological properties of Gibbs measures Appendix. Standard Borel spaces

Chapter 5 5.1 5.2

Specifications with symmetries

Transformations of specifications Gibbs measures with symmetries

Chapter 6 6.1

The existence problem

Three examples of symmetry breaking

Inhomogeneous Ising chains

11 11 15 18 25 26 30 35 39 44 44 49 54 57 58 60 66 71 73 81 81 85 94 95

XII

Contents

6.2 The Ising ferromagnet in two dimensions 6.3 Shlosman's random staircases Chapter 7 Extreme Gibbs measures 7.1 7.2 7.3 7.4

Tail triviality and approximation Some applications Extreme decomposition Macroscopic equivalence of Gibbs simplices

Chapter 8 8.1 8.2 8.3

Uniqueness

Dobrushin's condition of weak dependence Further consequences of Dobrushin's condition Uniqueness in one dimension

Chapter 9 Absence of symmetry breaking. Non-existence 9.1 9.2

Discrete symmetries in one dimension Continuous symmetries in two dimensions

99 106 114 115 125 129 136 140 140 153 164 168 169 178

Part II. Markov chains and Gauss fields as Gibbs measures

189

Chapter 10 Markov fields on the integers I

190

10.1 Two-sided and one-sided Markov property 10.2 Markov fields which are Markov chains 10.3 Uniqueness of the shift-invariant Markov field Chapter 11 Markov fields on the integers II 11.1 Boundary laws, uniqueness, and non-existence 11.2 The Spitzer-Cox example of phase transition 11.3 Kalikow's example of phase transition 11.4 Spitzer's example of totally broken shift-invariance Chapter 12 Markov fields on trees 12.1 Markov chains and boundary laws 12.2 The Ising model on Cayley trees Chapter 13 Gaussian fields 13.1 13.2 13.3 13.A

Gauss fields as Gibbs measures Gibbs measures for Gaussian specifications The homogeneous case Appendix. Some tools of Gaussian analysis

191 196 204 209 210 221 228 233 238 238 247 256 257 267 273 284

Contents

XIII

Part III. Shift-invariant Gibbs measures

289

Chapter 14 Ergodicity

290

14.1 Ergodic random fields 14.2 Ergodic Gibbs measures 14.A Appendix. The multidimensional ergodic theorem Chapter 15.1 15.2 15.3 15.4 15.5

15 The specific free energy and its minimization Relative entropy Specific entropy Specific energy and free energy The variational principle Large deviations and equivalence of ensembles

Chapter 16 Convex geometry and the phase diagram 16.1 The pressure and its tangent functionals 16.2 A geometric view of Gibbs measures 16.3 Phase transitions with prescribed order parameters 16.4 Ubiquity of pure phases Part IV.

Phase transitions in reflection positive models

Chapter 17 Reflection positivity 17.1 The chessboard estimate 17.2 Gibbs distributions with periodic boundary condition Chapter 18 Low energy oceans and discrete symmetry breaking

290 296 302 308 309 312 319 323 327 338 338 343 348 359 365 367 367 374 382

18.1 Percolation of spin patterns 18.2 Discrete symmetry breaking at low temperatures 18.3 Examples

383 394 398

Chapter 19 Phase transitions without symmetry breaking

408

19.1 Potentials with degenerated ground states, and perturbations thereof 19.2 Exploiting Sperner's lemma 19.3 Models with an entropy energy conflict 19.A Appendix. Sperner's lemma Chapter 20 20.1 20.2

Continuous symmetry breaking in N-vector models

Some preliminaries Spin wave analysis, and spontaneous magnetization

408 415 420 431 433 434 439

XIV

Contents

Bibliographical Notes Further Progress References References to the Second Edition List of Symbols Index

453 489 495 532 539 541

Introduction

The theory of Gibbs measures is a branch of Classical Statistical Physics but can also be viewed as a part of Probability Theory. The notion of a Gibbs measure dates back to R.L. Dobrushin (1968-1970) and O.E. Lanford and D. Ruelle (1969) who proposed it as a natural mathematical description of an equilibrium state of a physical system which consists of a very large number of interacting components. In probabilistic terms, a Gibbs measure is nothing other than the distribution of a countably infinite family of random variables which admit some prescribed conditional probabilities. During the two decades since 1968, this notion has received considerable interest from both mathematical physicists and probabilists. The physical significance of Gibbs measures is now generally accepted, and it became evident that the physical questions involved give rise to a variety of fascinating probabilistic problems. In this introduction we shall give an outline of some physical grounds which motivate the definition of, and justify the interest in, Gibbs measures. The physical background. Consider, for example, a piece of a ferromagnetic metal (like iron, cobalt, or nickel) in thermal equilibrium. The piece consists of a very large number of atoms which are located at the sites of a crystal lattice. Each atom shows a magnetic moment which can be visualized as a vector in IR3. Since this magnetic moment results from the angular moments, the so-called spins, of the electrons, it is also called, for short, the spin of the atom. The interaction properties of the electrons in the crystal imply that any two adjacent atoms have a tendency to align their spins in parallel. At high temperatures, this tendency is compensated by the thermal motion. If, however, the temperature is below a certain threshold value which is called the Curie temperature, the coupling of moments dominates and gives rise to the phenomenon of spontaneous magnetization: Even in the absence of any external field, the atomic spins align and thus induce a macroscopic magnetic field. In a variable external field h, the magnetization of the ferromagnet thus exhibits a jump discontinuity at h = 0. (As a matter of fact, a real ferromagnet falls into several so-called Weiss domains with different directions of magnetization. We ignore this effect which is superimposed on the above behaviour. In other words, we are only interested in the intrinsic properties of a single Weiss domain.) As a second example from Statistical Physics we consider the liquid-vapour phase transition of a real gas. On the macroscopic level, this phase transition is again characterized by a jump discontinuity, namely a jump of the density of the gas as a function of the pressure (at a fixed value of temperature). This

2

Introduction

analogy between real gases and ferromagnets also extends to the microscopic level, at least if we adopt the following simplified picture of a gas. The gas consists of a huge number of particles which interact via van der Waals forces. To describe the spatial distribution of the particles we may imagine that the container of the gas is divided into a large number of cells which are of the same order of magnitude as the particles. To each cell we assign its occupation number, i.e. the number of particles in the cell. (More generally, we could also distinguish between particles of different types and/or orientations.) We also replace the van der Waals attraction between the particles by an effective interaction between the occupation numbers. The resulting caricature of a gas is called a lattice gas. In spite of all defects of this reduced picture, one might expect that a lattice gas still exhibits a liquid-vapour phase transition. From a formal point of view, this transition is similar to the spontaneous magnetization of a ferromagnet: The cells in the container correspond to the ferromagnetic atoms, and the occupation numbers correspond to the magnetic moments. The mathematical model. How can a ferromagnet or a lattice gas in thermal equilibrium be described in mathematical terms? As we will show now, this question leads to the concept of a Gibbs measure. We shall proceed in four steps. Step 1: The configuration space. What are the common features of a ferromagnet and a lattice gas? First, there is a large (but finite) set S which labels the components of the system. In the case of a ferromagnet, S consists of the sites of the crystal lattice which is formed by the positions of the atoms. In a lattice gas, S is the set of all cells which subdivide the volume which is filled with the gas. Secondly, there is a set E which describes the possible states of each component. For a ferromagnet, E is the set of all possible orientations of the magnetic moments. For example, to design a simple model we might assume that each moment is only capable of two orientations. Then E = { — 1,1}, where 1 stands for "spin up" and — 1 for "spin down". In the case of a lattice gas, we can take E — {0,1,..., N}, where N is the maximal number of particles in a cell. In the simplest case we have E = {0,1}, where 1 stands for "cell is occupied" and 0 for "cell is empty". Having specified the sets S and E, we can describe a particular state of the total system by a suitable element co = (&>;);,= s of the product space Q = £ s . Q is called the configuration space. Step 2: The probabilistic point of view. The physical systems considered above are characterized by a sharp contrast: The microscopic structure is enormously complex, and any measurement of microscopic quantities is subject to statistical fluctuations. The macroscopic behaviour, however, can be described by means of a few parameters such as temperature and pressure resp. magnetization, and macroscopic measurements lead to apparently deterministic results. This contrast between the microscopic and the macroscopic level is the starting point of Classical Statistical Mechanics as developed

Introduction

3

by Maxwell, Boltzmann, and Gibbs. Their basic idea may be summarized as follows: The microscopic complexity can be overcome by a statistical approach; the macroscopic determinism then may be regarded as a consequence of a suitable law of large numbers. According to this philosophy, it is not adequate to describe the state of the system by a particular element m of the configuration space Q. The system's state should rather be described by a family (c;) ieS of £-valued random variables, or (if we pass to the joint distribution of these random variables) by a probability measure fi on Q. Of course, the probability measure /i should be consistent with the available partial knowledge of the system. In particular, /i should take account of the a priori assumption that the system is in thermal equilibrium. Step 3: The Gibbs distribution. Which kind of probability measure on Q is suitable to describe a physical system in equilibrium? The term "equilibrium" clearly refers to the forces that act on the system. Thus, before specifying a probabilistic model of an equilibrium state we need to specify a Hamiltonian H which assigns to each configuration œ a potential energy H(œ). In the physical systems above, the essential contribution to the potential energy comes from the interaction of the microscopic components of the system. In addition, there may be an external force. For example, in the case of a ferromagnet with state space E = { —1,1} it is reasonable to consider a Hamiltonian of the form (0.1)

H(co) = -

X

JiUficOiCOj -

^

{i,j}<=S

tot.

i E S

Here J(i,j) = J(j,i) > 0, and h is real number. The term — J(i,j)coi(yJ- represents the interaction energy of the spins co; and coj. This energy is minimal if a»; = coj, i.e. if a»; and coj are aligned, the interaction is thus ferromagnetic. The number h represents the action of an external magnetic field. (If h > 0, this field is oriented in the positive direction of the spins.) A Hamiltonian of the form (0.1) can also be used in the case of a lattice gas with state space E = {0,1}. In this case, the term — J(i,j')co;COy is only non-zero when the cells i and j are occupied; hence —J(i,j) is the interaction energy of the two particles in these cells, and the condition J(i,j) > 0 means that the particles attract each other. In the lattice gas context, h is to be interpreted as a chemical potential, i.e., h represents the work which is necessary in order to place a particle in the system. As soon as we have specified a Hamiltonian H, the answer to the question which was posed at the beginning of this step is provided by Statistical Mechanics: The equilibrium state of a physical system with Hamiltonian H is described by the probability measure (0.2)

n(dm) = Z1 exp [ - jBH (tö)] dœ

4

Introduction

on Q. In this expression, the notation deo refers to a suitable a priori measure on Q (for example, the counting measure if Q is finite), ß is a positive number which is proportional to the inverse of the absolute temperature, and Z > 0 is a normalizing constant. The above ß is called the Gibbs distribution (or, somewhat old-fashioned, the Gibbs ensemble) relative to H. (In the ferromagnetic context, n is the so-called canonical Gibbs distribution. In the lattice gas case, \i is the grand canonical distribution). The rigorous justification of the ansatz (0.2) is a long story which is still far from being finished. We just mention the key words "ergodic hypothesis", "equivalence of ensembles", and "second law of thermodynamics". In the meantime, the prescription (0.2) may be regarded as a postulate which is justified by its stupendous success. Step 4: The infinite volume limit. As we have emphasized above, the number of atoms in a ferromagnet and the number of microscopic cells in a lattice gas are extremely large. Consequently, the set S in our mathematical model should be very large. According to a standard rule of mathematical thinking, the intrinsic properties of large objects can be made manifest by performing suitable limiting procedures. It is therefore a common practice in Statistical Physics to pass to the infinite volume limit \S\ -> oo. (This limit is also referred to as the thermodynamic limit.) However, instead of performing the same kind of limit over and over it is often preferable to study directly the class of all possible limiting objects. In our context, this means that the finite lattice S should be replaced by a countably infinite lattice such as, for example, the d-dimensional integer lattice Zd. We are thus led to a study of systems with infinitely many interacting components, and we are faced with the problem of describing an equilibrium state of such a system by a suitable probability measure on an infinite product space like Q = Ezd. However, if S is an infinite lattice and the interaction is spatially homogeneous then a Hamiltonian like (0.1) is no longer well-defined, and formula (0.2) thus makes no sense. To overcome this obstacle we either might consider limits of suitable Gibbs distributions as S increases to an infinite lattice; this, however, turns out to be rather difficult in general. Alternatively, we might try to characterize the Gibbs distribution (0.2) by a property which admits a direct extension to the case of an infinite lattice. Such a characterization can indeed be obtained fairly easily, as we will now show. (In fact, this characterization will lead us to a result which is intimately connected with what can be obtained by suitable limits; cf. (4.17) and (7.30) in the text). To be specific we let £ = { — 1,1} or {0,1}, S be finite, and H be given by (0.1). We also let A be any non-empty subset of S and £ G EA and r\ G ES^K any two configurations on A resp. the complement S\A; the combined configuration on S will be denoted Çn. We consider the probability of the event "£ occurs in A" under the hypothesis "»/ occurs in S\A" relative to the probability measure \i in (0.2). (deo is counting measure.) Cancelling all terms which only depend on r\, we find that

Introduction

(0.3)

5

/i(C in A|»7 in S\A) = {i((n in S)/ii{n in S\ A) = exp [ - ßH&n)-] / X exp [ - 0ff (fr)] = Z A (>,r 1 exp[- i 8H A (Ç>,)].

Here «A(^)=-

£

J(i,Mitj-Ytti\h+

ji,j)cA

ieA

X 7(U)»/; , [_

J'ES\A

considered as a function of £, is the Hamiltonian of the subsystem in A with "boundary condition" n, and ZAfo) = I

fe£A

exp[-j8ff A (&)]

is a normalizing constant. Conversely, there is only one [i which satisfies (0.3) for all £, //, and A, namely the Gibbs distribution (0.2). (To see this it is sufficient to put A = S.) Since each A c S is automatically finite, we can conclude that the probability measure fi in (0.2) is uniquely determined by the property that each finite subsystem, conditioned on its surroundings, has a Gibbsian distribution relative to the Hamiltonian that belongs to this subsystem. Now the point is that the last property still makes sense when the lattice S is infinite. We are thus led to the following definition of an infinite-lattice counterpart of a Gibbs distribution: Consider a probability measure /i on a product space Q = Es, where S is countably infinite and E is any measurable space, n is called a Gibbs measure if, for each finite subset A of S and /i-almost every configuration n outside A, the conditional distribution of the configuration in A given n is Gibbsian relative to the Hamiltonian in A with boundary condition n. The family J — (ytAM)\,\ of all these Gibbsian conditional distributions is called the specification of /i. y describes the interdependencies between the configurations on different parts of S; these interdependencies are dictated by the interaction between the components of the system. Let us summarize the above paragraphs as follows: A Gibbs measure is a mathematical idealization of an equilibrium state of a physical system which consists of a very large number of interacting components. In the language of Probability Theory, a Gibbs measure is simply the distribution of a stochastic process which, instead of being indexed by the time, is parametrized by the sites of a spatial lattice, and has the special feature of admitting prescribed versions of the conditional distributions with respect to the configurations outside finite regions. As is evident from the last sentence, there is a formal analogy between Gibbs measures and Markov processes. This analogy contributes to the purely probabilistic interest in Gibbs measures. (As a matter of fact, there are some

6

Introduction

interrelations between Markov chains and certain particular Gibbs measures; this will be seen in Chapters 3, 10, and 11.) Phase transition as a non-uniqueness phenomenon. In the above, we argued that physical systems like ferromagnets and lattice gases in equilibrium are reasonably modelled by Gibbs measures. If this is true, the Gibbs measures should exhibit a certain kind of behaviour which reflects the physical phenomenon of phase transition. In order to find out what should happen we consider again the spontaneous magnetization of a ferromagnet at low temperature. How does this phenomenon become manifest? As a first experiment, we place the ferromagnet in an external magnetic field (which is oriented along one of the axes of the ferromagnetic crystal). Turning the field off and waiting until equilibrium, we find that the ferromagnet exhibits a macroscopic magnetic moment in the same direction as the stimulating external field. A second experiment with an external field in the opposite direction produces an equilibrium state with the opposite magnetization as before. The ferromagnet thus admits two distinct equilibrium states which are compatible with its internal structure (viz. the spin coupling) and the external conditions at the end of both experiments (temperature below Curie threshold, zero external field); these distinct equilibrium states happen to be distinguishable by their magnetization. A similar remark applies to the liquid-vapour transition of a gas. Consequently, as long as we ignore all dynamical aspects and confine ourselves to equilibrium physics, we may say that a phase transition of the above kind is equivalent to the existence of several distinct equilibrium states of a specific material in a particular experimental setting. Now, the intrinsic properties of the material and the experimental conditions are modelled by the system of conditional Gibbs distributions (i.e., the specification), whereas the distinct equilibrium states are modelled by suitable Gibbs measures relative to this specification. We thus expect that the physical phenomenon of phase transition should be reflected in our mathematical model by the nonuniqueness of the Gibbs measures for a prescribed specification. Fortunately and somewhat surprisingly, the model developed above is indeed realistic enough to exhibit this non-uniqueness of Gibbs measures in an overwhelming number of cases in which a phase transition is predicted by physics. This fact is one of the main reasons for the physical interest in Gibbs measures. In view of the above, the problem of non-uniqueness, and also the converse problem of uniqueness, are central themes of the theory of Gibbs measures and also of this book. It turns out that the size of the set ^(y) of all Gibbs measures for a given specification y depends in a rather sensitive way on the precise nature of y. In order to give a rough idea of what can happen we will present a table that shows the size of g(y) in four specific models. These models differ in the choice of the state space E. In the first model, the Ising model, we have E = { — 1,1}. In the second model, E is the unit circle S 1 c U2, whence this model is called a plane rotor model. The third model is often referred to

Introduction

7

as a discrete Gaussian model; its state space is Z, the set of all integers. The fourth model is Gaussian with state space E = R, the real line. Incidentally, all these choices for E are abelian groups and can be equipped with a natural a priori measure, namely a (suitably normalized) Haar measure (i.e., counting measure in models 1 and 3, arc length in model 2, and Lebesgue measure in model 4). In all four models we put S = Zd, the integer lattice of any dimension d > 1. Moreover, all four models are defined by the same kind of interaction: The interaction energy of the spins CÜ; and cOj at any two adjacent sites i,j e Zd is assumed to be given by the quadratic expression \<x>i — co-|2, and there are no other contributions to the energy. Finally, we assume that the inverse temperature ß is large enough. In each of the four cases, the above assumptions define a unique specification y which is characteristic of the model. In order to describe the size of the associated set ^(y) of Gibbs measures it is sufficient to note the number of its extreme elements. This is because ^(y) is a simplex. (The number 0 thus means that g(y) is empty, 1 means that &(y) is a singleton, and oo means that ^(y) is infinite-dimensional.) Here is the table of results. Table 1. The number of extreme Gibbs measures with quadratic interaction at low temperatures. Proofs and/or more details can be found in the following places in the text: (1) Ising, d = 1: Section 3.2, (8.39), (11.17); d > 2: Sections 6.2 and 18.3.1. (2) plane rotor, d = 1: (8.39); d = 2: Bibliographical Notes on Chapter 9; d > 3: (20.20). (3) discrete Gaussian, d = 1: (9.17), (11.19); d > 2: Section 6.3. (4) Gaussian model, d = 1: (9.17), (13.41); d = 2: (9.25), (13.41); d > 3: (13.43). Ising

plane rotor

discrete Gaussian

Gaussian

^\„^ state lattice^\space dimension ^^^^_^

{-1,1}

S1

Z

U

d= 1 d=2 d>3

1 2 oo

1 1?

0 00

00

00

0 0 oo

Parti General theory and basic examples

This part contains the basic elements of the general theory of Gibbs measures, together with some selected supplements which can also be considered fundamental. In Chapters 1 and 2 we shall introduce the central notions of this book, such as specifications, potentials and Gibbs specifications, as well as Gibbs measures for given specifications. The definitions will be accompanied by some preliminary results which contribute to a better understanding of these concepts. In Chapter 3 we shall digress from the general theory. This chapter will be devoted to the question of the sense in which the classical concept of a finite state Markov chain fits into the framework of Chapters 1 and 2. A historically important example is the one-dimensional Ising model. The analysis of this model will give some indications of how a phase transition can occur in higher dimensions. The abstract theory will be continued in Chapter 4 which deals with the existence problem for a Gibbs measure relative to a given specification. The construction scheme for Gibbs measures requires a topological investigation of specifications. Therefore we shall obtain, as a by-product, some compactness and continuity properties of Gibbs measures. A particularly important rôle both in theory and in applications is played by specifications with symmetries. A first discussion of symmetries will take place in Chapter 5. Its main subject will be the existence of Gibbs measures with prescribed symmetries. After the theoretical chapters 4 and 5, we shall again descend to earth: Chapter 6 will be devoted to the analysis of three specific models which show a phase transition, in that the associated Gibbs measures are not unique. In each of these examples, the phase transition will rest on the fact that some Gibbs measures fail to inherit all symmetries of the specification at hand. One of these examples will be the famous Ising model in two dimensions. Having established the existence of phase transitions, it will be natural to ask for the structure of the set of all Gibbs measures relative to a given specification. This question will be answered in Chapter 7 which is a cornerstone of the general Gibbs theory. The key words of this chapter will be: extremality of a Gibbs measure, triviality of the a-algebra at infinity, and extreme decomposition. In view of the examples in Chapter 6, it will also be natural to ask for conditions which imply the uniqueness of a Gibbs measure. This question will be studied in Chapter 8. The final Chapter 9 will be devoted

10

Part I

to a related question which is also suggested by Chapter 6: Under which conditions is it true that a group of symmetries of a given specification preserves all associated Gibbs measures? In answering this question we shall also obtain some sufficient conditions for the non-existence of Gibbs measures. As should be clear from what has been said above, there are good reasons for reading the chapters of this part successively as they stand. Nevertheless, a reader who only wants to be prepared for one of the later parts can take a short-cut through Part I according to the following scheme. (Some additional hints can be found in the introductions to the later parts.) The backbone of the abstract theory consists of Chapter 1, Sections 2.1 and 2.2, Chapter 4, and Sections 7.1 and 7.3. Whenever possible, one should also have a look at Chapter 5 and Sections 8.1 and 8.3. To get some flesh on the skeleton, one should read at least one section of Chapter 6, for example Section 6.2. The remaining material of Part I can be consulted as required. The essential dependences are shown in the diagram below. 1^2^4^7^8 4-

3

4-

•*•

5^9

1

6

Chapter 1 Specifications of random fields

In our context, a random field is a countably infinite collection of random elements of an arbitrary measurable space E. In order to specify the statistical dependence properties of these random elements we shall stipulate that a certain system of conditional probabilities admits a prescribed system of versions. A suitable system of such versions will be called a specification, and each random field which admits these versions of its conditional probabilities will be called a Gibbs measure relative to the given specification. The above-mentioned notions "specification" and "Gibbs measure" are the basic concepts of this text. Their precise definitions will be given in Section 1.2. Before this, in Section 1.1, we shall define some basic concepts and introduce some notations which will be used throughout. The main discussion will begin in Section 1.3. This section is devoted to specifications which are given by a system of density functions relative to a reference measure X on the state space E. Such specifications will be called ^-specifications. As a matter of fact, all examples of specifications which will be considered later on are ^-specifications. In particular, so are the Gibbs specifications which are to be introduced in Chapter 2. Our discussion of ^-specifications in Section 1.3 will be guided by the following basic question: Which consistency condition is necessary for a system of density functions to define a ^-specification, and what are the implications of this condition?

1.1

Preliminaries

Let S be a countably infinite set and (E, ê) any measurable space. (To have something definite in mind, one might think of the case where S = Zd, the integer lattice of any dimension d ^ 1, and E = U, the real line.) (1.1) Definition. A family (o-;),eS of random variables which are defined on some probability space (Q, 3F', /i) and take values in (E, S) is called a random field. The index set S is called the parameter set and (E, ê) the state space of the random field. We shall also use a second terminology which comes from the physical applications of the theory of random fields. In the language of Mathematical Physics, a random field is usually called a (classical) spin system or, if S is a

12

Specifications of random fields

lattice, a lattice model. Each element i of S is called a (lattice) site, and the associated random variable at is called the spin at site i. The space E is sometimes called the single spin space. (Of course, the term "spin" refers to the magnetical interpretation which was explained in the Introduction. In the lattice gas interpretation, the word "spin" has to be replaced by something like "occupation number".) According to a basic principle of measure theory, all information about a random field is contained in its (j° m t ) distribution on Es. Therefore we can and will assume that each random field is given in its "canonical version". That is, we agree to make the following specific choices: (1.2)

Q = Es = {m =

(mi)ieS:mieE},

the set of all possible configurations m of spins, (1.3)

& = Ss,

the product
o{. Q -> E CO - » C O ; ,

the projection onto the i'th coordinate. In this "canonical" set-up, a random field is just a probability measure on the product space (Q, 5F). The set of all random fields will thus be denoted by (1.5)

0(0, &).

Let us introduce some further notation. For each A c S we let (1.6)

aA: Q -> EA

denote the projection onto the coordinates in A. Thus
Next we put (1.7)

Sf = { A c S : 0 < |A| < oo},

where | A| is the cardinality of a set A. Note that Sf is countable. By definition (1.3), #" is the smallest
{
{he^,AeSA).

For each A c= S we shall also consider the
Preliminaries

13

and measure kernels. Let (X, SC) be a measurable space. We let Jt(X, SC) denote the set of all ^-finite measures \i on (X, SC) with ß(X) > 0, and 0>(X, SC) the set of all probability measures on (X,SC). We shall often use the functional analytic notation

(1.9)

M/) = J/d/i

for the integral of a measurable function / : X -> M. with respect to a measure ß on (X, SC) (provided this integral exists). Similarly, if & is a sub-(X, SC) then we shall write

(l.io)

n(f\m = E„(/I*)

for the conditional /^-expectation of/ given J1. In particular, p(A\#) = p(lA\&) will be the conditional probability of an event A e SC given &. Here (1.11)

\1 if x e A, Ux *w) = .„ ^ . [0 otherwise

is the indicator function of A. Furthermore, i f / i s a nonnegative measurable function on X then we let f\i denote the measure with Radon-Nikodym density / with respect to yU. That is, (1.12)

fn(A)±n(f\A)

= lfàn

(A e3T).

A

Finally, if v is a second measure on (X, SC) then the notation v « ft means that v is absolutely continuous relative to \i. Next we let {Y,<W) be a second measurable space. A function n: SC x Y-» [0, oo] is called a {measure) kernel from (Y, <&) to (X, SC) (or simply from ty to SC) if (i) n{• \y) is a measure on (X, SC) for all y e Y, and (ii) n(A\ •) is ^-measurable for each A e SC. If, in addition, n(X\ •) = 1 then n is called a probability kernel from ^ to SC. For example, if cp is a measurable mapping from X to Y then the function (A, y) -> l x o cp(y) is a probability kernel from ^ to SC. A kernel 7r from <& to SC maps each measure n on (7, ^ ) to a measure /m on (X, SC) which is defined by (1.13)

im(A) = \ànn(A\-)

(A e SC).

For example, if n is defined in terms of a measurable mapping (p: Y -> X as above then /m = cp(yu), the cp-image of yU. cp(n) is defined by the formula (1.14)

{A e SC).

Clearly, if n is a probability kernel and /j,e0>(Y, <&) then \m e 0>(X, SC). On the

14

Specifications of random fields

other hand, for each measurable function / : X ->• U we shall consider the measurable function (1.15)

nf±n(f\-)

= J n(dx | -)/(x)

on Y (provided it is well-defined). The symbol nf should not be confused with the next notation which is similar to (1.12): In the case / ^ 0 we let fn denote the measure kernel from <& to 3C which is defined by (1.16)

fn(A\-)±n(flA) = j7t(dx|-)/(x)

{AeT).

A

Next, if (Z,2£) is a third measurable space then the composition nxn2 of a kernel nt from 3£ to <& and a kernel n2 from <& to 3C is defined by the formula (1.17)

nln2{A\z) = \nl{dy\z)n2{A\y)

(Ae3C,zeZ)

and is a kernel from Jf to #". Finally, we let Si be a sub-c-algebra of #". A kernel 7t from Si to #" is said to be proper if (1.18)

7cC4riß|-) = n(,4|-)lfl

(ieä",ßeJ).

Clearly, this condition implies that n(fg) = (7t/)g whenever / is an 9Cmeasurable and g a ^-measurable bounded function. For probability kernels n, (1.18) is equivalent to the simpler condition (1.19)

n(B\-) = lB

(Béai).

Indeed, if A e % and B e Si then (1.19) implies n(AnB\-)^n(A\-)

A n(B\-) = n(A\-)lB

and n(A\B\-)^n(A\-)lXXB. This gives (1.18) because n(A flB| •) + 7tM\B| •) = 7tM| -)1 B + ^ 1 " ) W It is particularly convenient to use proper probability kernels as regular versions of conditional probabilities. One reason for this is stated in the remark below. (1.20) Remark. Let (X,2£) be a measurable space, âS a sub-u-algebra oîSC, n a proper probability kernel from Si to 9C, and /i e S*(X, 9C\ Then \i(A\S$) = 7t(^l|-)/i-a.s. if and only if /i7t = \i.

for all 4 e#"

Prescribing conditional probabilities

15

Proof: By definition, the first statement means that H(A H B) = J d/i K{A\ •)

{A £ SC, Be &).

B

As n is proper, the integral on the right equals \L%(A D B). The remark is thus obvious, D

1.2

Prescribing conditional probabilities

Let us return to the consideration of random fields. According to our physical intentions we are interested in random fields for which the spin variables al exhibit a particular type of dependence. To specify the dependence type we shall proceed in analogy to the definition of a Markov chain: We shall look at the conditional distributions of certain collections of spins when the values of all other spins are given, and we shall require that these conditional distributions are almost surely given by some prescribed probability kernels. However, in contrast to the definition of a Markov chain we are not allowed to use concepts like "future" and "past" which need a linear ordering of S. This is because our primary interest focuses on the case in which S is a multidimensional lattice such as the integer lattices Zd, d ^ 2. Instead, we shall follow the natural device of Dobrushin (1968) and Lanford and Ruelle (1969) which is to prescribe the conditional distributions of all finite collections of spins. In other words, we shall look at the conditional distributions relative to the external cr-algebras (1.21)

srK = ^ SXA ,

where A runs through the set 9" of all finite non-empty subsets of S. (By definition, 3~K is the cr-algebra which contains all information about the "boundary condition", i.e., the configuration of all spins outside the finite region A.) Plainly, these cr-algebras can be defined in general, without specifying any particular structure of S. (One might ask why we insist on A being finite. First of all, there is a practical reason: We should not prescribe too many conditional probabilities, for otherwise it will be hard to check if a random field with these conditional probabilities has any chance of existing. The main reason, however, is the following one which was already discussed in the Introduction: For finite A, the Gibbsian formalism of Equilibrium Statistical Mechanics provides us with a simple scheme for constructing probability kernels from 3~K to SF. This construction breaks down when A is infinite. So, in the Gibbsian case the restriction to finite A's is necessary. Since the abstract framework which we are going to develop here is guided by the Gibbsian prototype, we stick to the same restriction in general.)

16

Specifications of random fields

Taking the family ( ^ ) A s ^ of c-algebras as a basis, we can proceed just as in the case of a Markov chain. A dependence type for a random field is specified by a family y — (y A ) As ^ of probability kernels yA from (Q, STA) to (Q, !F\ and a random field /i is considered to exhibit the dependence type which is specified by y if it satisfies the condition (1.22)

n(A\fA)

= yA(A\-)

/i-a.s. for a l M e J F and A e ST.

Stated in words, this condition means that for each finite A the given kernel yA is a regular conditional distribution for JJL relative to the c-algebra STK. Suppose now we are given a family y as above. Suppose further that y is reasonable, in that there exists a random field ft which satisfies (1.22). Then y is bound to satisfy two conditions. Indeed, let A , A e y with A cz A, A e !F, and B e 2TK be given. The well-known properties of conditional probabilities (cf. Bauer (1981), assertions (54.16) and (54.18), e.g.) then imply that yA(Ar\B\-)

=

fi(Ar\B\^A)

= n(A\rA)lB

= yA(A\-)lB

A*-a.s.

and 7A7A(^|-) = M M ^ I ^ V ) I ^ ) = li{A\2TA) = yA(A\-)

H-2L.S.

That is, yA is "proper /z-almost surely", and yA and yA are "consistent /z-almost surely". However, we want y to be reasonable without knowing or requiring anything in advance about the supports of the random fields ft which satisfy (1.22). Therefore we need to require that the preceding equations for y hold everywhere rather than /i-almost everywhere. This fact motivates the following basic definition. (1.23) Definition. A specification with parameter set S and state space (£,
WfoF):

for allAeJf

fi(A\^A) = yA{A\ -)/*-a.s. and A e ^}

are then said to be specified or to be admitted by y. If y is a Gibbsian specification in the sense of Definition (2.9) below then each jx e 'Siy) will be called a Gibbs measure. Although this term should be reserved for the case of Gibbsian specifications we shall often follow a widespread practice and also use it in the general case. Let us observe that every specification y also satisfies the converse con-

Prescribing conditional probabilities

17

sistency relation yAyA = yA when A c A . This is because yA is proper and 2TA cz 2TA. Our next remark provides a simple characterization of &()>). A subset Sf0 of the index set Sf (which is directed by inclusion) is called cofinal if each A 6 if is contained in some A e y o . For example, if S = Zd for some d ^ 1 then the set ^o = { [ - n , n ] " n S : n ^ 1} of all centered cubes is cofinal. (1.24) Remark. Suppose y is a specification and ^ a random field. Then the following statements are equivalent: (a) ^ e <S(y). (b) [iyA = ii for all A e Sf.

(c) There is a cofinal subset £f0 of 5^ with [iyA = \i for all A e yo. Proof. The equivalence of (a) and (b) follows from Remark (1.20). (b) trivially implies (c), and (c) gives (b) because for each A e ^ there is some A e ^ 0 with A c A and therefore

The preceding remark shows that &(y) is nothing but the class of all probability measures that are preserved by some countable family of probability kernels. This fact already determines the simplex structure of &(y), as we shall see in Chapter 7. We conclude this section with four basic questions that arise in connection with Definition (1.23). (A) Is there any easy way of constructing specifications? If it is true that the Gibbs ansatz of Equilibrium Statistical Mechanics provides a construction scheme, what are then the characteristic features of the specifications which can be obtained by means of this ansatz? A first step towards these questions will be taken in Section 1.3 below, and a full discussion will follow in Chapter 2. (B) Is &(y) non-empty? This will turn out to be not necessarily the case. Sufficient conditions will be provided in Chapter 4. (C) For which specifications y is $(y) a singleton? This question will be taken up in Chapter 8. (D) Are there any examples of specifications y that admit several distinct random fields? What is then the structure of &(y)7 Are there any general principles behind these examples? These last questions are the main theme of this book. Three basic examples of non-uniqueness will be presented in Chapter 6. Further examples will follow in Parts II to IV. As we have emphasized in the Introduction, interest

18

Specifications of random fields

in the problem of non-uniqueness arises because of its physical interpretation as a phase transition.

1.3

A-specifications

Having introduced the notion of a specification we need to establish that specifications do exist. This section contains some first steps in this direction. Our first remark below will show that it is easy to construct certain trivial specifications. After that we shall discuss the problem of whether non-trivial specifications can be obtained from the trivial ones by means of a suitable perturbation procedure. The above-mentioned trivial specifications are those which specify the independent identically distributed random fields. In the "canonical" setting that was adopted from (1.2) to (1.4), the i.i.d. random fields are nothing but the product measures Xs with X e 0>(E, S). The remark below states that for any given A e SP{E, S) the standard versions of the conditional probabilities Xs (• \STA\ A e £f, constitute a specification. (1.25) Remark. Let X e 0>{E, S) be given. Define XA(A\œ) = XAx Ô^JA) = X*(Ç e £ A : Oo svv e A). Here 0 # A <= S, A e SF, œ e Q, and <5WsXA is the Dirac measure on (E, S)S^A at (os\A- Then each XA is a proper probability kernel from STK to &. Moreover, for any two non-empty subsets A, A of S we have XAXA = XAUA. In particular, the family X. = (XA)AE^ is a specification. A. satisfies

W

= {Xs}

and is called the independent specification with single spin distribution X. Proof. By definition, each XA(-1co) is a probability measure on (Q,êF). On the other hand, if/ is a bounded measurable function on Q then, as a consequence of Fubini's theorem, the function XAf = \

XA(dOMaSXA)

is ^-measurable. Hence XA is a probability kernel from 2TA to 3F. Further, if i e J * and B e ^A then XA(Ar\B\co) = j m ) U C c o S X A ) U C c o s X A ) = XA(A\co)lB(co) because ÇcoSXA e B if and only if co e B. Hence XA is proper.

^-specifications

19

Next, let A, A c S be non-empty. Then for each bounded measurable function / and all co e Q we have

^ A / M = 1 ^ A (d0 1 A<WMCco SXA ) SXA ) = jA A \ A (dC)j^(d/ 7 )/( f7 Cc ÜSX(AUA) ) = ^AUA/M-

Thus /lA/lA = /lAUA. Finally, for each p e ^(Q, J^) and A e ^ w e have /UA = Xs on J*A. Thus if ß e ^(A.) then /x = Xs on each J ^ and therefore, by the monotone class theorem, on 3F. The assertion Xs e ^(X) is obvious, D Of course, we are primarily interested in dependent specifications. Such specifications are easily constructed from independent specifications by means of suitable families of densities. In fact, this construction is not restricted to the case when X is a probability measure, and it will be useful later on to also include the case of an infinite reference measure X. (We also might associate a separate measure Xi to each site i e S, but for notational convenience we shall refrain from this obvious generalization.) (1.26) Notation. For any given measure XeJi(E,i) we let X. = (XA)Ae^ denote the family of measure kernels XA from STA to 3F which are defined by XA(-\œ) = XAxôa^

(Ae^coeQ).

A glance at the proof of (1.25) shows that each XA is proper, and XAXA = XAUA for all A, A e Sf with A n A = 0. Suppose now we are given a family p = (p A ) Ae y of density functions. We then can define the kernels pAXA according to equation (1.16), and we would like to know whether these kernels constitute a specification. This question gives rise to the following definition. (1.27) Definition. Let X e M{E,g). A X-modification is a family p = (p A ) Ae y of measurable functions p A : Q -> [0, oo [ such that the family pX_ = {pAXA)Ae£/> is a specification. A X-specification is a specification y of the form y = pX_ for some ^-modifications p. A /-modification p (and also the associated ^-specification pX) is called positive if pA > 0 for all A e ^ . (1.28) Remarks. (1) Each probability kernel of the form pAXA is proper because XA is. Therefore a family p = (p A ) A e ^ of measurable functions on Q is a A-modification if and only if the kernels pAXA are normalized, i.e. ^APA

= J ^ A ( < M - ) P A M = 1,

and consistent, i.e. ^A(PA ^A(PA U )

= ^A(PA 1J

(Ae &)

20

Specifications of random fields

for all A , A e y with A c A . The second equation will be analysed in Proposition (1.30) below. (2) Let y = pX. be a ^-specification. Then (1.29)

9(y) = {n e 0>(Q, &)\\L = p A (Mv)

for all A e ^ } .

This follows immediately from the equation

pA(fiXA)(A) = j p(dco) J XA(dC\œ)pA(0 = MM) A

which holds for each fi e ^(Q, IF), A e IF, and A e if. In particular, if p is positive then for all \i e <&(y), A e if, and B e SA we have n(aA e B) = 0 if and only if ^A(B) = 0. Thus any two measures in ^(y) are mutually absolutely continuous on J*A for all A e if. (3) If I = rX for some measurable function r > 0 on E then p defined by pA(co) = pA(co) / F] rfa)

(A e if, œ e Q)

/ leA

is a 1-modification with pl_ = p^,.. In particular, we can choose a function r > 0 with ^,(r) = 1 because X e Jt{E, S). Thus each ^-specification is a 1specification for some X e 3P(E, S). (4) Suppose y is a specification. A necessary condition for y to be a Xspecification is that yA(-\a>) « XA(-\œ)

(A e if,œ e Q).

As yA and XA are proper, (1.29) is equivalent to the condition yA(aA e -\œ)«XA

(A e if, œ e Q).

In general, this condition is not sufficient for y to be a ^-specification. This is because the density functions pA which are provided by the Radon-Nikodym theorem are only known to be partially measurable, in that the functions C ->• PA(CACOS\A) a r e measurable for every œ e Q. However, if S is countably generated then a constructive martingale approach to the Radon-Nikodym theorem (see, for example, Bauer (1981), Section 61 III) shows that y is a ^-specification whenever y is absolutely continuous with respect to X in the sense above. (5) Suppose E is countable and X is counting measure. Then each specification y is a ^-specification. The associated ^-modification is given by P/Sfo) = ytX{aK = (oK}\(o)

(Aeif,œeQ).

y is positive if and only if yA(^4| •) > 0 for each atom Aoî ^FA and all A e if. (6) For many of the most popular choices of an uncountable state space (E, S) there is a natural "a priori" measure X on E such that the /(.-specifications are of primary interest. For instance, if E = W for some n ^ 1 then Lebesgue measure on E is such a measure. If £ is a compact group then (normalized)

A-spécifications

21

Haar measure is a natural choice of X. Or if E is some sphere then the surface measure on E might serve as an a priori measure, o Because of the last two remarks we will henceforth confine our attention to 2-specifications. In order to obtain examples of dependent 2-specifications we need to construct non-trivial 2-modifications. To this end the following characterization of 2-modifications will be useful. (1.30) Proposition. Let X e Jt{E,S) and p = (p A ) Ae y a family of nonnegative measurable functions on Q. with XApA = 1. Then the following statements are equivalent: (a) p is a À-modification. (b) For all A, A e 9 with A c A and all œ e Q pA = pA XApA

XA(- \co)-almost surely.

(c) For all A, A e 9 with A c A, all a e Q and XA\A(- \a)-almost allcoeQ we have PA(C)PA(»7) =

PA(I)P\(0

for XA(- |co) x XA(- \co)-almost all (Ç,n) e Q x Q. Proof. We put y = pÀ_ and fix two sets A, A e 9 with A c A . 1) First we will show that yAyA = yA if and only if ^ A ( P A ^ P A ^ A P A I ' ) = 0.

This will prove the equivalence of (a) and (b). The latter property is equivalent to the identity PA^A = (PA 'UPAMA-

Thus we only need to prove that 7A7A = (PA ^APAMA

or, equivalently, that yAy*f) = 4 ( / P A ^APA) for each nonnegative measurable function / on Q. We fix such an / and let g and h be two further such functions. As XA is proper and XAg and XAh are ^-measurable, we have

Choosing g = fpA and h = pA we conclude that ^A(/PA 4 P A ) = 4 ( P A 7A/)-

On the other hand, we know that XA = XA\AXA. Consequently,

22

Specifications of random fields -*A(/PA /UPA) = ^A(PA yAf)

= TAÏA/-

2) Next we show that (b) is equivalent to (c). Let a e ß . Then (b) implies O = ^A(IPA-PA/UPAI)0X)

= 1 ^A\A(deo|a)/lA(|pA - pA AApAj)(o>). Therefore we have for XA\A(- |a)-almost all co PA

= PA ^APA = PA ^ A P A M

^A("

M-almost surely

and thereby PÀQP/M) = PA(C)PA(>7UAPAM =

PÀQpM

for XA(- |co) x XA(- |cu)-almost all ((, rç). Conversely, integrating over rj and using the equation XA\AXA = XA we see that (b) follows from (c). D Statement (c) of the proposition above suggests the following definition. (1.31) Definition. Let us agree to call a family h = (hA)Ae9. functions hA: Q, -> [0, oo [ a pre-modification if

of measurable

hA(C)hA(r,) = MOM*/) for all A , A e y with A c A and all (, r\ e £2 with (SXA = rçsyA. It follows from Proposition (1.30) that if E is countable and X is equivalent to counting measure then each A-modification p is a pre-modification. More generally, the same conclusion holds under the topological conditions (i) to (iii) listed below: (i) £ is a second countable topological space and S its Borel a-algebra. (The existence of a countable base of the topology is necessary to make sure that U e SA whenever A c S and U c EA is open in the product topology.) (ii) X e Ji(E,S) is everywhere dense, i.e., X(U) > 0 for all non-empty open subsets U of E. (This also implies that, for all A <= S, XA is everywhere dense in EA.) (iii) For all A , A e ^ with A c A and all a e f i , the function ( -> pA((a>s\A) on EA is continuous. The simple proof is omitted because we shall only need the following converse result: (1.32) Remark. Suppose h is a pre-modification and X e Ji{E,S) is such that 0 < XAhA < oo for all A e y . Then p = (hA/XAhA)Ae6? is a ^-modification. Proof. Since AA/JA is J^-measurable, we have XApA = XA(hA/XAhA) = XAhAj XAhA = 1, and p is also a pre-modification. A fortiori, p satisfies assertion (c)

^-specifications

23

of Proposition (1.30). Forifcü e Q then the product measure XA(- \œ) x XA{- \co) is supported on the set {(£, r/) e Q x Q: £S^A = ns\A = coS\A}. The remark thus follows from Proposition (1.30). G A large class of pre-modifications will be introduced and investigated in Chapter 2. Here we include a theorem that deals with the following question. To what extent is a specification y (and the associated set @(y)) determined by its "singleton part" (?{;}); eS ? This question is a counterpart to Remark (1.24)(c) above which asserts that ^(y) depends on the yA's for large A's only. In spite of the theoretical interest in this question, the theorem below will be used only occasionally. On a first reading, one should therefore skip the proof and proceed directly to Chapter 2. (1.33) Theorem. Let X e Jt(E, ê), and suppose y is a specification which enjoys the following property: For each i e S there is a measurable function p^ > 0 on Q such that y^ = p^X^y Then there exists a positive X-modification p with y = pX_. y is uniquely determined by (P{i))ieS and X, and %{y) = {pe 0>{Q, #"): py{i} = p for all i e S} = {p. e 0>(Q,^y. p = p{i}{pX{i]) for all i e S}. Proof 1) In view of Remark (1.28)(3), we can assume without loss that X(E) = 1. For each A e ^ w e shall define a positive measurable function pA in terms of {p{i])ieA and X such that p = pA(pXA) for any p e 0»(Q, J*) satisfying W{.} = V- f° r a ll i e A. This will prove the theorem. Indeed, since yA(- \(o)y^ = yA(- \co) for all i e A and œ e Q we then can conclude that y A (-1eo) = p A (y A (- \œ)XA) = pAXA(- \co)

for all œ e Q. Hence yA = pAXA. This shows that (i) p = (pA)Ae<^ is a Xmodification and y = pX_, (ii) y is uniquely determined by (p{;})ieS and X, and (iii) if p e &(Çl, #") is such that py{i} = p for all i e S then p = pA(pXA) = pyA for all A e ^ and thereby p e &(y). 2) To define the pA's we shall proceed by induction on | A|. The case | A| = 1 is trivial. So let A e if be the union of two disjoint sets A 1 ; A2e^, and suppose p Ai and pA2 have already been constructed. We define =

JPA.AWPA.PÄ,)

[l

if

^A1(PA1PÄ21)<°o» otherwise.

Then 0 < pA < oo, and pA is measurable. To show that pA has the required property we let p e 0»{Q, #") be such that py{i] = p for all i e A. By the induction hypothesis, p = pAk(pXAk) for k = 1, 2. Thus we can write, letting / ^ 0 be an arbitrary measurable function,

24

Specifications of random fields

PK(f) = P^A2(^AJ) = MPÄ* *AJ) = I^ASPA.PÄI =

KJ)

^A1(/^A1(PA,PÄ21))

= M/PA~MA 1 (PA 1 PÄ 2 1 ))-

In particular, putting / = 1 we see that AAi(pAip^) < oo /i-almost surely. Hence n^A(f) = p{fp\X) for all / ^ 0. It follows that \i = pA(/UA), and the proof is complete. (In general, the above definition of pA depends on the choice of Aj and A 2 . To obtain an unambiguous definition of pA we thus need to choose a definite strategy for exhausting A site by site.) D

Chapter 2 Gibbsian specifications

In Section 1.3 we have seen how a specification can be defined by means of an a priori measure X and a A-modification p. Here we shall use the BoltzmannGibbs Ansatz to define a A-modification p® in terms of an interaction potential O. The resulting A-specification y® is called the Gibbs specification for O, and each n e ^(O) = ^{f) is called a Gibbs measure. In addition to the physical significance of these concepts (which was already discussed in the Introduction), there is also a purely mathematical reason for studying Gibbsian specifications. Namely, each A-specification which exhibits some natural extra properties is necessarily Gibbsian. This follows from the Gibbsian representation theorem (2.30). Thus, even a complete layman in Statistical Physics will be led to consider Gibbsian specifications (provided he is interested in random fields), and I hope that this book will convince him that it is worthwile doing so. In Section 2.1 we shall introduce the basic terms potential, Gibbs distribution, Gibbs specification, and Gibbs measure. We shall also define particular classes of potentials such as uniformly convergent, absolutely summable, finite range, nearest neighbour, and pair potentials. All these types of potentials will be considered repeatedly throughout this book. Section 2.3 contains the Gibbs representation theorem which was mentioned above. It rests on the notion of quasilocality which will be discussed in Section 2.2 and will also be important later on, especially in Chapter 4. Physically speaking, quasilocal functions are nothing but the microscopic quantities, and the notion of a quasilocal specification is suggested by the physical assumption that each spin has a microscopic horizon of interaction. From a mathematical point of view it is natural to consider quasilocal specifications because it is trivial to construct non-quasilocal specifications with many mutually singular Gibbs measures. In the final Section 2.4 we shall deal with equivalence: many different models that are designed to describe completely different physical systems may be mathematically equivalent, in that the associated potentials define the same Gibbs specification. Using suitable normalizations we shall be able to single out a unique representative from each class of equivalent potentials. Whilst the definitions in Sections 2.1 and 2.2 are fundamental for this book, the results of Sections 2.3 and 2.4 will be used only occasionally in later chapters. For a reader who is willing to accept the Gibbsian specifications on physical grounds it might thus be preferable to concentrate on the first two sections. He can return to the other sections when the need arises.

26

Gibbsian specifications

2.1

Potentials

The equilibrium distribution of a finite system of Statistical Physics is generally believed to be the Boltzmann-Gibbs distribution. Its density relative to a suitable a priori measure on the phase space is usually written in the form Z" 1 exp(-ßH). Here H is the Hamiltonian of the system. It assigns to each phase point œ the total energy H(a>) which is made up by the energy contributions of all subsystems. These contributions are described by a potential . O^(co) is the excess energy of subsystem A when the energy of all proper subsystems of A is disregarded. The number Z > 0 above is a normalizing factor. The coefficient ß > 0 is related to the absolute temperature T, in that ß = 1/kT, where k is Boltzmann's constant. From a mathematical point of view, there is no loss in setting k = 1. We shall do so throughout this book. In fact, when the temperature is held fixed we can and shall incorporate the factor ß into the Hamiltonian. This amounts to setting ß = 1. As stated before, the Boltzmann-Gibbs distribution does not admit a direct extension to infinite systems. However, when dealing with infinite systems we can still look at finite subsystems provided the "rest of the world" is held fixed. Indeed, starting from a potential we can define for each finite subsystem A a Hamiltonian H® which includes the interaction of A with its fixed environment. The associated Boltzmann-Gibbs distribution can be viewed as a probability kernel which maps each environment to the corresponding equilibrium distribution of subsystem A. The collection of all these kernels constitutes a specification, the Gibbsian specification for O. (Historically, it was this fact which gave rise to the notion of a specification.) Let us turn to the details. As in Chapter 1, we are given a measurable space (£, $), the state space, and a countably infinite set S, the set of sites. We look at the product space (Q,&) = {E,é)s. As in (1.7) we let y be the countably infinite set of all non-empty finite subsets of S. We shall often be concerned with infinite series which are indexed by an infinite subset of 5f. (2.1) Convention. Suppose t?0 is an infinite subset of y and let a: ^0 -> U be any function. We shall say that the infinite sum

exists and equals seU, and we shall write

s = X a ( A )' Ae^o

if the net I

£

\,Aey0,AcA

a(A) I /Aey

converges to s.

Potentials

27

(2.2) Definition. An interaction potential (or simply a potential) is a family $ = (®A)Aey of functions $,,: Q -> U with the following properties: (i) For each Ae y,Q>A is J^-measurable. (ii) For all A e y and co e Q, the series (2.3)

HX(a>)=

£

OA(
exists.

H%(co) is called the (total) energy of co in A for <5, and H * the Hamiltonian in A /or <5. We shall also use the abbreviation (2.4)

h*((u) = exp[-fl?((u)].

h* is called the Boltzmann factor. The negative sign in (2.4) corresponds to physical convention, although it is a nuisance in practice. Condition (2.2) (i) above implies that, for each potential $ and all Ae £f,®A may be thought of as a measurable function on (EA, SA), and we will do this occasionally to simplify notation. The next proposition provides a key to the construction of dependent specifications. (2.5) Proposition. For each potential <5 the family modification.

(II*)AE^

is a positive pre-

Proof. Let 0 ^ A <= A e
H%-H»=

£

^\-

®A

An&^0,AnA=0

and property (2.2) (i) of <5. D Combining Proposition (2.5) and Remark (1.32) we obtain a simple method for the construction of dependent specifications, as we shall show now. Let A e Jl{E, S) be an a priori measure. We shall think of A as being a fixed quantity just as E and S are. Therefore we shall suppress the ^-dependence in the notations to be introduced below. Let us agree to call a potential $ A-admissible if (2.7)

Z ? M A AAh*(co) = J AA(dC)exp[-/f*(Co)SXA)]

is finite for all A e y and co e Q. ZA(cu) is called the partition function in A for <5, <x> (and A). For each A-admissible potential <1> we may define the functions (2.8)

Pt

= h*IZ»

(Aey)

28

Gibbsian specifications

on Q. According to Proposition (2.5) and Remark (1.32) the family p® = i s a ^-modification. We thus arrive at the basic definition of this text.

(PA)AS^

(2.9) Definition. Suppose is a A-admissible potential and œ eQ, A e £f. Then the probability measure A^yZ{A\a>)±P*lA{A\a>) = Zïiœy1

f ^dOexp[-i/*(ÇœsvJ]UÇœSXA)

on (Q, #") is called the Gibbs distribution in A with boundary condition coS\A, interaction potential , and single spin measure X. The ^-specification y* = (y^Ae^ = P*7'-. i s called the Gibbsian specification for and X. Each random field /i e ^() = ^(y*) is called a Gifefes measure or a Gj'fefes random field for ) of all Gibbs measures for O can, and should, be regarded as a proper mathematical description of the set of all possible equilibrium states for a physical system that consists of a huge number of components which are coupled together by O. (In fact, the Gibbs measures exhibit the rather strong equilibrium property that even each microscopic, i.e. finite, subsystem is in conditional equilibrium when the "surrounding world" is frozen.) According to this interpretation, the nonuniqueness of the Gibbs measure for a given O means that the physical system may "choose" from several equilibrium states. Such a "free choice" is characteristic of a physical system which undergoes a phase transition (of the specific kind that was considered in the Introduction). This fact suggests the following terminology. (2.10) Definition. A potential will be said to exhibit a phase transition if |0(O)| > 1. It should be mentioned that the non-uniqueness of the equilibrium state is not the sole critical phenomenon of physical interest. An example of a different sort of transition is the so-called Kosterlitz-Thouless transition which is characterized by a change from an exponential decay of correlations to a power law decay; cf. Fröhlich and Spencer (1981). In this text, however, we shall concentrate on phase transitions in the sense of the above definition. In the remaining part of this section we shall introduce some particular classes and properties of potentials. This will give us some familiarity with the fundamental notion of a potential. (2.11) The space 3fl. A potential O is said to be absolutely summable if for all

Potentials

29

ieS

(2.12)

\m\i=

Z

IIOJ

is finite; here || • || is the sup-norm. Clearly, this condition implies property (2.2) (ii) of a potential. It even implies that O is uniformly convergent in the sense that, for all A e ^ , the net (H^A)âkey converges uniformly to if®. Here (2.13)

fl*A=

X

0„.

We let ^ denote the linear space of all absolutely summable potentials. The functions || • ||; on ^ are seminorms and define a complete metrizable locally convex topology on aß. Thus ^ is a Fréchet space. The assumption of absolute summability only makes sense when X is finite. This is because

(2.14)

HflJII S Z HO», ie A

for all O e ^ and A e ^ . Therefore O e ^ is A-admissible if and only if X is finite. A further comment is in order. From a mathematical point of view, it might seem natural to replace \Q>A\ by the Lco(A'4)-norm of H £. However, there is no advantage in doing so because we can always choose a sufficiently nice version of <S>A. (2.15) Finite range potentials. A potential O is said to be of finite range if for each i e S there is some A,- e !? such that <$>A = 0 unless A a A; for all i e A. In particular, if S = Zd for some d ^ 1 then we shall say that O has bounded range if sup {diam A i e ^ O ^ 0} is finite, and we will say <1> has range R if diam A ^ R for all A e !? with Q>A =£ 0. Here diam .4 is the diameter of A. Clearly, each finite range potential O is uniformly convergent, and we have O e M if and only if all 2. (There is no restriction on A(a>) = <J(i,i)'Hcol) [ 0

iîA = {i,j}, i # j , itA = {i}, otherwise,

where J: S x S -> M and q>: E x E -> 1R are symmetric, and ç> and \j/: E-* U

30

Gibbsian specifications

are measurable. Clearly, pair potentials (in particular those of the preceding special form) are most appealing intuitively. However, we will repeatedly see that it is advantageous to work with general potentials also. (2.17) Nearest-neighbour potentials. Suppose S is the vertex set of a simple graph. By definition this means that there is a distinguished non-empty subset Bof {bczS:\b\ = 2}, each element b = {ij} e B being called a bond or an edge between i and j . Two sites i,j e S are said to be adjacent or nearest neighbours if {i,j} e B. A potential O is called a nearest-neighbour potential or a Markov potential if <5>A = 0 except when A is a complete subgraph of (S, B), i.e., b e B for all b c A with \b\ = 2. (Notice that singletons are complete subgraphs.) In particular, we will often think of the integer lattices Zd, d ^ 1, as simple graphs by taking B={{i,j}czZ":\i-j\

= l}.

The only complete subgraphs of Zd are the singletons and the sets in B. Therefore the nearest-neighbour potentials over Zd are exactly the pair potentials of range 1. (2.18) Self-potentials. By definition, these are all potentials O such that <&A = 0 unless \A\ = 1. Thus each self-potential is of the form ^

J

[0

if|^|>l,

with measurable functions R. If O is an arbitrary potential, its restriction to the set of all singletons will be called the self-potential part of <ï>. Clearly, a Gibbsian specification for a potential O with a non-trivial self-potential part (
2.2

Quasilocality

Having introduced the notion of a Gibbs specification we aim at showing that Gibbs specifications are not as particular as they might seem at a first sight.

Quasilocality

31

We shall do this in two stages: First we shall introduce the concept of a quasilocal specification, and we shall argue that quasilocality is a natural condition. Then, in the next section, we shall show that quasilocal, positive pre-modifications are necessarily Gibbsian. The motivation for introducing quasilocal specifications arises from the physical idea of a strict separation of microscopic and macroscopic quantities: A microscopic part of a system does not possess any information about the macroscopic state of the system. Let us describe the microscopic and the macroscopic quantities. A real function / o n f i will be considered as describing a macroscopic observable if / is measurable with respect to the tail a-field (2.19)

f ^ n ^ A = n

&S\A-

2T is also called the a-algebra at infinity. The reason for this interpretation is obvious: The ^"-measurability of a function / j u s t means that the value of/ is not affected by the behaviour of any finite set of spins. (At first sight it might seem that ST = {Q, 0 } . Quite on the contrary, &~ is very rich. A typical example of a tail event is < lim |A„| _1 Y, ai exists and belongs to B >, where B e S and (A„) is a cofinal sequence in y ; of course, this is assuming that E is equipped with a measurable additive structure.) The condition of properness in the definition of a specification implies that every specification y preserves the macroscopic observables, in that yAf = /whenever A e ^ and / is bounded and tail measurable. In particular, y maps the set of all bounded tail measurable functions into itself. Next we ask for a precise meaning of the term "microscopic quantity". It is natural to say that a function / on Q is a microscopic quantity if / is arbitrarily close to functions which only depend on finitely many coordinates. There is, however, no canonical interpretation of the word "close". The simplest meaning is "close in the uniform norm". This leads to the concept of a quasilocal function. (2.20) Definition, (a) A real function / on Q is called a cylinder function or a local function if/is J^-measurable for some A e ^ . For each A e ^ w e write ifA for the linear space of all bounded immeasurable functions, and we let if = I J A S ^ - ^ A denote the set of all bounded local functions. (b) A function / : Q -> M will be said to be quasilocal if there is a sequence (/Jnâi °f l° c a l functions /„ such that lim,,^ \\f — f„\\ = 0. Here || • || is supnorm. We write if for the space of all bounded quasilocal functions. Clearly, if is the uniform closure of if'.

32

Gibbsian specifications

(2.21) Remarks. (1) A measurable function / is quasilocal if and only if (2.22)

lim

sup

| / ( O - / f o ) l = 0.

(The notation lim means that the limit is taken along the directed set y.) Clearly, each quasilocal / satisfies (2.22). Conversely, (2.22) implies that / is the uniform limit of the local functions œ ->/(coArçs\A) when r\ e Q is arbitrarily fixed and A runs through a cofinal sequence in y . In particular, we conclude from (2.22) that a non-constant tail-measurable function can never be quasilocal. (2) Suppose E is a metric space and d is any of the usual metrics on Q. that induce the product topology. Then each uniformly continuous function / : Q —> R is quasilocal. To see this, let e > 0 be given and 8 > 0 be such that | / ( 0 — f(r))\ < e whenever d(Ç,j}) < 8. By the definition of the product topology, there exists some A e y such that ÇS\A = VS\A implies d((,, TJ) < 8. This yields (2.22). In particular, if E is a compact metric space then every continuous function on £2 belongs to SI. (3) If E is finite and Q is endowed with the product topology of the discrete topology on E then / : Q -» IR is quasilocal if and only if / is continuous. Because of the preceding remark we only need to verify that each quasilocal function is continuous. But this follows easily from (2.22) and the fact that

is a metric for the product topology on Q; here n: S -» N is any bijection.

o

We started from the question in which we asked for a condition on a specification y expressing the idea that a microscopic part of the system only has a microscopic horizon. Now we are ready to introduce such a condition. Formally, this condition will remind the reader of the notion of a Feller kernel. (2.23) Definition. Let us say that a specification y is quasilocal if, for each A e y , / e S implies yAf e <£. Clearly, to verify that a given specification y is quasilocal we only need to check that yAf e S when A G y and / e if. Also, it is easily seen that each independent specification is quasilocal. The next proposition deals with the quasilocality of ^-specifications. (2.24) Proposition. Let X e Ji(E, S) be an a priori measure. (a) Suppose y = pX is a X-specification. If each pA is a local function, or if X is finite and each pA is quasilocal then y is quasilocal. (b) Suppose 0 is a X-admissible potential such that all Hamiltonians H^ are quasilocal. Then y® is quasilocal.

Quasilocality

33

(c) Conversely, suppose E is finite, X is counting measure, and y is a quasilocal specification. Let p be the unique X-modification with y = pX_. Then pA e S£ for all A e y . Proof, (a) Let A e y and / e if. If pA is local then so is fpA and therefore y A / = XA(fpA) e if. So let us suppose pA is quasilocal and X is finite. For given £ > 0 we choose a local function g with ||pA — g\\ < e. Then XA(fg) is local and

\\yAf-Ufg)\\

S\\f\\X(Ers.

As e was arbitrary, y A / e if. (b) Let A e y and e > 0. By hypothesis there is a local function u with || JïA — u|| ^ e. We put v = e~u. Then XAv :g e£ZA < oo. We consider the local function g = v/XAv. We have

K\pt-g\ = K(\hflxAht-vixAv\) ^ AA{\h* - v\)ßAh* + XAv\\/XAht

- l/XAv\

Z2XA{\ht-v\)/ZZ = 2y?(|l-exp(fl*-u)|) ^ 2(e£ - 1). Now if / e if then XA(fg) is local and htf

- AA(/0)|| ^ 11/11 ||AA(|p* -
As e was arbitrary, y A / e i£. (c) For each A e y we may write PA=

Z

!K=<;} 7 A 1 {*,=<;}>

Çe£A

and we have l{ffA=ç} e i£ and yAl{ffA=C} e S. Thus pA e if'.

o

In the case of a finite state space £ there is also a converse to assertion (b); see Corollary (2.31) below. We stress some particular cases of Proposition (2.24). (2.25) Example. Let X e Ji{E,é>) and $ be a uniformly convergent Xadmissible potential. Then (2.24) (b) shows that y® is quasilocal. Under additional assumptions we have the following results (which are stronger because of (2.24) (a)). (i) If $ even has finite range then each pA is local. (ii) If , besides being uniformly convergent, has bounded Hamiltonians HA then pAeJ£ for all A e y . (This case is only possible if X is finite. It occurs, for instance, when $ is absolutely summable.) o

34

Gibbsian specifications

We conclude this section with the observation that examples of specifications y with \y(y)\ > 1 can easily be constructed when y is not assumed to be quasilocal. In this case, however, the non-uniqueness phenomenon cannot be interpreted as a kind of phase transition. It corresponds rather to a transition which is induced by a change of the experimental conditions. (2.26) Remark. Let (X,9C) be a measurable space and (fix)xeX be a family of random fields. Suppose (i) there is a tail measurable function £: Q -• X with /ix(£ = x) = 1 for all x e X;and (ii) there is a family (yx)x e x of specifications such that the functions (eo, x) -* y£(A |co) are measurable and jux e ^(y x ) for all x e X. Then there is a specification y satisfying {fix: x e X} c 3(y). Proof: We put yA(4 | ta) = yi (w) (^ |
( A e ^ , / l e f , ( a e Q).

It is easily checked that y is a specification and satisfies fixyA = fixy£ = \ix for all A e if and x e X. G The conditions of the preceding remark are met whenever X is countable and (fix)xeX is a family of random fields which are pairwise mutually singular on 2T and specified by suitable specifications. (The latter is true whenever (E,&) satisfies a weak regularity condition; see Sokal (1981).) Let us now present an example with an uncountable X. (2.27) Example. Let E = { 0 , 1 } , S = N , X = [0,1], and 3C be the Borel «r-field on X. For x e l w e let fix denote the Bernoulli measure on Q with probability x for "success", i.e., fix = (Xx)s with lx = xô1 + (1 — x)ô0. The supports of the fix,s are separated by the tail measurable function 1 " £, = liminf- £ n~*<x>

f;;

<*- i —1

this follows from the strong law of large numbers. From Remark (1.25) we know that fix e ^(yx), where yx = Xx is the independent specification with single spin distribution Xx. Clearly, the functions (co, x) -> y£(A\a>) are measurable. Thus Remark (2.26) provides us with a (non-quasilocal) specification y such that @(y) contains all /i x 's. In fact, we have g(y) = {J w{Ax)nx: w e &{X,&)}. For " ZD " trivially holds, and if fie &{y) then for all A e if and A e 3FK we have

Gibbs representation of pre-modifications

35

H(A) = pyA(A) = J ii(dco)yfœ)(A\œ)

= J p(dœ)p^(A) = J C(ß)(dx)px(A). Applying de Finetti's theorem we thus can conclude that ^(y) consists of all exchangeable random fields, i.e. of all random fields which are preserved by all permutations of finitely many spins. De Finetti's theorem will be proved later in Example (7.31). o

2.3

Gibbs representation of pre-modifications

In Section 2.1, Gibbs specifications were introduced on physical grounds. Here we will provide a mathematical justification for considering Gibbsian specifications. We shall consider any /l-specification y = pX,, and assume that the associated /l-modification p is a quasilocal positive pre-modification. (We have seen before that these assumptions are natural.) Under these conditions, the Gibbs representation theorem (2.30) below will state that y is Gibbsian for some potential O. In addition, O can be chosen to satisfy a normalization condition. The significance of this normalization will be discussed in Section 2.4. Examples can be found in (2.38). (2.28) Definition. Let a e ^(E, i) and define probability kernels aA from STA to !F (A cz S) as in Remark (1.25). We say a potential O is ^-normalized if *B*A

= I aB(dO
whenever 0 # B cz A e y . In particular, we say O is a gas potential with vacuum state a e E if O is normalized by the Dirac measure 8a at a. (2.29) Comments. (1) O is a-normalized if and only if aBQ>A exists when 0 # B cz A e £f, and a.^Q>A = 0 whenever i e A. In particular, O is a gas potential with vacuum state a if and only if Ox(a>) = 0 whenever coi = a for some i e A. (2) The terminology "gas potential" and "vacuum state" comes from the interpretation of a random field as a lattice gas: For x # a, the event [oi = x] means that the site i is occupied by a particle of type x, whereas {ui = a) is thought of as the event that i is empty, o We are now ready to state the Gibbsian representation theorem. We say a family (p A ) A e ^ of functions on Q is quasilocal if each pA is quasilocal. (2.30) Theorem. Let X e Ji{E, S) be an a priori measure, and suppose p is a positive quasilocal pre-modification such that XApA = 1 for all A e Sf. Then for each a e E there is a unique X-admissible gas potential Q>" with vacuum state a

36

Gibbsian specifications

such that p = p°a. If, in addition, logp A is bounded for all A e £f then for each a G âP{E, ê) there exists a unique uniformly convergent ^.-normalized À-admissible potential O" with p = p°". Proof 1) Let a G 3P(E, S). For each A e Sf U {0} we define

PA/= I (-D MXC W/ CŒA

whenever / : Q -> M is a measurable function for which all functions a s \ c / exist. The operator pA has the following properties. (i) PA/ i s ^A-measurable. This is because, for each C, <xs\cf is measurable with respect to SFC. (ii) For all A G SP U {0}, Z

«S\A/ =

PA/-

AŒA

This comes from the inclusion-exclusion principle. (iii) For all 0 # ß c A, a ^ / ) = 0. It is sufficient to check this when B = {i}. In that case we have ««(P^/) =

E

( - l)|i4NC|«{i}(«s\c/ - «s\<cu{i}/) = 0.

For if i £ C then a {l} a sxc = asXc = a{l}aSX(CU{i}), by Remark (1.25). 2) For A e SP we put u^ = logp^ and Q>A = —pAuA. <&A exists when a = ôa for some a e E or uA is bounded. It follows from properties (i) and (iii) of pA that <&A is ^-measurable and a-normalized. 3) Next we claim that OA = —pAuA whenever 0 # .4 c A e y . As p is a pre-modification, M O - MxM = " A ( 0 - " A M for all Ç, to G Ü with ÇSV4 = c a ^ . Integrating over Ç with respect to aA(-\œ) we obtain a^u^ — uA = a^uA — uA. Thus for each C c i w e have <XS\C(UA -

" A ) = C*S\CVA(UA -

" A ) = a S («A - " A ) -

Since

E (-ir ci = o, CcX

we conclude that pAuA = pAuA. 4) Finally we look at the partial sums H* A which were defined in (2.13). For A c A w e have (J^^cA

= - Z P^MA + .4c:A

0ftAcA\A

Z P^"A .4c:A\A

Gibbs representation of pre-modifications —

— a

S\AUA +

a

=

«S\A(aAUA -

"A)

=

«S\A(«A"A -

"A)

37

S\(A\A)UA

where vA = aAuA — uA. The third equation follows from property (ii) of pA, and the next to last equation was proved in Step 3) above. Suppose now that a = öa for some ae E, and let a e Q denote the constant configuration taking the value a. Then for each co e Q we have «S\A^AM =

lo

gPA(

and the quasilocality of pA implies lim H ^ A M = lim aSSAvA(co) A

A

= l°gpA(aAtoSNA)/pA(to) = vA((o). If a is arbitrary but uA is bounded then the quasilocality of pA entails that of uA and vA. Therefore lim ||fl* A - vA\\ £ lim j a s ^ ( d Q sup K K C S \ A ) - vA(œ)\ = 0. A

A

(y

Thus in both cases HA exists and satisfies # A = "A = « A " A - l o g P A -

Equivalently, h* = p A exp( — aAuA). Integration yields ZA =^A^A = exp(-a A u A ) because aAuA is ^-measurable and XApA = 1. Hence PA = PA5) The uniqueness of $ will follow from Theorems (2.34) and (2.35) (a) below, a The preceding proof shows that Theorem (2.30) remains true when the hypothesis of quasilocality is replaced by the weaker condition that lim sup IPA(WACS\A) - P A M I = 0 A

Çefi

for all co e Q and A e y. Let us look at two particular cases. (2.31) Corollary. Suppose E is finite, X is counting measure, and y is a positive quasilocal specification. Then for each a e &{E, S) there is a unique anormalized k-admissible potential with y = y®. is uniformly convergent, and H^eS for alike
38

Gibbsian specifications

Proof. Let p be the unique ^-modification with y = pi.. By assumption, pA > 0 for all A e y . From the paragraph below (1.31) we know that p is a premodification, and Proposition (2.24) (c) asserts that p is quasilocal. By Remark (2.21) (3), each pA is continous with respect to the natural product topology on Q. Hence so is log pA. As Q is compact, each of the functions log pA is bounded. Thus the result follows from Theorem (2.30). o Next we ask for ^-modifications which are Gibbsian for a nearest-neighbour potential in the sense of (2.17). (2.32) Corollary. Let p and a satisfy the conditions of Theorem (2.30) and
{i}{J{jeS:{i,j}eB}.

Then $ is a nearest-neighbour potential. Proof. Suppose A e Sf is not a complete subgraph of (S, B). Then there are distinct sites i,j e A such that; <£ B(i). By definition, ® ^ = - Z (-irc|as\cUx, CaA

where uA = logp^. We split the last sum into contributions corresponding to the intersection of C and A\{i,j}. Each D c A\{i,j} corresponds to the four subsets ö i = ö , ß 2 = ö U {i}, D3 = flU{i},D4 = DU {i,j} of A. The contribution of D to the sum in question is thus equal to a

S\DiMX

—

a

S\D2UA ~

a

S\DiUA +

a

S\D4UA

= xS\D2(ct{i]uA - uA) - CS\DM{Ï\UA - UA)As p is a pre-modification, a

{i}UA

— UA = a{i}U{i}

~

U

{i}-

Combining this equation with the hypothesis that u^ is J^ (i) -measurable, we see that the contribution of D is equal to a

B(i)VD! M {»} ~

a

B(i)\D2U{i}

— aB(i)\D3u{i}

+ aB(i)\D4M{i}-

This expression vanishes because B(i)\D1 = B(i)\D3 and B(i)\D2 = ß(i)\Ö4Thus <S>A = 0, and the proof is complete, o

Equivalence of potentials

2.4

39

Equivalence of potentials

A given Gibbsian specification y can be defined in terms of many different potentials. For example, let O be a /l-admissible potential and c — (cA)AeS? a family of real numbers cA such that X \CA\ < oo A3i

for all i e S. Define ¥ by ¥ x = OA + c^ (A e 9>). Then xnA#0

for all A e y , and the expression on the right is a finite constant. This shows that ¥ is a /l-admissible potential, and p* = p®. To summarize this observation, we may state that a Gibbsian specification remains unaffected by the addition of constants to the potential. But more than this is true. Indeed, to conclude that p* = p* we do not need that H* - H% is constant. It is sufficient that tf£~* = H* - H% be immeasurable. This is because every ^-measurable factor of h® reappears in the partition function Z* and thus cancels in p*. This fact suggests the following definition. (2.33) Definition. Let O and ¥ be two potentials. We say O and ¥ are equivalent and write O ~ ¥ if, for all A e Sf, HX^ is ^-measurable. Clearly, " ~ " is an equivalence relation, and if O ~ ¥ then O is /l-admissible if and only if so is ¥ . The significance of" ~ " is explained by the next theorem. (2.34) Theorem. Let I e Jf{E, S) and suppose O and ¥ are À-admissible potentials. Consider the following statements: (i) O ~ ¥ 00 Pq> = pv (Hi) y® = y*

(iv) #(<&) n ^ ( ¥ ) # 0. Then the implications (i) o (ii) => (Hi) hold. Suppose further there is a second countable topology on E such that S is the Borel o-algebra, X is everywhere dense, and the functions A — X¥A are continuous with respect to the product topology on Q. Then (iii) implies (ii). If in addition, $ — ¥ is uniformly convergent and 9(Q) U <SQ¥) # 0 then (i) o (iv). Proof, (i) => (ii). Let A e y . As h*"* is ^-measurable, the equation /i* = KT^tiX implies Z* = hJ^^Zj. Combining both equations we obtain (ii).

40

Gibbsian specifications

(ii) => (i). For each A G y we have p* = pA and therefore

HZ-v = log{ti*/hX) = \og(ZX/ZZ). Z* and Z* are ^-measurable. Thus $ ~ *¥. (ii) => (iii). Obvious. (iii) => (ii). Let A e ^ and c o e f i b e given. The equation y%(- \a>) = yA(- \a>) is equivalent to lA{AA) = 0, where AA = {C 6 £ A : P ? ( C « S \ A ) / P I ( C « S \ A ) # 1}.

Our topological assumptions imply that lA is everywhere dense and AA is open. Hence AA = 0 and thus p^co) = PA (ct)). (iii) => (iv). Under the hypothesis ^(O) U ^(*F) ^ 0, this implication is trivial. (iv) => (i). By assumption, there exists some \i e ^(
for l x us\A(/i)-almost all (C,co) e £ A x £ S \ A . The expression on the right is independent of £. Hence for XA x /lA x HA~4'(Ca)) is a uniform limit of continuous functions and thereby continuous. Moreover, the measure lA x lA x oS\A{n) is everywhere dense. Indeed, we have assumed that A is everywhere dense, and this implies that p. is everywhere dense. (To see this it is sufficient to note that every non-empty open subset of Q contains a non-empty open cylinder set, and aA(n) is equivalent to XA for all A e ^ ; cf. Remark (1.28)(2).) We thus can conclude that H^iÇco) = flj-'faca) for all Ç, r\ e £ A and co G £ S \ A . This proves that H* - 4 ' is ^-measurable, n Of course, the topological assumptions of Theorem (2.34) are trivially satisfied when £ is countable and I is equivalent to counting measure. In Section 4.4 we shall see that the assumption ^(cD) U ^(*F) ^ 0 is satisfied if, for example, and *F are uniformly convergent with bounded Hamiltonians and the state space (£, ê) is standard Borel. The equivalence relation " ~ " has very large equivalence classes. At the beginning of this section we have observed that each potential *F of the form X ¥A =
Equivalence of potentials

41

(2.35) Theorem, (a) Suppose $ — *F is normalized by some a 6 0*(E,
«SXAtf£=

X

as^d>A =

X

<^.

Indeed, the interchange of integration and summation is trivial when a is a Dirac measure, and in the alternate case it is justified by the uniform convergence of 3>. The second equation in (2.36) can be verified as follows. If A <= A then <xs\AQ>A = <&A because
*A=

I

(-D

W

\A4

Moreover, since <1) ~ 0 and 3> is a-normalized we have «SNAA*

= «*(«£) =

I

s a

(
ADAÏ0

Hence 0> = 0 = *P. (b) Let *F be given by

^ = - 1 (-ir% vc iogp*. Theorem (2.30) then states that *F is a-normalized and p T = p*, and Theorem (2.34) yields *F ~ G>. Let us derive a simpler expression for *F. As the partition functions ZA are ^-measurable, we have

^,= x (-irvcfl? = 1 1

(-i^ c W®.-

BTM^O C<=A

If there is some i e A\B then the inner sum equals X

( - i r c | ( a s x c d > B - aSX(cu{i})(DB) = 0

CŒAJÏC

because <5B is J^-measurable. Hence *F is given by

42

(2.37)

Gibbsian specifications

(-ir c | a S X c O B .

VA= X I B=>A CŒA

The reader is invited to check directly that ¥ , as defined by (2.37), has the required properties, o We emphasize the following consequence of formula (2.37) for the unique a-normalized potential ¥ with ¥ ~ O: ¥ is supported on the subsets of the sets in the support of O, i.e., ¥,, is identically zero unless there exists some B z> A for which O B is not constant. In particular, if O is a pair potential or a nearest-neighbour potential then so is ¥ . We conclude this chapter with an example concerning the equivalence of magnetic models and lattice gas models. (2.38) Example. Let £ = { — 1, +1} and X be equidistribution. We think of £ as the state space of a magnetic spin having only two possible orientations, up or down. For each A e £f, the product

°A = n « ie A

indicates the parity of down spins in A. Therefore it is reasonable to look at potentials O of the form

(AeSf),

where J is a real function on £f. (The minus sign is due to convention.) Such potentials are called spin potentials. It can be immediately checked that each spin potential is normalized by X. By the way, there is also a purely mathematical reason for looking at spin potentials: The functions aA are precisely the characters of Q = Es, considered as a compact group with coordinatewise multiplication. Thus for each spin potential O the J(v4)'s are the Fourier coefficients of the associated Hamiltonians H®. Now let us switch to a lattice gas interpretation of E. A site i is thought of as being occupied by a particle if and only if u, = 1. This suggests considering potentials ¥ of the form r,

,

\K(A)

ifffj = 1 for a l l i e d

T.-Kwrjc+w-y otherwise; here K is any real function on Sf. Clearly, such a ¥ is a gas potential with vacuum state — 1. For which choice of ¥ is the lattice gas model equivalent to a spin model with interaction ? Suppose O e J , i.e., X \3{Ä)\ < oo for a l l i e S. Asi

Theorem (2.35) and equation (2.37) assert that there is a unique gas potential

Equivalence of potentials

43

*P ~ $ which is given by (2.39)

K(A)=-YJ

I

(-l)^c|7(B)(-l)|B\q

= _ 2 W X ( - lfVI J(B). B=>.4

*P is uniformly convergent but does not necessarily belong to &. For we have the estimate \\V\\t = I

|KG4)| ^ I 2W I

^31

A 3l

|J(B)|

B =) yl

= 2 X l^)^11"-1 B3i

with equality when ( — 1)|B|J(B) has constant sign. Conversely, let us start from a gas potential *Pe J . Then (2.37) provides us with a unique equivalent spin potential which is given by (2.40)

-J(A)=

X

X (-1) MVC| 2-' B \ C| K(B)

= X K(ß)2-Ißl X (_l)H\ci2lci = X K(B)2-IBI. B=>^4

<1> satisfies

||«D||^ X X |K(B)|2-'BI = ^11^11, ^3i

B=>A

^

for each i e S, hence cD e ^ . For a better understanding of formula (2.37) it might be helpful to check directly that the potentials and *P which are related by equations (2.39) and (2.40) are indeed equivalent. This is most easily done by expressing the function f\ie A iai + l)/2 in terms of the CTB'S, or vice versa, o

Chapter 3 Finite state Markov chains as Gibbs measures

This chapter should be viewed as an intermezzo. Its contents will not be needed for an understanding of later chapters, although it will occasionally reappear in some remarks and examples. There are, however, two reasons for inserting this chapter at this early stage. First, a reader who is familiar with Markov chains will ask if the usual description of Markov chains by one-sided conditional probabilities is related or even equivalent to a description by means of Gibbsian conditional probabilities. In the case of a finite state space E, this question has a positive answer which will be given in Section 3.1: There is a one-to-one correspondence between the set of all positive transition matrices P and a suitable class of nearest-neighbour potentials
3.1

Markov specifications on the integers

Throughout this chapter we choose the integers for the parameter set, i.e., we put S = Z. We also fix a finite non-empty state space E. Of course, we let ê be the power set of E, and the a priori measure X on E is counting measure. (3.1) Defînidon. Let y be a specification with parameter set Z and state space E. We say y is a positive homogeneous Markov specification if there is a function

Markov specifications on the integers

45

g{-,-,-) > 0 on £ 3 such that 7{iM = y\co) =

flf^i-i.y.cOj+J

for all ieZ, y e E, and <w e Q. (3.2) Comment. Formally, the preceding definition only refers to the singleton part of y. But because of the consistency of y each condition on its singleton part has consequences for all yA's. In fact, Theorem (1.33) implies that each positive homogeneous Markov specification is uniquely determined by the corresponding function g. Therefore g will be called the determining function of y. o We are going to show that each positive homogeneous Markov specification y admits a unique Gibbs measure p. In fact, pi is a Markov chain with a positive transition matrix which can be computed explicitly in terms of y. To state the result we introduce some notation. Let P = (P(x,y))x,yeE be a stochastic matrix with non-zero entries. Then we let pip e é?{Q, êF) denote (the distribution of) the unique stationary Markov chain with transition matrix P. pP is uniquely determined by the condition (3.3)

/lp((7; = X0,<Ti

=

+ 1=X1,...,Oi+n

=

Xj

ccP(x0)P(x0,x1)...P(xn_1,xn).

Here i e Z, n e Z+, x0, ..., x„e E, and aP e ]0,1 [ £ is the unique probability (row) vector satisfying the equation otPP = aP. (A proof of existence and uniqueness of aP will be given in Appendix 3.A.) For each A c Z w e let (3.4)

SA = {i e Z\A: \i - j \ = 1 for some; e A}

denote the boundary of A. (3.5) Theorem. The relation establishes a one-to-one correspondence y<-^P between the set of all positive homogeneous Markov specifications and the set of all stochastic matrices on E with non-vanishing entries. For given P the corresponding y is determined by the equation (3.6)

yA(
(A e Sf,œ e Q, Ç e EA).

Conversely, P can be expressed in terms of the determining function g of y as (3.7)

P(x,y) = Q(x,y)r(y)/qr(x)

(x,y e E).

Here Q(x,y) = g(a,x,y)/g(a,a,y) for some arbitrarily fixed a e E, q is the largest positive eigenvalue of Q = (Q(x,y))x yeE, and r e ] 0 , oo[ £ a corresponding right eigenvector.

46

Finite state Markov chains as Gibbs measures

Before entering into the proof we give some comments and draw a conclusion. (3.8) Comments. (1) The expression on the right of (3.6) can be expressed explicitly in terms of P as follows. Each A e y is of the form A

= u k + !>•••»'*+ »*}• k= l

for some n ^ 1, where ik e Z and nke N are such that the sets {ik,...,ik are pairwise disjoint. For such a A (3.3) gives

+ nk}

Hp{aA = C\(T8A = œ8A) n

= EI pKk^
Here P m is the w'th matrix power of P. (2) Equation (3.7) establishes a well-defined mapping from the set of all positive functions g on E3 to the set of all positive stochastic matrices on E. Indeed, let a e E be fixed and g: E3 -> ]0, oo [ be given. Then Q=

(g(a,x,y)/g(a,a,y))XiyeE

is a positive matrix. Therefore a well-known theorem of Perron and Frobenius (cf. Appendix 3.A) guarantees that Q has a unique eigenvalue q > 0 with the following properties: (i) all other eigenvalues have modulus strictly less than q, (ii) there exists a strictly positive right eigenvector r corresponding to q, and (iii) each right eigenvector corresponding to q is a multiple of r. Consequently, the matrix P=

(Q(x,y)r(y)/qr(x))x,yeE

is uniquely determined by Q and therefore by g. Clearly, P is stochastic,

o

As a by-product of Theorem (3.5) we obtain the following characterization of the positive homogeneous Markov specifications. A nearest-neighbour potential O is called homogeneous if there are two functions q>1 : E -> U and q>2 : E x E -> M such that A

Woi.rç+i)

iM = { U + l } .

(3.9) Corollary. /I specification y is a positive homogeneous Markov specification if and only if y is Gibbsian for some homogeneous nearest-neighbour potential O. Proof. Suppose y is a positive homogeneous Markov specification. By Theorem (3.5), there exists a positive stochastic matrix P such that y can be written

Markov specifications on the integers

47

in the form (3.6). In view of Comment (3.8) (1) this means that y is Gibbsian for the homogeneous nearest-neighbour potential 0

= J

(-log P(at,ai+1) [ 0

iîA = {i,i + 1}, otherwise.

The converse is obvious, D Combining Corollary (3.9) with Theorems (3.5) and (2.35)(b) we obtain the following result: Given any a e 3?{E, S\ the relation

establishes a bijection <&<->P between the set of all a-normalized homogeneous nearest-neighbour potentials and the set of all positive stochastic matrices. In the particular case when E = { — 1,1} and a is equidistribution, this bijection will be studied in detail in Section 3.2. We now turn to the proof of Theorem (3.5). Proof of Theorem (3.5). For the sake of brevity we let IT denote the set of all stochastic matrices on E with non-vanishing entries and T the set of all positive homogeneous Markov specifications. 1) If P G II and y is defined by (3.6) then y e T and nP e &(y). To show this we fix some A e Sf. It is easily checked that HP(aA = Ck MA = cöANA) = i*p(oA = CkdA = mdA) for all C G EA, œ e Q, and all A e Sf with A => A U 3A. Thus we have

M K = C}n^) = jd/i p r A K = ci-) A

A

for all C e E and all cylinder events A'\-a.3~K and therefore for al\Ae^~A. shows that /zP = nPyA. Hence fiP e ^(y). To show that y is a specification we only need to prove that (3.10)

This

yAyA(crA = C|-) = yAK = C|-)

whenever 0 ^ A c A e Sf and ( e EA. ((3.10) implies the consistency equation yAyA = 7A because yA and yA are proper.) As pip e &(y), the argument preceding definition (1.23) shows that (3.10) holds almost surely with respect to nP. But both sides of (3.10) are #3A-measurable and fiP charges each atom of J^ A . Hence (3.10) holds everywhere. Finally, formula (3.6) immediately shows that y has the determining function (3.11)

g(x,y,z) =

P(x,y)P(y,z)/P2(x,z).

Hence y e T, and we have shown that equation (3.6) defines a mapping 6: n -> T such that /iP e <${â{P)) for all P e n . 2) Next we show: If P G IT and y = S(P) then P can be recovered from y by means of formula (3.7). In particular, 6 is injective. Let a e E be fixed. By

48

Finite state Markov chains as Gibbs measures

definition, the determining function g of y is given by (3.11). Thus for all x, y e £ we have from the definition of Q Q(x,y) = g(a,x,y)/g(a,a,y)

= P{a, x)P(x, y)/P(a,a)P(a, y).

Putting q = l/P(a, a) and r(x) = P(a, x) this equation can be rewritten as Q(x,y)r(y) = qr{x)P{x,y). Summing over y we see that q is an eigenvalue of Q and r a corresponding right eigenvector. Moreover, if q' e C\{q} is any other eigenvalue of Q with eigenvector r' e C £ then q'/q is an eigenvalue of P with eigenvector (r'(x)/r(x))xeE. As P is stochastic we may conclude that \q'/q\ < 1 (see Appendix 3.A). Hence q is the Perron-Frobenius eigenvalue of Q. This shows that P is obtained from Q by (3.7). 3) Let g: £ 3 -> ]0, oo [ be the determining function of some y e Y, and let P e n be defined in terms of g and some a e E by means of (3.7). Then g satisfies (3.11) if and only if the equation (3 12)

g(*' y>z) d(a> x > a ) g(x, a, z) g(a, a, a)

=

d(a, x, y) g(a, y, z) g (a, a, y) g (a, a, z)

holds for all x, y, z e E. Indeed, (3.11) is equivalent to the statement g(x,y,z) g(x,a,z) for all x,y,ze equals

^P(x,y)P(y,z) P(x,a)P(a,z)

E. According to the definition of P, the expression on the right

Q(x,y)Q(y,z)/Q(x,a)Q(a,z) =

g(a,x,y) g{a,y,z) lg(a,x,a) g{a,a,z) g(a, a, y) g (a, a, z)\ g (a, a, a) g (a, a, z) '

Thus (3.11) is equivalent to (3.12). 4) The injection 6: Il -> Y is surjective. For let y e Y have determining function g, and let P e n be defined in terms of g and some a e E via (3.7). We will prove that y = 6(P). Let y = 6(P). Then Theorem (1.33) asserts that y = y as soon as y^ = y^ for all i e S. But the latter statement is equivalent to equation (3.11) and therefore, by stage 3) above, to (3.12). We are thus left with proving (3.12). To this end we put A = {1,2}. We also fix any z e E and write [xy] = yA(
(x,ye E),

where to e Q is such that a>0 = a and co3 = z. The consistency equations ÏA

=

yAy{i} = y A y { 2 } t n e n

[xy] = g(a,x,y)

ta

^e

form

£ [wy] = g(x,y,z) ueE

Hence

tne

£ [xt>] veE

(x,yeE).

The one-dimensional Ising model

g(a,x,y)/g(a,a,y)

= [_xy]/[ay]

g(x,y,z)/g(x,a,z)

= [xy]/[xa]

49

and for all x, y e E. Consequently, the expressions on the left and on the right of (3.12) both coincide with the ratio [xy]/[aa]. This proves (3.12). Hence

y = *(n 5) It remains to show that <&(y) a {fiP} when P e l l and y = S(P). This is an immediate consequence of a uniqueness theorem in Chapter 8, but there is also a direct proof using the ergodic theorem for Markov chains. (One of the simple proofs of the ergodic theorem is included in Appendix 3.A). Let i e Z, n ^ 1, A = {i,i + 1,..., i + «}, k ^ 1, and A = A(fc) = {i- k + l,...,i + n + k - 1}. For each Ç G EA and coeflwe have y A K = £|co) =

Y

HP(.VA

= iWdA = WOA)

= Pk(œt.k, QP(U C i+1 )... P(Ci+n-,, £•+.) x Pk(Ci+n, œi+n+k)/Pn+2k(œ^k,

œi+n+k)

The ergodic theorem for Markov chains states that lim Pk(x, •) = aP indepenk œ dently of x G E. Hence ^ lim y A W K = Ç\co) = a , ( Q j P K i , C i + i ) - i > K i + 1 , - i . U = M ^ A = 0Therefore, if ^ G ^(y) then the dominated convergence theorem gives n{oA = 0=

!i m 1 K^co)yMk){aA = £|co) fc->ao

Thus ^ = fip. The proof is thus complete, D To conclude this section, we note that Theorem (3.5) admits a straightforward modification which covers the case when S = N and fiP is replaced by the Markov chain with transition matrix P which starts at time 0 at a fixed initial point a G E. The details are left to the reader.

3.2

The one-dimensional Ising model

The Ising chain is a simplified model of a ferromagnetic or antiferromagnetic substance. It was suggested by W. Lenz and first investigated by E. Ising in his thesis in 1924. The model is characterized by the following assumptions:

50

Finite state Markov chains as Gibbs measures

(i) The substance consists of spins with two possible orientations + 1 and — 1 ("up" and "down"). (ii) The spins form an infinite linear chain, i.e., they are located at the sites ofZ. (iii) The interaction energy of two spins at and cr,-with \i — j \ = 1 is — Ja^j, where J e IR is a coupling constant. (1/|J| is proportional to the absolute temperature. In the case J > 0 any two adjacent spins have minimal energy if and only if they are aligned in that they have the same sign. This means that the interaction is ferromagnetic. On the other hand, if J < 0 then any two adjacent spins prefer to point in opposite directions. Thus in this case we have a model of an antiferromagnet.) (iv) There is no interaction between non-adjacent spins. (v) There is a constant heU describing the action of an external field (directed upwards when h > 0). We read these assumptions as an invitation to investigate the set @(<&J,h) of Gibbs measures for a particular potential (&Jh. We have to choose E = {-1,1},S = Z, and (3.13)

®JAh

-J<Ti<Ti+l if A = {i,i + 1}, if A = {i}, = < -hoi 0 otherwise.

Q>J,H is called the I sing potential with interaction J and external field h. (Thus the Ising potentials are just the homogeneous nearest-neighbour spin potentials.) We look for an explicit description of <S(<&J,H). Clearly, the Gibbsian specification yJ,h for 0>J,H is a positive homogeneous Markov specification with determining function (3.14)

g(x,y,z) = e*h+Jx+Jz)/2cosh(h

+ Jx + Jz),

x, y, z e { — 1,1}. Theorem (3.5) thus shows that (3.15)

9{&>») = {^, h },

where fiJh is the distribution of the stationary Markov chain with transition matrix P, h which is defined in terms of (3.14) via (3.7). Let us compute Pj h. We put a = 1 and write + and — instead of + 1 and •l.Then Q

g(+, -,-)/g( + + ,-) g(+, + ,-)/g(+ + , - ) e" 2 " l

e

g( + ,-,+)/g( g( + ,+,+)/g(

-2*-4J\

1 I

The largest solution of the characteristic equation (e" 2 '' - q)(l -q)-

e~2h-*J - 0

+ ,+,+) + ,+,+\

The one-dimensional Ising model

51

ofßis 1 4- e~2h

(3.16)

/ i _ e" 2 ''^ 2

I

,Jtfc = i ± i — +

/ e — +

= e ''(coshfc + V e

4J

fl

+ sinh2/i).

Hence P,,*(-, - ) = e"2*^,1*. JJ.A( + > + ) = «7,* and therefore P-i/J

ij.fc- I . _

-1

-!

A short computation gives

(3.18) W D - i f n 2\

Sinh

"

V e ^ J + sinh2/jy

Using (3.3), (3.17), and (3.18) we may compute the \ij^-probabilities of cylinder events. In particular, we have (3.19)

nJih(<Ti) = (e _ 4 J + sinh2 h)'1/2 sinh h

for all i e Z. In the case J = 0, of no interaction, we see that q0 h = 1 + e~2h and therefore both rows of P0 h coincide; hence fi0 h is a product measure. In the rest of this section we shall investigate the "low temperature limit". O J '' 1 will be endowed with a factor ß, the inverse absolute temperature, and we shall ask for the behaviour of <#{ß<&J'h) = ^ ( O ^ ' 1 ) = {ßßJ,ßh} as ß tends to infinity. This behaviour will turn out to be closely related to the set of ground states for O7''1. A configuration co e Q, is called a ground state of O J '' 1 if for each i e Z the pair (o}i,o}i+1) is a minimal point of the real function \j/\ (x,y) -»• — Jxy —fc(x+ y)/2 on { - 1 , l} 2 . (Note that the potential y

x

=

Moi,oi+i) [0

if

^ = {U + 1}. otherwise

is equivalent to 0. Then lim qß ßh = 1, /)-»oo

J i m œ P ^ = (o i and lim <xP h (l) = 1. Hence ßßhß 0->oo

converges weakly towards <5+, the Dirac

52

Finite state Markov chains as Gibbs measures

measure at the constant configuration co+ e Q defined by u>? = 1, i e Z. Notice that co+ is the unique ground state of O 1 \ Similarly, if h < 0 then œ" = —œ+ is the unique ground state of O1'*, and the Dirac measure cL at a>~ is the weak limit oi ßßßh as ß -* oo. In the case h = 0,O 1 ' 0 has precisely two ground states, namely u>+ and <x>~. On the other hand, we have aP = (j, ^) and

as ß -»• oo. Hence L L hm ^ , 0 =-<$+ +x<5-. /j->00

^

^

Consequently, the limiting measure is the equidistribution on the set {co+, co~} of ground states. In particular, the non-uniqueness of the ground state implies an asymptotic loss of tail triviality: Each of the Gibbs measures nß< 0 satisfies a 0-1 law for all events in 2T, but the limiting measure (S+ + S_)/2 does not. (In order to check the former claim we can anticipate an argument of Chapter 7. Let us write \i = ßßh and y = yß,h, and suppose A e 2T is such that \x{Ä) > 0. Putting v = niA)'1 \Aß we have for each A e «S'7 vyA = n(A)'1 lA(nyA) = v because yA is proper. Hence v e @(y) = {ß} and therefore ß(A) = 1. On the other hand, the limiting measure (<5+ + S_)/2 is not trivial on ST because it assigns probability 1/2 to the tail event [at = 1 for infinitely many i e Z}.) We will see later that the loss of tail triviality is far more dramatic for the Ising model in two or more dimensions. In higher dimensions the set <&(Q>ß'°) is so strongly attracted by the two ground states a>+ and a>~ that for sufficiently large (but finite!) ß there exist two distinct Gibbs measures for 0^'° which are close to ö+ resp. S_. Thus for these ß a phase transition occurs, and Theorem (7.7) will show that this is equivalent to the existence of (many) Gibbs measures for O^'0 with a non-trivial tail cr-field. Case 2. Ising antiferromagnet in the low temperature limit. Here we put J = — 1, and we distinguish the cases h > 2, h = 2, \h\ <2, h= —2 and h < — 2. In the case h > 2 we have

lim a P j h (l) = 1, and therefore lim nßJ<ßh = ö+.a>+ is the unique ground state ß->00

'

/)->00

of O J ' \ Similarly, if h < — 2 then a>~ is the unique ground state of Q>Jh and

/)->00

Next we suppose that \h\ <2. Then <5)Jh has precisely two ground states, namely the configurations a>+~ and a>~+ with

The one-dimensional Ising model

,

a >

,

f

53

1 if i is even,

i ~ = ^ = [_1

if.isodd

We let <5+_ and <5_+ denote the Dirac measures at co+~ and co~+. Inspection of (3.16) and (3.17) shows that lim aPtJfh(l) = 1/2 and /J->00

s p ^ = (l 0 Hence lim Hßj,ßh = ~ö+__ +-ô_

+.

Thus, in this case we also observe an asymptotic loss of tail triviality. This phenomenon is again a pale reflection of a phase transition in two or more dimensions. Finally, we consider the case h = 2. (The remaining case h = — 2 is similar.) It is then obvious from equation (3.16) that lim qß~}t2ß = (\/5 — l)/2, the ß golden ratio. Thus ~*w lim PßJt2ß = F^\

(^ß

- lV

,/5 - 1

Since lim aP/(j2/)(l) = (1 + yj 5)12^5 = a f (l), we conclude that lim nßJt2ß = UP, the stationary Markov chain with transition matrix F. The letter F reminds us of Fibonacci. Indeed, F is the unique stochastic matrix satisfying F(-, - ) = 0 and F{ + , -)F(-, +) = F( + , +)F( + , + ). Consequently, for all [' e Z, n ^ 1, and x 0 , . . . , x„ +1 e {— 1,1} we have ^fi^i

+l

=

x

l>

• • • >CTi + n

'l/fln+(*o+*»+i)/2 [0

=

x

n\(7i

if x

=

x

X

0^(Ti

+n+l ~

X

n + l)

( ;> ;+i) # (-1» - 1 ) otherwise,

for

all 0 ^ j ^ n

where (a0,a1,a2,...) = (1,1,2,3,5,8,13,...) is Fibonacci's sequence. In particular, ftp can be thought of as the equidistribution on the infinite set {co e Q: (co;, (oi+1) ^ (— 1, — 1) for all i e Z}. This set is precisely the set of all ground states of O-7'2. fiF is trivial on 3~. (This is a well-known consequence of the ergodic theory of Markov chains and will also follow from Example (10.24) (2) and Theorem (10.35) (iii) below.) Thus in spite of the existence of uncountably many distinct ground states there is no asymptotic loss of tail triviality. This comes from the fact that the set of ground states for O-7'2 does not exhibit a long range order: If co is a ground state and i e Z then (Of remains unknown even if the restriction of co to the set {j e Z: \j — i\ > 1} is known. A ground state

54

Finite state Markov chains as Gibbs measures

degeneracy of this kind cannot be expected to give rise to a phase transition, even in higher dimensions. The preceding discussion can be summarized as follows: In the low temperature limit ß -» oo, the unique Gibbs measure nßJtßh for the Ising potential (frßj,ßh c o n v e r g e s weakly to the equidistribution on the set of ground states for Jh. In two cases we have found an asymptotic loss of tail triviality. These cases are candidates for a phase transition in the multidimensional Ising model.

3.A

Appendix: Positive matrices

Let £ be a finite set with cardinality |£| ^ 2. A matrix Q = (Q(x,y))x yeE is called positive (in the strict sense) if Q{x,y) > 0 for all x, y e E. The following theorem of Perron and Frobenius was used in Section 3.1. (3.A1) Theorem. Each positive matrix Q has a distinguished eigenvalue q > 0 with the following properties. (i) \z\ < q for all eigenvalues z ^ q of Q. (ii) There is a right eigenvector r corresponding to q such that r(x) > 0 for all x e E. (Hi) q is simple, i.e., if r, r' are right eigenvectors corresponding to q then r' = cr for some c e C . q will be called the PF-eigenvalue of Q. Proof. We introduce the set R of all column vectors r e [0, oo [ £ \{0} and define a function / : R -» ]0, oo [ by fir) =

min

Qr(x)/r(x).

xeE:r(x)>0

Heregr(x) = £ Q(x,y)r(y)./is

continuous and satisfies the equation f(tr) =

y eE

f(r) for all t > 0. Hence f(R) coincides with the set / ( < r e R: £ r(x) = 1 •

and the latter set is compact. We put q = max f(R). q is an eigenvalue of Q. For let r e R be such that q = f(r), and suppose Qr 7^ qr. Then Qr(x) ^ qr(x) for all x e E, with strict inequality for at least one x. Therefore the vector Q(Qr — qr) has strictly positive coordinates, and this implies f(Qr) > q which is in contradiction with the definition of q. Consequently, q is an eigenvalue of Q, and r is a right eigenvector for q satisfying r(x) = q^Qrix) > 0 for all x s E.

Appendix. Positive matrices

55

Next we consider an arbitrary eigenvalue z of Q. Let u e C £ be a right eigenvector for z and a = minQ(x,x) > 0. Let / denote the identity matrix. The equation (Q — al)u = (z — a)u shows that for all x e E X (Q(x,y) - al(x,y))\u(y)\ ^ \z - a\\u(x)\ ye E

and therefore X Q(x,y)\u(y)\^(\z

- a\ + a)\u(x)\.

ye E

Consequently, \z — a\ + a ^ max/(i?) = q. Hence either z = q or \z\ < q. Finally, we let r e R be as above and v be an arbitrary right eigenvector corresponding to q. We may assume that v is real. (Otherwise we may look at its real and imaginary part separately.) Let c = min v(x)/r(x). xe E

Then v(x) ^ cr(x) for all xe E. This implies v = cr because otherwise v(x) - cr(x) = q'1

£ Q(x,y)(v(y) - cr(y)) > 0 ye E

for all x e E which is in contradiction with the choice of c.

n

Next we consider positive stochastic matrices. (3.A2) Remark, (a) The PF-eigenvalue of a positive stochastic matrix P is 1. (b) For each positive stochastic matrix P there is a unique probability (row) vector a on £ satisfying OLP = a (and therefore a(x) > 0 for all x e E). Proof. Let q be the PF-eigenvalue of P. Applying Theorem (3.A1) to the transposed matrix PT, we see that P admits a left eigenvector ( for q such that £(x) > 0 for all x e E. Thus Zf(x)P(x,y)

= qt(y)

(y e E).

xe E

Summing over y we obtain that q = 1. This proves assertion (a). To prove statement (b) it is sufficient to note that ( is unique up to a factor, D A second tool needed in Section 3.1 was the ergodic theorem for positive stochastic matrices. (3.A3) Theorem. Let P be a positive stochastic matrix and a the unique probability vector with otP — a. Then

56

Finite state Markov chains as Gibbs measures

lim P"(x,y) = oc{y) n-*oo

for all x, y e E. Proof. For s ^ O w e put \j/(s) = s log s — s + 1. I/MS strictly convex and attains its minimum 0 at s = 1. For an arbitrary probability vector n on E we define the relative entropy Jf(7i|a) = X oc{x)iJ/{n{x)/a{x)) xe E

of n with respect to a. We claim that Jf(nP\oc)< Jf(n\oc) for all n ^ a. (For 7t = a equality holds.) Indeed, for n ^ a we have xTk xTk

V^TE

,ti

a

W

a(x)

«W/ \a(j;)/

= Jf(7r|a)

because ip is strictly convex, a(-)P(-,x)/a(x) is a probability vector, and 7t( •)/(*(•) is not constant. Now let x G £ and s > 0 be given. Let (5 > 0 be a lower bound of the positive continuous function 71-> J4f(n\a) — Jf(7tP|a) on the compact set Ke of all probability vectors n with £ |7T(y) - a(y)| ^ e. }>e£

Suppose P"(x, •) e K£ for infinitely many n ^ 1. Then W + 1 ( x , -)|a) ^ jr(P"(x, -)l«) - <5 for infinitely many n. This is impossible because (Jf(P"(x, -)la))«ai *s decreasing and nonnegative. Hence X \Pn(x,y) - x(y)\ < e ysE

for sufficiently large n. This proves the theorem,

a

A further proof of Theorem (3.A3) will be given in Example (7.15), and a far more general ergodic theorem will be proved in (10.34).

Chapter 4 The existence problem

The first question which we will ask concerning a given specification y will always be: Does y admit any Gibbs measure \i e ^(y)? In many cases we shall pose an even stronger question: Does <&(y) admit two (or more) distinct measures which are distinguished by certain additional characteristic properties? In order to give a positive answer to these questions we need to construct one, or even two (or more), elements of ^(y). There is a natural device for doing so: Introduce a topology on 0*(Q, SP), pick one (resp. two or more) boundary conditions co e Q, and show that (I) the net (yA(-\co))Ae belongs to <&(y), and

(III) each cluster point of (yA(-|co))Ae^ exhibits the desired additional properties. Whilst problem (III) is the most interesting in concrete models, the existence of a Gibbs measure merely requires us to solve problems (I) and (II). The latter is the objective of this chapter. Clearly, any solution to problems (I) and (II) depends crucially on the choice of the topology on é?{Q, SP). In fact, (I) and (II) impose conflicting demands on this topology. Problem (I) is trivial when ^*(Q, S?) is compact, whereas (II) is trivial when the evaluation map \i -> /x(/) is continuous for each bounded measurable function / on Q. In other words, problem (I) requires a coarse topology, whilst (II) requires a sufficiently fine topology. It will turn out that a practicable compromise is provided by the so-called topology of local convergence. This topology will be introduced in Section 4.1. If the state space E is finite, the topology of local convergence coincides with the weak topology. So, in this case it is metrizable. In general, however, the topology of local convergence does not do us the favour of being metrizable. This will necessitate our working with nets instead of sequences. (A standard reference to the theory of nets is the monograph of Kelley (1975).) For the chosen topology of local convergence, none of the statements (I) and (II) holds in general. In fact, there are simple and innocent looking examples of specifications y for which
58

The existence problem

space shall be used for the first time. (It will also be needed later on, mainly in Section 7.3.) For the convenience of those readers who are not familiar with this notion, some basic properties of standard Borel spaces are stated and proved in Appendix 4.A. On the first reading, however, it might be preferable to skip the technicalities of Section 4.2. In fact, there is no loss in doing so for anyone who is only interested in the case of a finite state space E. For, it is well-known that in this case the weak topology on ^(Q, #"), and thus the topology of local convergence, is compact, and this implies that assertion (I) holds automatically without any condition on y and œ. In Section 4.3 we shall turn to a discussion of problem (II). We shall see that the topology of local convergence is tailor-made for (II) to hold for every quasilocal specification y. In fact we shall obtain a more general result than (II): the boundary condition co may be replaced by a net (va) of random boundary conditions, and we can deal with a convergent net (ya) of specifications instead of a single specification y. Among other things, this generalization implies some important continuity properties of the correspondence that maps a potential to its set of Gibbs measures, as will be shown in Section 4.4. In this section we shall also combine the results of the preceding sections to state a general answer to the existence problem. (See Theorem (4.22).)

4.1

Local convergence of random fields

Let (E, S) be an arbitrary measurable space, S a countably infinite set, (Q, #") = (£ s , 8s), and ^(Q, J^) the set of all probability measures on (Q, #"). We will choose a topology on ^*(Q, #"). If nyA = ß for a given specification y then we have some information on the behaviour of ß on SFK, but nothing can be said about the /i-probability of tail events. Therefore we need a topology which only depends on the local behaviour of the probability measures. If E is finite (and thought of as a discrete metric space) then the weak topology on ^(Q, 2P) is such a topology. For, it is well-known that for finite E a sequence (/On gi m ^ A ^ O converges weakly to some /i e ^>(Q, #") if and only if lim n„(A) = n(A) for each cylinder event A; cf. also Remark (4.3)(3) below. n-*oo

There are two possibilities to deal with the case of an infinite E. We could either impose topological conditions on E and use the weak topology on ^(Q, SF) (but if applied to Gibbs measures this would force us to require suitable continuity properties of specifications and potentials), or simply might stick to the topology of convergence of cylinder probabilities. We choose the latter possibility. We thus consider the algebra (4.1)

J F ° = U J^A

of all cylinder events in D.

Local convergence of random fields

59

(4.2) Definition. The topology of local convergence (or if'-topology for short) on ^"(Q, &) is defined to be the coarsest topology on ÉP(Q, ^) for which the evaluation maps v -> v(A), A e J^ 0 , are continuous. Equivalently, the iftopology is defined by the requirement that for each ß e ^"(Q, &) the sets lve^(Q,JF): max \v(Ak) - ß(Ak)\ < el I lSkS» J

with A1,..., Ane J^ 0 , e > 0, and n ^ 1 form a base of neighbourhoods of \i. Thus a net (na)aeD in ^(Q, J^) converges locally to /i if and only if limna(A) = D ß(A) for all A e J^°. (In this case we will write ßa -g ß.) From now on we will always assume that ë?(Q., 3?) is endowed with the £?topology. (4.3) Remarks. (1) For each y, e 0>(Q, J*) we let /i° = \i\SFQ denote its restriction to J^ 0 . According to Caratheodory's extension theorem, the projection p: /i ->• n° is a bijection from ^>(Q, &) onto the set ^(Q, J^°) of all u-additive normalized nonnegative set functions on J*0. ^(Q, J*70), considered as a subset of the compact Hausdorff space [0,1]^°, is equipped with the topology of setwise convergence. The if-topology on ^(Q,J^) is precisely the initial topology of p and turns p into a homeomorphism. In particular, ^(Q, &) is a Hausdorff space. (2) The if-topology is the coarsest topology such that for each / e S the evaluation map v ->• v(/) is continuous. Thus a net ( A O „ 6 D converges to some /i in the if-topology if and only if lim ßa{f) = ß{f) for a l l / e J?. (This justifies D _

the name if-topology.) For let / e if, /i e ^(Q, J*), and e > 0. Then there are some A e ^ and a step function n k= l

in ifA such that At,..., put

à = eilt /

A„ e &K are pairwise disjoint and \\f — g\\ < e/4. We

Kl-

k= l

Then for each v e ^>(Q, &) with max |v(i4k) - ß{Ak)\ < Ô we have Iv(/) - n(f)\ ^ 21|/ - srll + f

| f l k | |v(^k) - / i ( ^ ) | < e.

k= l

Therefore the mapping v ->• v(/) is continuous.

60

The existence problem

(3) Suppose E is a metric space and Q is equipped with the product topology. Then the y-topology is finer than the weak topology on ^{Q., J5"). This follows from Remarks (2.21)(2) and (4.3)(2) together with the so-called portmanteau theorem which states, in particular, that the weak topology is already induced by the integrals of all uniformly continuous bounded functions. Moreover, if E is countable with the discrete topology, each local function is continuous, so that the weak topology coincides with the Jf-topology. However, the „Sf-topology is only metrizable if J5"0 is countable, which happens only if E is finite. In this case, a standard choice for a metric inducing the „Sf-topology is d(fl, V) = J ] 2"" Y^

where (A(/i))

4.2

lM(0rA(»i) = Ç)-v((TA(#i) = OI.

j is any cofinal sequence in y .

o

Existence of cluster points

This section is devoted to problem (I) of the introduction to this chapter. More generally, we consider an arbitrary net (jj a ) J€D of random fields. We ask for conditions which ensure that (ßa)aeD has a cluster point relative to the ^-topology. This question is trivial when E is finite. For in this case the ^-topology coincides with the weak topology which, as is well-known, turns ^(Q, #") into a compact space; see also (4.11)(2) below. Consequently, a reader who is only interested in the case of a finite state space E can (and is advised to) skip this section and proceed directly to Section 4.3. To begin with, we observe that (na)aeD can only have a cluster point when (4.4)

lim Mm'mîna(Am) = 0 m-*oo

aeD

for all sequences (Am)mèl in #"° with Am \ 0 as m \ oo. For let / i b e a cluster point and £ > 0. As n is cr-additive, there is some m ^ 1 with n(Am) < e. na is frequently in the open neighbourhood {v e ^(Q, #"): v(Am) < e} of ß. Thus lim inf na{Am) < e. aeD

As (Am)m^1 is decreasing, (4.4) follows. Conversely, standard methods show that (/i a ) I6D has a cluster point whenever (Hx)aeD is equicontinuous in the sense that (4.5)

lim lim sup ßx{Am) = 0 m-*cc

aeD

for all sequences (Am)mil in J r ° with ^4m | 0 ; cf. Proposition (4.9) below. Unfortunately, the latter condition is difficult to verify in general. This is

Existence of cluster points

61

because the set of coordinates on which Am depends is allowed to become arbitrarily large. We thus resort to the following concept. (4.6) Definition. A net (na)a€D in ^(Q, 3F) is said to be locally equicontinuous if for each A e ^ and each sequence (Am)m^1 in !FA with Am j 0 lim lim sup na(Am) = 0. Under suitable topological conditions the local equicontinuity of a net will be sufficient for the existence of cluster points. (4.7) Definition. A measurable space (£, S) is called a standard Borel space if there exists a metric d on £ which turns £ into a complete separable metric space and is such that S is the Borel c-algebra with respect to d. (4.8) Comment. Some results concerning standard Borel spaces will be collected in Appendix 4. A. In particular, we will show there that if £ is a Borel subset of a complete separable metric space X and S the set of all Borel subsets of E then (£, S) is standard Borel. (In general, the corresponding metric on £ will not be topologically equivalent to the given metric on X.) o (4.9) Proposition. Let (E,S) be a standard Borel space. Then every locally equicontinuous net in £?{&., 3F) has at least one cluster point in £?(Q., 2F). Proof. Let ()Ua)a6D be a locally equicontinuous net in ^(Q,#"). Its restriction (Ha)aeD t o ^° is a n e t in the compact Hausdorff space [0,1]-^°. Therefore there is a subnet (fxa )peD> such that (n° )peD- converges (setwise) to some [i° e [0, Vf°- Clearly, fi° is finitely additive. In fact, [i° is c-additive on each 3FA. For if Am e !FA (m ^ 1) are such that A„10 then lim fi°(A„) = lim lim naß(AJ m^oD

m-*oo ß e D'

^ lim lim sup ßa(Am) = 0. m-»oo

oceD

We thus conclude that the mappings A -> n°{aA e A) (A e l A , A e y ) constitute a consistent system of marginal distributions. As (£, S) is standard Borel, a well-known theorem of Kolmogorov implies the existence of a unique fx e &>(Çl, &) with n\^° = n°; see Theorem 62.3 of Bauer (1981), for example. This n is the local limit of the subnet (fxa)ßED-. • In the following, we will provide some sufficient conditions for the local equicontinuity of a net (ixa)aED. An obvious sufficient condition is the local uniform domination of(/xa)aeD which means that for each A e y there exists a finite measure vA on !FK with following property: Given any e > 0, there is some ô > 0 such that lim sup fia(A) < s for all Ae^A with vA(A) < Ô. This oteD

62

The existence problem

domination property stands behind the more specific conditions which will be considered below. As a warm-up, we state a simple corollary. (4.10) Corollary. Suppose (E, S) is a standard Bor el space, and for each A e ^ we are given a finite measure vA on J \ . Then the set JT = {/i e ^(Q, 3F): p(A) ^ vA(^4) for all A e !FA and A e £f} is compact. Proof. X is closed and each net in J T is locally equicontinuous. D (4.11) Examples. (1) Let (E, S) be a standard Borel space and X G Jl(E,ê) a finite a priori measure. Suppose y = pX. is a ^-specification with bounded pA's. (For example, y might be Gibbsian for an absolutely summable potential.) The set 1ß{y) is then contained in a set JT as considered in the corollary. Hence &(y) is relatively compact. (2) Suppose E is finite. Then the set 3?{£l, &) of all random fields is compact (and metrizable, cf. (4.3)(3)). This follows from (4.10) by putting vA(erA e A) = |yl|foralM c £ A a n d A e y . o We will now prove a theorem on the local equicontinuity of nets of Gibbs measures or Gibbs distributions with random boundary conditions. On the one hand, this theorem will provide an answer to problem (I) of the introduction to this chapter. In fact, this problem will correspond to the particular case where y" = y, va = Sa, and (AJ a e D = (A) Ae ^. On the other hand, choosing va = p.ae ^(y11) we shall obtain a statement on the existence of cluster points of nets of Gibbs measures for possibly varying specifications. (This statement extends the above Example (4.11)(1).) To state the result we introduce some notation. Given a net (A a ) aeD in £f, we shall write Aa -g S if (A0L)aeD is cofinal in £f, i.e., if for each Ae £f there exists an a 0 G D with A^ => A for all a ^ a 0 . (4.12) Theorem. Let {E,S) be a standard Borel space and Xe Jt(E,S). Let (/OaeD be a net of random fields of the form px = vay%a, where (va)aeD is a net in £P{Çl, 3F\ (yx)aeD a net °f X-specifications y" = p"X„ and (A a ) a e B a net in y with Aa -g S. Suppose that for each A e y and e > 0 there are sets ß e f and BA G êA such that B c {oA G ß A }, XA(BA) < oo, lim sup sup pA(co) < oo, and lim sup pa(Q\B) ^ e. xeD

Then the net (/OaeD is locally equicontinuous and therefore has a cluster point. Proof. Let A G y and (Am)mil be a sequence of sets Am = {aA G AmA} G J \ with Am 10. Let e > 0 be given and B e J* and BA G
Existence of cluster points

63

Then lim sup nMm) ^ I™ sup na(Am D 5) + e for all m ^ 1. Since Aa -^ S there is some a 0 with Aa => A for all a ^ a 0 . For a ^ a 0 we have /*« = v«TÂa = VsJÂJÂ = ^«TÂ = P Â ( M A ) and therefore /xa(i4m H Ä) =

fiakA{p*lAmnB)

^IIPAlBIIMA(Keylm,AnßA}) = ll P ^i B II^V m ,Anß A ) for all m ^ 1. Thus

Hm sup na(Am n ß) ^ cAA(>imiA n ß A ) IED

for all m ^ 1 and some constant c < oo. Since kA(BA) < oo and kA(Am A D BA) -> 0 as m -> oo. The theorem is now evident, o

AmA[0,

The hypotheses of the preceding theorem are trivially satisfied when k is finite and the pA's are bounded uniformly in a. These conditions are natural whenever E is bounded in some sense. In general, however, we can only require that the pA's be bounded on suitable subsets of Q. This case is treated in the corollary below. To fix the ideas the reader might think of the specific case E = R,k = Lebesgue measure, K( = [ — tj~\, £ ^ 1. (4.13) Corollary. Let {E,S) be standard Borel, k e Ji(E,S\ (0")« eD a net of k-admissible potentials, (AJ a s D a net in y with Ax -^ S, (vj a e l ) a net in ^*(Q, J r ), and p.x = vayA" for all a e D. Suppose there is a sequence {K()(il in S such that the following conditions are satisfied: (0 0 < k{K() < oo for all £ ^ 1; (ii) lim lim sup /ia(<7; £ K() = 0 for all i e S; and /-•oo

ae D

(Hi) each A e y is contained in some A e y swc/i t/iat /or all £ ^L 1 lim sup

sup

|ffA'(a>)| < oo.

TTzen t/ie net (/OasD is locally equicontinuous and therefore has a cluster point. Proof We verify the hypotheses of Theorem (4.12). Let A e y and e > 0 be given. Assumption (iii) implies the existence of some A e y with A c A such that for all { ^ 1 c(0 = limsup "ED

sup 0)6K'X

\H%(co)\ ES\"

64

The existence problem

is finite, and hypothesis (ii) guarantees that there is some £ ^ 1 with X Hm sup na(ot i Kf) < 8. ieA

a-»oo

We put B = Kf x £ S \ A and BA = K A . Then /lA(flA) = /l(K^)|A| < oo, B c lim sup /za(Q\B) ^ lim sup £ M«^ <£ K/) < e, cteD

aeDi'eA

and lim sup sup p£"(co) ^ lim sup sup /zf(co)//lA(/z*° l K e B j | c o ) ^ e 2 c ( ^ ( K , r | A | < oo. The net (na)aeD thus satisfies the hypotheses of Theorem (4.12). D (4.14) Comments. (1) The assumptions (i) to (iii) of the preceding corollary are trivially satisfied when A is finite and lim sup ||
for all i e S and <&A is bounded on Ks( for all { ^ 1 and Ae SP. o We conclude this section with a proposition which is of general interest but will not be needed later on. Its proof should be skipped on a first reading. (4.15) Proposition. Suppose ^ e ^"(Q, J^) is a cluster point of a locally equicontinuous sequence (^„)„èi in ^ ( ü , J*). Then there is a subsequence (n„k)k^i which converges to [i. Proof. We put nil

and for each n ^ 1 and A e S? we let hn A denote a Radon-Nikodym density of n„\^A with respect to v\!FA. We write séA for the (7-algebra which is generated by the functions h„ A, n ^ 1, thus «s/A c J*A. We let s/A denote the system of all finite intersections of sets of the form {h„ A ^ c} with n ^ 1 and rational c. «s/A generates «s/A, and s/° = (J «s/A is countable. We write

Since n is a cluster point of (^„)„ai> there exists a strictly increasing sequence ("k)kêi such that max \n„k{A() - n(A,)\ < 1/fc

Existence of cluster points

65

for all k. Clearly, lim n„k{A) = fi(A) k-*oo

for all A e sé°. We show that n = lim fi„k. k For fixed A e ^ w e define ^°° s*k = y-es*K-

KA) = Um fi„k(A)l.

We will show that séA = séA. As stf\ contains the generator s#A of séA and s/° is stable under finite intersections, it is sufficient to prove that s/A is a Dynkin system. By definition, this means that (i) Q e srfA, (ii) if A, B e stfA with A a B then B\A e stfA, and (iii) if (-B^^i is a sequence of pairwise disjoint sets in sé\ then B = [J Be e ^ . (i) and (ii) are trivial. To prove (iii) we put Cm = [j Be,m^l. As

Clearly, Cm e sé\ for all m ^ 1. Let e > 0 be given.

(AO«ai is equicontinuous on J \ and B\Cm J. 0, there exists m ^ 1 such that lim sup n„(B\Cm) < e and fi(B\Cm) < e. «-•QO

This implies lim sup fi„k(B) < lim /i„k(Cm) + e = MQ.) + £ ^ KB) + e and limhu>„ k (ß)^ lim fi„k{CJ k->co

k-*<x>

= KCJ > n{B) - e. As e was arbitrary, B e r f ^ . Hence séA is a Dynkin system, and this implies that sé\ = séA; see Theorem 2.3 of Bauer (1981), for example. By standard extension arguments we conclude from the identity sé\ = séA that lim n„k(f) = n(f) f o r all bounded ^-measurable functions /. But if Ae^A

andfA is a version of the condition probability v(A\jrfA) then H„(A) = v(h„tAlA) = v(hn,AfA) = nn{fA)

for all n ^ 1. Therefore lim n„k(A) = n(A) for all Ae&A. arbitrary, the proposition follows.

•

Since A was

66

The existence problem

4.3

Continuity results

Here we turn to problem (II) of the introduction to this chapter. We start with an example of a specification y which violates (II). y has a finite state space. Thus for each <x> e Q the net (yA(- |eo))Aey> has cluster points. In fact, this net even converges to a limit. But this limit cannot belong to ^(y) because ^(y) is empty. Intuitively, y describes a lattice gas which consists of a single particle with a completely random position. (4.16) Example. Let E — {0,1} and S be arbitrary. For each a £ S we look at the configuration co" which is defined by 1

f 1 if i = a, | 0 otherwise,

and we let 0 denote the constant configuration taking the value 0. For each A £ y we define a proper kernel yA from STA to ?F by TIM- 1 X h(coa) yA(A\œ)=i <-A

ifcoSXA = 0sVA, otherwise;

LU(0AU>S\A)

here A e 3F, co e Q. The system y = (y A ) Ae y is a specification. Indeed, let 0 ^ A c A £ y , / £ =£?, and co e Q. IfcoSXA = 0SXA then WA/MHAI-1

1

X yAf(œa) ae A

HAP x lAr1 £/•(">•') +iAr x / K ) UËA

ieA

aeA\A

= TA/MSimilarly, if cos\A ^ 0syA then MA/M

= yA/(°A«s\A) = /(0A»S\A) = y A /M-

Hence yAyA = yA. It is easily seen that vAyA -^<S0 for each net (v A ) Ae y in ^(Q,#"). But <S0 does not belong to ^(y). In fact, ^(y) is empty. For suppose there exists some ji £ ^(y). Then

/'(j>i> i ) = /'(y>= o >= 1 }) = Z Wfij}^ = o-j = i) = o and

=

"fe« °)

= WA({0})

I

Mdeo)y A K = 0A|co) = 0,

Continuity results

67

where A e SP is arbitrary. Finally, /

\ieS

neS

because /.({Co«}) =

W A ({CO"})

= I A P V K A = 0SXA) S lAp 1

whenever a e A e y , Consequently, ^(Q) = 0. As this is impossible, $(y) = 0. o In the example above, the non-existence comes from the fact that yA(-\co) depends on the behaviour of co at infinity which cannot be controlled by the topology of local convergence. Thus, in order to exclude this source of nonexistence we need to rule out y having a tail dependence. As we know from Section 2.2, this can be done by requiring that y is quasilocal in the sense of Definition (2.23). The theorem below will show that the condition of quasilocality is tailor-made to ensure property (II) of the introduction to this chapter. We need to introduce a concept of convergence for specifications. To avoid technicalities we choose the simplest. Let (yx)aeD be a net of specifications. We sa Y (ya)oLED converges uniformly in the J5f-topology to a specification y, and write ya -g y, if lim||tf/-yA/||=0 for all A e S? and / e if ; here || • || is the sup-norm. Clearly, this condition implies that yA(-\œ) -g yA(-\œ) for all A e Sf and coeQ. (4.17) Theorem. Suppose y is a quasilocal specification and (y")^,, a net of specifications with y* -$ y. Let {Ax)xeD be a net in if with Aa-^ S and (v a ) aeD a net in 0>(Q, &) such that vay^ -$ \i for some \i e 0>(Q, &). Then \i e <&(y). Proof. Let A e ^ and / e if be given. By hypothesis, yAf e Ï£. Thus KVA/)

= l i m \Jl

(V*f)

and therefore IWA(/)

- M/)l = Hm \\y%m(yj) - vay*A(f)\ xeD

= lim|v a y Aa (y A /)-v c< y A (y A /)| IES

^lim \\yAf - y°Af\\ = 0.

68

The existence problem

The second equality follows from the fact that yA = yA yA as soon as Aa =3 A. Since / and A are arbitrary, it follows that p. e ^(y). 6 (4.18) Comment. The above theorem provides an answer to problem (II) of the introduction to this chapter: If y is quasilocal and p is any cluster point of the net (yA( • | a>))A e y for some boundary condition coeQ then p e ^(y). This follows from (4.17) by considering a subnet (yA (• \(o))aeD which converges to p, and setting ya = y and va = <5m, the Dirac measure at co. More generally, suppose we are given a net (&x)aeD of non-empty finite subsets of & such that A« ^ (1 {A: A e # . } ^ 0 and A« -^ 5. Let (v„)eD be any net in 0>(Çl, &) and ha = \&a\~1 I

vayA

(cceD).

Ae«,

If y is quasilocal then each cluster point of the net (/OaeD belongs to <$(y). This follows from (4.17) because pa = payA for all a e D. (As we shall see in (5.18), nets (pa)aeD of the above form can be used to construct Gibbs measures with symmetries.) o In order to pave the way for some further applications of Theorem (4.17) we will now provide a sufficient condition for the uniform ^-convergence of Gibbsian specifications. (4.19) Proposition. Let X e Ji{E, i) and (a)aeD be a net of X-admissible potentials. Suppose there is a potential (D such that lim WHf-fW = 0 otsD

for all A e y . Then
s AA(ifcr - ^D/z* + w i 3v z r3 - i/zfl 5 = yA (i/Ir*-ii) + izr-zA i/zA ^27^(1^"*-H) ^2\\hX~9-

1||

g2(exp||iï*-*l|-l). Therefore, if/ e i ? then

Continuity results

69

p r / - ^ / i i = ii^(/(pr-p^))ii ^ 11/11 UAUPA - PA 1)11

^2||/||(exp||Hr _ *l|-l)By hypothesis, the final expression tends to zero as a runs through D.

a

Proposition (4.19) implies the following particular result: For any finite a priori measure A, the mapping Q> -> y® from the space J 1 of all absolutely summable potentials into the set of all Gibbsian specifications is continuous. There are two standard construction of nets (^"Xeo which converge to a given potential Q> in the sense of Proposition (4.19). These constructions are characterized by two particular types of boundary conditions outside a given region A e y , namely "free boundary conditions" and "periodic boundary conditions"; see the examples below. (4.20) Examples. Suppose À e Ji{E, S) is finite and
J0>. i f ^ A , [0

otherwise.

A

Suppose O is 1-admissible. (This certainly holds when O e f and A is finite.) We then can look at the associated Gibbs specification y">". The restriction of }>*4(-\a>) to J ^ does not depend on the choice of co e Q and is called the Gibbs distribution for O in A with free boundary condition. Since lim sup \\HT~°\\ £ lim

£

||0>J = 0

for all A e y , we can conclude from Proposition (4.19) and Theorem (4.17) that each cluster point of the net (y*4(- \co)Aey belongs to ^(O). Incidentally, the free boundary condition coincides with a configurational boundary condition whenever $ is a gas potential with a vacuum state a e E. For in this case we have, letting a denote the constant configuration with value a, O A = 0 > 4 a S V i ) for all A e y , and thereby yf(- \ is shift-invariant, i.e., Q>A{co) =
A = S n r ] [mk,mk +

pk-l]

k=l

with m — (m 1 ,...,m d )e S and p = (p1,...,pd)e

Nd.

70

The existence problem

We fix such a A and think of it as a torus; that is, we identify A with the factor group S/p • S. Accordingly, we write i = j if i, j e S are such that ik = jk (modp k ) for all 1 ^ k Q denote the periodic continuation of the projection erA, i.e., for each œ G Q we put <xA(co) = (cOj(i))< e s> where 7(1) is the unique element of A with 7(0 = i- The shift-invariance of ,, o
— •. Assuming <5A is /l-admissible, we can look at the specification y*4. The restriction of y£*{-\co) to 3FK is independent of <x> G Q and is called the Gibbs distribution in A for $ with periodic boundary condition. Now let co G Q be given. We claim that each cluster point of the net (yTi'l^&esr belongs to ^(3>). This follows from Proposition (4.19) and Theorem (4.17) provided we can show that Hi/*4"*!! -> 0 for all A G ¥ as A runs through 5^D. So let A G y be given and A G 5^D so large that A D A . For each A a A with A D A ^ 0 we can assume without loss that B D A ^ 0 for all B G £(4). Then

I

I

AczA BeM(A) 4 Ar\Ayt0 B<=A

#B ° 5A =

£
and thus

||flf-*||g

I

I

||0„|| + I ||OJ

.4<=A BeM(A) AnAïQ B\A#0

A\A^0 .4nA#0

Û2 X ||0>J. .4\A#0 .4nA#0

Under our hypotheses on <S>, the last expression tends to zero as A runs through Sfu. The claim is thus proved, o We conclude this section by mentioning a simple consequence of Theorem (4.17). Occasionally, Gibbs measures are defined by the condition \iyK = /jon 3FK (rather than #") for all A e y . Clearly, this condition implies \iyK -g fi.

Existence and topological properties of Gibbs measures

71

Hence, if y is quasilocal it follows from Theorem (4.17) that \i e ^(y). We thus have the following result. (4.21) Remark. Suppose y is a quasilocal specification. Then \i e ^(y) if and only if /i(^) = fy A (/l|-)d/i for all A e ^ A and all A e ^ .

4.4

o

Existence and topological properties of Gibbs measures

We now combine the results of the preceding sections to state a general though somewhat vague existence theorem. (4.22) Theorem. Let (E, S) be a standard Borel space and y a quasilocal specification. Suppose there is a locally equicontinuous net (/J.a)aeD of random fields of the form /j.a = vayA ,oceD, where (va)a eDis a net in 0>(Q, #"), (y ")„ e D a net of specifications with ya-j}y, and (AJ J 6 f l a net in if with Aa-gS. Then ^(y) contains a cluster point of ( j j j ^ , , and is therefore non-empty. Proof According to Proposition (4.9), (na)xeD has a cluster point \i. /i is the limit of a subnet (fia ) ß £ i y oï{^.a)a,eD. Applying Theorem (4.17) to this subnet we see that fi e ^(y). D Suppose we want to apply the preceding theorem to a given specification y. Then, of course, our main task is to find a locally equicontinuous net (/OaeD of the required form. Unfortunately, there is no general rule which allows us to make a clever choice of ( / J J I E B . Rather this choice depends on the particular properties of the specification y at hand. However, in general we are not interested in the mere existence of a Gibbs measure but in the existence of a Gibbs measure with certain additional properties. We then need to choose a net (na)aeD such that all its cluster points have these properties. In concrete models, there is often a natural candidate for such a net, and we are only left with the problem of verifying its local equicontinuity. As a tool then we can (and will) use Theorem (4.12) or Corollary (4.13). As a matter of fact, we shall proceed in this way in the proofs of Theorem (6.21), Proposition (18.12), and Proposition (20.7). There are no existence problems when y is a quasilocal ^-specification for some finite A such that all pA's are bounded. For in this case each net ( / j j , , ^ of the form na = vayA with va e ^(Q, SF~) and Aa -^ S satisfies the hypotheses of Theorem (4.12). In particular, this holds when y is Gibbsian for some

72

The existence problem

absolutely summable potential. The next (and final) theorem deals with this important particular case. As in (2.11) we let 0& denote the Fréchet space of all absolutely summable potentials, and for a given finite X e Jt(E, ê) we look at the correspondence $ from â& to 0>(Q., 3?) which maps each
(c) The graph {(0,/i): O e ^ / j e &($>)} of the correspondence & is closed. (d) & is upper semicontinuous, i.e., for each closed subset F of 0>(Q, !F) the set <$-\F) ^ { O e f : #(0)flF # 0} is closed. Proof, (a) Let O e ^ and coeQ.. Comment (4.14)(1) and Corollary (4.13) show that the net (y®(- \(o))Ae9> is locally equicontinuous. Therefore Example (2.25) and Theorem (4.22) imply that ^(A) = A. Then y°. Moreover, Aä ^ S and / ^ :

= / i ä ^ /i.

Theorem (4.17) thus shows that /i e ^(O). (We note that statement (c) does not rely on the hypothesis that (E, S) is standard Borel.) (d) Suppose O is the limit of a sequence (O n ) nè x in ^ _ 1 (F). (As J 1 is metrizable, we only need to look at sequences.) We will show that O e ^ _ 1 ( f ) . By definition, there is a sequence (/x„)nai m F with /i„ e ^(cD"). (/Ongi has a cluster point \i. This follows from Comment (4.14)(1) and Corollary (4.13) because lim ||
Appendix. Standard Borel spaces

73

(actually of a subsequence, by Proposition (4.15)). In particular, lim (<J>"% fin) = (<1>, /i). Since the graph of ^ is closed, fi e ^(<1>). As F is closed, n e F. Hence £(0) fi F ^ 0 and thereby
4.A

Appendix. Standard Borel spaces

This appendix is devoted to a proof and some consequences of a fundamental result on standard Borel spaces, the isomorphism theorem (4.A6). This theorem has not been used in Chapter 4, but one of its consequences will be needed later on in Chapters 7, 14, and 15. Of course, the isomorphism theorem also contributes to a proper understanding of the notion of a standard Borel space. (4.Al) Definition. Let (X,SC) and (Y,%/) be two measurable spaces. We say (X, SC) and ( Y, <W) are isomorphic and write (X, SC) ~ ( Y, <&) if there is a bijection f:X-> Y such t h a t / a n d / " 1 are measurable. We write (X, SC) £ (Y, <W) if there is some B e <& such that (X,SC) ~ (B,<W\B), where <W\B = {A e <&: A c B}. (4.A2) Remarks, (a) The relation " £ " between measurable spaces is transitive. (b) If S ^ 0 is a countable set and (X, SC), ( Y, <30 are measurable spaces with {X,SC)^{Y,^) then (Xs,iFs)£(Ys,^s). (c) If (X,^) and {Y,<&) are measurable spaces then {X,SC) ~ {Y,<&) if and only if (X,SC)^{Y,<&) and (Y, «0 S (*,#")• Proof. Only the "if" part of (c) needs to be proved. Suppose there are sets B G <W, A e SIC and bimeasurable bijections / : X -» B and 3: Y -» A We define ^„ e <2T, ß„ g
F(x)H

i f x e / l U U ^ ^ i ) .

3 _1 (x)

i f x e {J

'f-l(y)

i f ^ e ß U U (ß 2 n + i\ß2 n + 2 ),

(A2n+1\A2n+2)

and n>0

G(y)=^ g(y)

if ye [j «Ï0

(B2n\B2n+1).

74

The existence problem

It is easily checked that G o F = id x and F o G = id y . Hence F is an isomorphism of (X,9£) and (Y,<&). • We will now look at standard Borel spaces; cf. Definition (4.7). First, we note that a measurable space (E, S) is a standard Borel space if (and only if) (E, S) ~ (X, 3ß{X)) for some complete separable metric space X with its Borel er-algebra 3ß(X). This is because in this case it is possible to define a metric on E in such a way that the isomorphism between E and X becomes an isometry. Next we observe that the configuration space of a spin system is standard Borel whenever the single spin space is standard Borel: (4.A3) Remark. If S ^ 0 is countable and (E, S) is a standard Borel space then (Q, #") = (Es,£s) is a standard Borel space. Proof. The case of finite S is easy. So we assume without loss that S = N. Let d be a complete metric on E which generates S, and D a countable dense subset of E. Define d(co, to') = X 2~" 1 A d(œn, to'n)

(to, to' e Q).

nil

It is easily seen that dis a complete metric on Q which induces the product topology on Q. For each fixed a e D, {to e Ds: to„ = a eventually} is a countable dense subset of Q. Each J-open set is a countable union of sets of the N

form P| {d(an,ton) < e} {to„ e F,iV ^ l,e > 0) in
K = {0,1}N

with the cr-algebra, @)(K), which is generated by the cylinder events. It is well-known, and easy to check, that &(K) is the Borel cr-algebra with respect to the metric dK(to,co')= X 2~"|co„-co;|

(to, to' s K)

on K. (K, dK) is a compact, and therefore complete separable, metric space. The space (K, @>(K)) is idempotent in the following sense: (4.A5)

(Ks, @(Kf) ~ (K, âS(K))

for each countable set S i= 0. This is because each bijection tp: S x N -» N defines an isomorphism / from (K,&(K)) to (Ks,@(Kf) by /((co„) Bèl ) = ton e {0,1}. The main properties of standard Borel spaces are collected in the next theorem. We let ^ß(M) denote the set of all subsets of a set M.

Appendix. Standard Borel spaces

75

(4.A6) Theorem. Let (E, S) be a measurable space, and suppose that (E, S) £ (X, 08(X)) for some complete separable metric space X and its Borel a-algebra 08(X). Then ~(K,08(K)) UN,<$(N)) if E is _({1,...,|£|},
(E,S)~

uncountable, countably infinite, finite.

In particular, (E, S) is standard Borel, and for each standard Borel space (E, S) there exists a metric on E such that E is compact and ë is the corresponding Borel a-algebra. The proof of Theorem (4. A6) will be preceded by two propositions. (4.A7) Proposition. Let (X,d) be a complete separable metric space. Then (X,08(X)))Z(K,08(K)). Proof 1) (X,08(X)) £ (H,38(H)), where H = [0,1[ N and 08(H) is the Borel (T-algebra for the metric dH(co,a>') = £ 2~"\co„ - m'n\

(co,co'e H)

«ei

on H. Let {z„:n^

1} be a dense subset of X and define / : X -> H by

V1 + d(x,z„)Jnèl It is easily seen that / is injective and continuous and thereby measurable. To prove that f(X) e 08(H) and f~u.f(X) -> X is measurable it is sufficient to show that the system {A cz X: f(A) e 08(H)) contains all closed sets. To this end we first observe that for each x e X and e > 0 there is some 0 < ô(x, e) < e such that for all y e X with dH(f(y), f(x)) < ô(x, s) we have d(y, x) < e. For a given closed set A cz X we consider the open sets Ge= (J

{dH(-,f(x))<ô(x,e)},

xe A

e > 0. Clearly, f(A) cz GE for all e > 0. We show that f(A) = f] G1/m. (This mal

will imply f(A) e 0S(H)) Let to e f] G1/m. Then for each m ^ 1 there is some mal

xm e A with dH(co,f(xJ) < ô(xm, l/m) < l/m. We define a sequence (mk)kil recursively by m, = l,mk^ 2k, l/mk+1 < ô(xmk, l/mk) - dH((o,f(xmJ), k^\. Then for all k ^ 1 we have dH(f(xmkJ, f(xmJ) < ô(xmk, l/m J and therefore d(xmhi,xmh) < \/mk ^2 k. Hence (xnJk â l is a Cauchy sequence and

76

The existence problem

converges to some x e A. As / is continuous, f(x) = lim / ( x m J = co. Hence k x co e f(A). ^ 2) ([0,1[,#([0,1[))£(K,#(K)). To see this we define a function g: [0,1 [ ^ K by g(t) = (t„) n â l with tn = [2"t]mod2, 0 g t < 1. Thus (t„) nàl is the unique dyadic expansion of t with tn = 0 infinitely often. It is well-known that ôf is injective and measurable. #([0,1 [) is the complement of the countable set {co e K: co„ = 1 eventually}, and co -» £ «„2"" is a continuous inverse of g. "âl 3) From 2) and Remark (4.A2)(b) we obtain (H, 08(H)) S (KN,<%(K)N). Combined with 1) and relation (4.A5) this implies (X, &(X)) S (K, @(K)). a (4.A8) Proposition. Let (E, S) be a measurable space with uncountable E, and suppose that (E, S) £5 (X, &(X)) for some complete separable metric space (X, d) and its Borel o-algebra âS(X). Then (K,@(K)) £ {E,S). Proof. 1) We consider the set si of all subsets A of X with the property that there is a continuous bijection from a complete separable metric space onto A. We will show that si => 0&(X). The proof rests on the following four observations. i) si contains all open subsets of X. For if A <= X is open then the metric d'(x,y) = d(x,y) + |d(x,.XV4)-1 — d(y,X\A)~l\ (x,y e A) turns A into a complete separable metric space. ii) si contains all closed subsets of X. This is because the restriction of d to a closed set A a X is complete, and choosing one element of each nonempty set of the form {x e A: d(x,y) < 1/n} (n ^ 1 and y e D, a countable dense subset of X) we obtain a countable dense subset of A. iii) si is stable under countable disjoint unions. iv) se is stable under countable intersections. For let /„: Y„ -» An (n ^ 1) be continuous bijections from complete separable metric spaces Y„ onto subsets AnoîX. Then Y = |(y„)„ à l e f i r»: L(yn) = MVx)

for all n ^ l |

is a closed subset of the complete separable metric space Y\ Yn (cf. Remark (4.A3)) and therefore, by ii), a complete separable metric space. Moreover, / : Y -» p | An defined by /((yJng i) = fiiy^ is a continuous bijection. Now we look at the set si* = {^1 e ^/: X\/l e si}. According to i) and ii), si* contains all open sets. Moreover, if (An)n^1 is a sequence in si* then [J A„ e J / * . This is because iii) and iv) imply that [j An = (J (y4„n"ff X \ y 4 J e j ^ and X \ y A„ = f] X\Anesi. n è l \

fc

=l

/

"al

(T-algebra. This proves that si* =3 ^(X).

nâl

Hence si* is a

Appendix. Standard Borel spaces

77

2) Suppose (Y,d) is an uncountable complete separable metric space. We construct a continuous injection g: K -> Y. We let °ll denote the countable set of all open balls in Y with rational radii and centers in a countable dense subset of Y. % is a countable base of Y. The set Y0 = (J {U: U e W, U countable} is countable. Hence Yj = Y\ Y0 is uncountable, and each neighbourhood of any point in Yx is uncountable. For each n 2: 1 and each (co l5 ..., co„) e {0,1}" we choose a point y(co1,...,con) e Yt and a neighbourhood [ 7 ( Q J 1 , . . . , Q J B ) 6 $f of y(coj,...,
1) CZ [7 ( « ! , . . . , (U„),

and d(f/(cu u .. .,co„,0), [ / ( c o j , . . . , ^ , 1)) > 0. This can be done by recursion on n because if y (co !,..., coj and U{oiu...,(a^ are already chosen then {7(0»!,..., co„) D Y1 is uncountable and therefore contains at least two distinct points (which then are called y(œ1,...,œn,0) and y(co1,...,con,l)). Now let co — (co1,co2,...) e K. For each n ^ 1 we have d{y(co1,...,con+1), y(coy,...,con)) :g 2~". Hence {y(co1,... ,co n ))„ äl is a Cauchy sequence and converges to some éf(co) e Y. This gives us a mapping g: co -> g(co) from X to Y. g is injective and continuous. For.let co # co' and n be the smallest integer with con # a»;,. Then the closures of U{co1,...,con) and U(œ'1,...,œ'n) are disjoint and contain g{co) and éf(co') respectively. Hence g(co) # fl(co'). On the other hand, d(g(co),g(co')) ^ 2"«"-1» ^ 2dK(co,o/) because [/((u 1 ,...,co n _ 1 ) contains g(co) and öf(co'). 3) The results 1) and 2) above imply the proposition as follows. By hypothesis, there is a set A e 3&{X) with (E,ê) ~ (A,@{X)\A). According to 1), there is a complete separable metric space Y and a continuous bijection / : Y -y A. As E is uncountable, so is A and therefore Y. Thus 2) applies, and / o g : K - > , 4 i s a continuous injection. As X is compact, / o g is a homeomorphism from X onto a compact subset of A. Hence (K,0(K)) S (A,@(X)\A) ~ (£,«f), and this implies the proposition, D Proof of Theorem (4.A6). We distinguish two cases. Case 1. £ is uncountable. Propositions (4.A7) and (4.A8) give (£,
78

The existence problem

{/(x)} is closed and thus {x} = / _ 1 ({/(x)}) e g. This implies S = ty{E). Thus each bijection from E onto {n e N: n <; |£|} establishes an isomorphism of (E,S) and ({n e M: n S \E\}, ty({n e N: n ^ |£|})). Clearly, if | £ | < oo then {1,..., |£|} with the discrete metric is a compact space, and its power set is the corresponding Borel cr-algebra. On the other hand, the set N can also be turned into a compact metric space. For example, we may take the metric determined by the conditions d(m, n) = |m _1 — n_11 for m, n ^ 2 and d{\, n) — n _ 1 for n ^ 2. ty(N) is the Borel cr-algebra for this metric, D In Chapter 7 we shall need a property of standard Borel spaces which follows readily from Theorem (4.A6). (4.A9) Definition. Let (Q, F) be a measurable space. A countable subset # of J* will be called a countable core of J* if ^ satisfies the conditions below: (i) # generates F and is stable under finite intersections. (ii) If (/Ongi is a sequence in ^(fi, J*) such that lim |I„(J4) exists for all n->oo

Ae^ then there is some (necessarily unique) /i e ^"(Q, J^) with /i(yl) = lim ßn(A) for all A e
(4.A10) Comments. (1) Suppose 2F has a countable core <$ = Then d{ß,v)=

{A1,A2,...}.

X 2--|A*(-4„) - v(X„)| nil

is a metric on 0>(Q, F). In this metric, 0>(Q, F) is compact. For let (/!„)„èl be a sequence in ^(fi, F). Then the diagonal method gives a subsequence (/i„k)i,àl which converges setwise on c€. Thus condition (ii) provides us with some ß e 0>(Q, F) such that d(n„k, /i) -> 0. (2) Suppose (Q, !F) and (fi', #"') are measurable spaces, and q>: Q -> Q' is a surjection such that J* = (p~lF' = {(^T1^': .4'e J*'}. If <<£" is a countable core of #"' then (p_1<^' is a countable core of F. In particular, if (Q, !F) ~ (Q', F') then #" has a countable core if and only if 2F' has. Moreover, J* has a countable core if and only if its canonical image on the space of all atoms of #" has a countable core, o The last sentence implies that in order to know all u-algebras with a countable core we only need to look at cr-algebras containing singletons. For such (j-algebras the existence of a countable core is equivalent to the standard Borel property. (4.A11) Theorem. Let (Q, SF) be a measurable space. If (Q, IF) is standard Borel then !F has a countable core. Conversely, if'!F has a countable core and contains all singletons then (Q, IF) is standard Borel. (The converse will not be needed later on.)

Appendix. Standard Borel spaces

79

Proof. 1) Because of Comment (4.A10)(2) we only need to look at the three particular standard Borel spaces occuring in Theorem (4.A6). Case 1. Q finite, & = ^ß(Q). Then
n-*ao

finitely additive. In fact, fi0 is cr-additive. For let (Ak)kèl be a sequence in # with Ak i 0. As each Ak is compact, y4fc = 0 for sufficiently large k. Hence lim fi0{Ak) = 0- Carathéodory's extension theorem therefore implies that fc-*ao

fi0 can be extended to some fi e ^(O, •&r). 2) Suppose J^ has a countable core <€ and contains all singletons. If <€ is finite then so are SF and Q. Therefore we assume <€ is infinite. We write ^ = {Cl5 C 2 ,...}. According to Theorem (4.A6) it is sufficient to prove that (Q, J*0 S (X, #(K)). We define a mapping / : Q -> X by /(CB) = ( l C k M ) t à x. / is injective. For if /(co) = /(co') then the Dirac measures at co and co' coincide on 9% and therefore on 3F. As {co} e J^, this implies co = co'. To show / is bimeasurable we put C 1 = C, C° = Q\C when C e9S. For all n ^ 1 and s l 5 . . . , s„ e {0,1} we have /

>

= ( 4 Ï I ^ : Ï I

= SI

x, = U ) = c j ' n . . . n q " .

Hence / is measurable. On the other hand, for each A e 9% U {£2} we have (4.A12) f(A) =

Ç]\xeK:Ar\f]Cï"^0\. nil

I

fc=l

J

For, /(y4) is clearly contained in the expression on the right. Conversely, let n

x e K be such that for each n ^ 1 there is some co„ e A C\ f] Ckk. Let /i„ be k=l

the Dirac measure at co„. Then lim n„{A) = 1 and lim n„(Ckk) = 1 for n-*co

n-»ao

all k ^ 1. In particular, lim fi„ exists on 9%. Hence there is some /i e

^(Q,^)

with fi(A) = 1, /z(Q") = Tfor allfc^ 1 and therefore ^ M H f] Q " ) = 1. Thus there exists some co e A with co 6 C£k for all k ^ 1, i.e. f(co) = x. Consequently, x € f(A). The right side of (4.A12) is a countable intersection of cylinder events in K and therefore measurable. Hence {A a Q: f(A) e &(K)} is a cr-algebra containing (€. This proves that f~1:f(Çl) -> Q is measurable, D (4.A13) Corollary. Suppose that (Q, #") = (£, S'Y for some standard Borel space

80

The existence problem

(E, S) and a countably infinite set S. Then J2" admits a countable core %> which consists of cylinder events. Proof. {E, S) can be chosen to be one of the particular spaces which appear in Theorem (4.A6). In the case of a finite set E we let # be the algebra of all cylinder events (cf. Case 3 of the proof of (4.A11)). If (E,&) = (K,@(K)) then (Q,.F) = ({0,l},$({0,l})) N x S , and we can again take the algebra of all cylinder events in ^({0, l}) N x S . In the case £ = Nwe identify S with N and define <£ = {0} U {{
A*(^{i

jv} = CI) = A*(o-{i

jv-ij = 0 -

Z A*(o"{i

AT} = W

X>1

when £ e N Î 1 - - ^ - 1 } . For each AT, ji determines a unique probability measure onfj! j,). Kolmogorov's extension theorem therefore implies that \i can be extended uniquely to a probability measure on 3F. D

Chapter 5 Specifications with symmetries

This chapter is devoted to a preliminary study of specifications which are preserved by a group of transformations on the configuration space. As might be expected, the investigation of symmetries plays a prominent rôle in the theory of Gibbs measures. Of course, this is suggested by the physical applications. Consider, for example, a piece of iron or nickel. Its atoms are located at the sites of a crystal lattice with certain axes of symmetry. Accordingly, the interaction of the atomic spins remains invariant under a transformation group including some spatial translations and reflections as well as the reversal of all spin orientations. Plainly, a reasonable mathematical model of such a physical system should share these invariance properties. This is one reason for studying specifications with symmetries. A more important motive is illustrated by the phenomenon of spontaneous magnetization: Below Curie temperature, the spin system takes one of several possible equilibrium states each of which is characterized by a well-defined direction of magnetization. In particular, these equilibrium states fail to be preserved by the spin reversal transformation. In other words, the equilibrium states break the spin reversal symmetry of the interaction. This phenomenon of symmetry breaking is a wide-spread and important special type of phase transition and thereby provides one of the main motivations for studying symmetries. Needless to say, there is also a practical reason: The presence of symmetries often permits a great simplification of the mathematical analysis of a model. The phenomenon of symmetry breaking will be discussed extensively in later chapters. Here we address ourselves to the converse problem, the existence of Gibbs measures which inherit the symmetries of a specification. There are two possibilities for obtaining such Gibbs measures: either by means of a symmetrization procedure, or as a limit of averaged Gibbs distributions with suitable boundary conditions. These possibilities will be described in Section 5.2. In Section 5.1 we shall introduce the class of transformations of Q with which we are dealing, and discuss the relationship between the invariance of specifications, modifications, and potentials.

5.1

Transformations of specifications

As before, we are given a countably infinite product (Q., #") = (£,
82

Specifications with symmetries

of S and/or transformations of E. Specifically, we let T denote the set of all transformations x: Q -» Q of the form (5.1)

T:(Tj
(co e Q).

Here T^ is any bijection of S, and the xt are invertible measurable transformations of E with measurable inverses. Thus each x e T is a composition of a spatial transformation T^ (which transports the spin at a site; to the site T„J) and the spin transformations xt, i e S, which act separately at distinct sites of S. To specify these components of x we shall occasionally write x = (x^; xt, i e S). Clearly, each x e T i s measurable. More precisely, if A c S and / is 3FKmeasurable then / o x is ^-immeasurable. It is also evident that T is a group. For example, the inverse of x = (T^; xt, i e S) is x = (T" 1 ; T~} , i e S). We mention three particular transformations in T which will be frequently used later on. (5.2) Examples. (1) Let S — Zd for some A ^ 1. Then for each) e S the transformation 0 ; :û)-»(û)j_,), e S

(coeQ)

of Q is called the s/n/t or translation by j . 0 = (Ö,)jeS is an abelian subgroup of T. (2) Suppose £ is a symmetric Borel subset of U. Then the transformation T:GO-»(-<0,-),6S

(coeQ)

in T is called the spin flip or spin reversal. (3) Let £ be a rotationally invariant Borel subset of some UN (e.g., the unit sphere), and M e SO(N) a rotation. Then the mapping w->(MeO;)ieS

(coeQ)

in T is called the spin rotation by M.

o

Suppose next we are given an a priori measure A e Jt(E, S). We shall then say that a transformation T = (x^x^i e S) e T is A-preserving if all xt are Apreserving, in that xt(A) = A for all i e S. (Recall that x^A) = A o T^ 1 stands for the T,-image of A.) It is the object of this chapter to introduce and to study the action of the transformations in T on potentials, specifications, and Gibbs measures. Let x = (T^; T;, i e S) e T be given. For each family q> = (
T((P) = ( f t - l A O T - 1 ) A e y .

Note that this definition applies to potentials and ^-modifications. Clearly, if is a potential then so is T(<5). If p is a ^-modification then Proposition (5.6) below will show that x(p) is a ^-modification, too. Next we define the x-image

Transformations of specifications

83

x(y) = (x(y)A)Ae^ of a family y = (yA)Aey of measure kernels on & by (5.4)

T(y)A(^|co) = y t , lA (T-M|T- 1 a;)

or, equivalently, x(y)ZtA(xA\xco) = yA(i4|ca); here A e y , A e S', and <x> e Q. It can be immediately checked that the r-image of a specification is a specification again. We also note that equation (5.4) is equivalent to the property that (5-5)

(T(y)t.A/) o T = yA(fo x)

for all A e y and all bounded or non-negative measurable functions / on Q. The next proposition shows that the action of x on potentials, Xmodifications, and specifications is defined consistently. (5.6) Proposition. Let X e Jt{E, S) be an a priori measure and x e T a Xpreserving transformation. Then the following assertions hold. (a) x{X) = X.. (b) For each X-modification p, x(p)X_ = x(pX). (c) If 0 is a X-admissible potential then x(<&) is X-admissible, and pT(Q>) = x(p°). Proof (a) This follows readily from the assumption that x is A-preserving. (b) Let A e if and / be a bounded measurable function on Q. Then (5.5) gives M P / U . A / ] ° I = ^A(PA f°

T)

= AA((T(p)ttA/) o T) =

[^.AWPW)]°T

= [ ( T ( P ) A . U / ] O x.

The third equation comes from (5.5), applied to y — A. = T(A.). As T is invertible and A and / are arbitrary, (b) follows. (c) For each A e if we have ^ ' ^ =

I

T(0)tti4OT = H j

/I e . ^ : / I D A 5*0

and therefore hffl o T = /i*. Applying (5.5) to y = A. = T(A.) we obtain Z£A° ° * = ( ^ A O ° T = K(KS

° T) = z*.

This shows that T(O) is A-admissible, and looking at equation (2.8) we find Pl^ox

= pt =

This implies assertion (c). D

x{p%tAoX.

84

Specifications with symmetries

We now turn to the main subject of this chapter, namely potentials and specifications which remain invariant under the action of a transformation in T. (5.7) Definition. Let t = ( T ^ T , , ie S)e T and I a T. (a) A function q> on Q. is called x-invariant if q> o x = q>. More generally, a family

(A e ^ J eS,Ae

&,coe£l);

here A + j = {i + j : i e A}. Similarly, a potential $ is called shift-invariant or translation-invariant or homogeneous if <^+; o 0,. = 0>x

04 e ^ J e S).

The set of all shift-invariant absolutely summable potentials will be denoted by 3$@. Plainly, 3&@ is a || • ||0-closed subspace of the space 3& of all absolutely summable potentials (cf. (2.11)), and the norm

ilfcllo = X I M

(<*>e^0)

turns 0&@ into a Banach space. The shift-invariant bounded range potentials are dense in 8ß@. o The next corollary to Proposition (5.6) clarifies the relation between the T-invariance of a potential and the t-invariance of the associated Gibbsian specification. (5.9) Corollary. Let X e M(E, S) and a X-preserving x e T be given. (a) If p is a x-invariant À-modification then the corresponding X-specification pX_ is x-invariant. Conversely, suppose E is countable, S = ty(E), X is equivalent

Gibbs measures with symmetries

85

to counting measure, and y is a x-invariant specification. Then the unique X-modification p with y = pXm is x-invariant. (b) Let <ï> be a X-admissible potential. If <ï> is x-invariant then so is p*. Conversely, suppose <ï> is normalized by some a e 3P{E, S) which is preserved by x, and either a is a Dirac measure or <ï> is uniformly convergent. Then the x-invariance of p* implies the x-invariance of <ï>. Proof, (a) The first statement follows immediately from Proposition (5.6)(b). The converse is a consequence of Remarks (1.28)(5) and (3) and the fact that Î ' I . A K . A = (T<Ü),„A|TCO) = yT,A(T{(7A = COA}\xCO) = y A ((7 A =

(0A\(0)

for all A G if and œ e Q. (b) The first assertion is obvious from Proposition (5.6)(c). For the converse we note that T(<1>) is a-normalized because

for all 0 # S c ,4 e y . Therefore, if p* is i-invariant then Proposition (5.6)(c) and Theorem (2.34) show that T(<Ï>) ~ <ï>, whence Theorem (2.35)(a) implies T((I>) = (I). D In the case S = Zd, d^.1, the shift-transformations 6j preserve each a e &(E, S). Therefore a normalized potential <ï> is shift-invariant if and only if p* is shift-invariant, and for countable E and positive X the latter occurs if and only if y* is shift-invariant.

5.2

Gibbs measures with symmetries

We will now turn to the problem of existence of Gibbs measures with prescribed symmetries. We start with a simple observation. (5.10) Remark. Let y be a specification and xeT.lffie &(y) then x(n) e ^(x(y)). In particular, ^(y) is invariant with respect to all symmetries of y. Proof. For all A e if we have T(MM?)A

= \ Md<ü)t(y)A(-

|TO>)

= J^(dû))y t f , A (T _ 1 -|û)) = T(WI.-»A) =

T

M-

n

(5.11) Corollary. If &(y) = {n} then n is preserved by all symmetries of y. What about the case of non-uniqueness of ^(y)? Does then &(y) contain a measure which is invariant with respect to all symmetries of y? This question

86

Specifications with symmetries

will be answered using two different approaches. These will be described in the two theorems (5.15) and (5.19) below. Later on, we shall only use the second of these. Thus Theorem (5.15) and its examples can be skipped on a quick first reading. For each subset / of T we let (5.12)

^ ( Q , ^ ) = {/ie ^(Q, &): x(\i) = \i for all x e /}

denote the set of all /-invariant random fields. Clearly, if [/] = {id} U (J {T„ O ••• • o T l : T, e / or if 1 e / (1 ^ i g n)} nil

is the group generated by / then ^/(O, IF) = ^/](Q, 3F\ We also notice that ^j(Q, J^) is closed in the if-topology. This is because fo x e if for all fe i£ and T e T, and /j, is /-invariant if and only if n(f° x) = n(f) for all fe if and re/. We are interested in the set (5.13)

9I(y) =

9(y)r\PI(a,&)

of all /-invariant Gibbs measures for an /-invariant specification y. Our first approach to the existence of such objects consists of a symmetrization procedure which enlarges the symmetry group of a Gibbs measure. Let / 0 , /j c T and x e T. We write (5.14)

Iy o / 0 = {Tl o T 0 : TX e / ^ T Q e / 0 } .

In particular, we put t o / 0 = {i}o / 0 and / 0 o x = / 0 o {T}. If / 0 and ^ are subgroups of T then / 0 is a normal subgroup of [/0 U / t ] if and only if Tt o / 0 = / 0 o i j for all Tt e / 1 ; and the latter implies [/0 U/j] = / 0 o / x = /i o /0.

(5.15) Theorem. Let / 0 and It be two subgroups of T such that xt o / 0 = I0 o xt /or a// Tj e / t , ara/ /et y be an It o I0-invariant specification. Suppose that either (i) there is a topology on lY which turns 1^ into a compact topological group and is such that the evaluation map e: (x, a>) -> xco from Iy x Q onto Q is measurable {provided It is equipped with the Baire o-algebra); or (ii) yio(y) is compact (in the &-topology), and for all T 1( T 2 e / t there exists some T 0 e I0 (possibly the identity) such that I , O I 2 = I 2 O I , O T 0 . Then %o(y) # 0 implies 9IlOl0(y) * 0. Proo/. 1) We first assume that hypothesis (i) holds. We take any v e <SIo(y). If I\ is a finite group then it is natural to consider the measure A* = Uil" 1

Z *i(v).

Similarly, in the general case we let m denote the normalized Haar measure

Gibbs measures with symmetries

87

on (the Baire a-algebra on) J^ ; an existence proof for m can be found in Cohn (1980), e.g.. We then define fi = e(m x v ) = J m(dx)T(v). We show that /i e ^IlOl0(y). (5.10) implies

First of all, \i is specified by y because Remark

WA = 1 m(dx)T(v)yA = J m(dx)T(v) = /i for all A e y . Next we consider a transformation Tt o T0 e Ix o I0. For each fe ££ we have M / ° T i ° T 0 ) = J m(dr)v(/o T ^ ^ O T) = J m(dx)v(/o Ti o T) = J m(dr)v(/o T)

proving the Ix o 70-invariance of /i. The second equality follows from the hypothesis that for each x e Ix there exists some f0 e I0 such that x0 O T = x o f0. For this gives v ( / o Xl o T0 o

T)

= v(/o

Tl

o T o f 0 ) = v ( / o Tt o

T)

because v is 70-invariant. The third equality comes from the fact that m is preserved by the translation x ->• TX O T of / t . 2) Turning to the case of hypothesis (ii) we only need to show that

*W(y) * 0 for each finite subset F of I1. For in this case we may use the compactness of the closed subsets ^IoUF(y) of ^Io(y) to conclude that 0*

H ^oUF(y) = ^ 0 u/ 1 (y) = ^ /l o /o (y). Fc/, |F|
Let v e &Io(y) and F = { T 1 5 . . . , T „ } c /j be given. For each i = (i 1 ; ...,i n )e Z" we write X' =

x[l

o •• • o T^ n .

As t ' e / i , Remark (5.10) ensures that x'(v)e^(y). T'(V) e &Io(y) because T 0 o T'(V) = Xl

O f 0 (v)

=

We even know that

T'(V)

for all T 0 e I0 and suitable x0e I0. We further note that T1' O TJ'(V) = xi+J(v) for all i, j G Z". Indeed, it is sufficient to check this when only one coordinate of i, say ik, is different from zero. In this case we have, setting;' = (j1,...,jk,0,...,0) and v = TJ'~J"(V),

88

Specifications with symmetries T1' 0 TV'(V) -

Ti+J(v)

= T'kk O TJ"(V) -

Tj' O T ^ ( v ) = 0 .

For, our hypothesis on It implies that It acts commutatively on the 70-invariant measure v. We now use the argument of the Markov-Kakutani fixed point theorem. For each N ^ 1 we consider the cube A^ = Z" fl [ — N, NY and define ^ = lA^r1

x T'(V). ieA„

Since ^Jo(y) is convex, /j,N e ^Io(y). As ^Io(y) is compact, the sequence (fJ.N)„äl has a cluster point /i in ^Jo(y). We show that \i is F-invariant and therefore ^ o uF(7) = ^ o ( 7 ) n ^ F ( y ) ^ 0 . Let j e Z", / e if, and e > 0. Because foxje AT ^ 1 such that IM/) - M / ) l + I M / °

TJ)

- Kf°

if, there are infinitely many *j)\ < B.

Thus \fi{f) — fi(fo Tj)\ < e provided we can show that lim | M / ) - M / °

T>)|

= 0.

JV->oo

But M / ° ^ ) = | A A , r 1 X V(/OT^') = |AN|-1

X

V(/OT ; )

and therefore IM/î-M/^Ol^lAjvr1 ^ 11/11

X

V(/OT ; )

\ANA(AN+j)\/\AN\-^^0.

The proof is thus complete, D (5.16) Corollary. Let I be a subgroup of T and y an I-invariant specification with &(y) =£ 0. If I is a compact group and e is measurable, or if ^(y) is compact and I is abelian then ^j(y) =£ 0. Proof. Take I0 = {id}, Ix = I in Theorem (5.15). D For more complicated subgroups I oi T the existence of /-invariant Gibbs measures may be proved by an iterated application of Theorem (5.15). For instance, this gives ^7(y) ^ 0 when y is /-invariant and / is a direct product of finitely many compact or abelian normal subgroups of /. (5.17) Examples. Let S = Zd for some d ^ 1 and y an /-invariant specification such that &(y) is non-empty and compact. Consider the following choices of/.

Gibbs measures with symmetries

89

(1) / = ® = {dy.jeS}, the shift-group. As 0 is abelian, Corollary (5.16) gives %(y) # 0. (2) / = R o 0 , where 0 is the shift-group and R the finite group which is generated by the mirror reflections in the planes {x e Ud: xk = 0} (1 ^ fe ^ d) and {x e M.d: xk = exf} (1 ^ k < £ ^ d,e = ± 1). Note that R contains all lattice rotations. Each x e R is of the form xco = ( (( —l)' 1+ "' + ' d co;) ieS the spin flip on the odd sublattice of S, and T 0 :CO-> — x^co the spin flip on the even sublattice of S. The finite set F = {id,TQ^^TQ OTJ} is a group of involutions. Let R, 0 be as above and / = R ° 0 o F. We show that %„ 0 o F (y) # 0. By Corollary (5.16), %{y) # 0. As 0, ° F = F o B} for ally e S, Theorem (5.15) gives ^@oF(y) =£ 0. Finally, for all x e R we have x o 0 o F = 0 O T O F = 0 O F O T . Hence a second application of Theorem (5.15) ensures that^Ro@oF(y)#0. o The preceding approach to the existence of invariant Gibbs measures consisted of two steps: First we needed to find an element v e &(y), and then a symmetrization of v gave an invariant Gibbs measure \i. As we have seen, the latter step requires some commutativity properties of the transformation group at hand. This restriction can be avoided by following another approach in which an invariant Gibbs measure is constructed directly from the specification. This second procedure is preferable whenever there is no interest in non-invariant Gibbs measures. The basic observation is contained in the next proposition. (5.18) Proposition. Let x e Tbe a transformation and D a directed set. Suppose for each a e D we are given a transformation t a = (Ta*;Tai,i e S) e T, a xainvariant specification y", a xa-invariant random field va, and a non-empty finite subset mx of Sf. If ll/°T,-/°T||-i?0

for all fe Se

and

!{Ta,A:Ae«a}AfapJ^0 then each cluster point ß of the net

90

Specifications with symmetries

/*« = i#.r z \ii

(^D)

is x-invariant. Proof. For each / e S£ we can write |/x(/o T) - /x(/)| ^ lim sup |/x„(/o T) - /x a (/)| IED

= lim sup |/x„(/o T„) - n„(f)\ D

= lim sup D

v.fWr1 Z

= lim sup

*.r

z

[>A(/° O

- yÂ/])

KA/-7X/])

A E I ,

/

£ 11/11 l i m s u p l ^ r M I ^ A i A e ^ j A ^ J = 0. The third equality follows from equation (5.5) and the 1,,-invariance of y" and v„. Since / was arbitrary, we conclude that T(/Z) = p. a Here is a counterpart to the general existence theorem (4.22). (5.19) Theorem. Let (E,ê) be a standard Borel space, I cz T a set of transformations, and y an I-invariant quasilocal specification. Suppose there exists (i) a net (ya)xeD of I-invariant specifications with y" -ft y; (ii) a net (^„)lieD of non-empty finite subsets ^a of Of such that Aa = f] A -ft S and Ae«.

\{z,A:Ae<Xa}A<Xa\/\<Xa\-ftO for all T e I; and (Hi) a net (vj a e i ) in ^ ( Q , J*) such that the net

Ha-W.r1

Z WÂ

(aefl)

AE«,

is locally equicontinuous. Then ^(y) contains a cluster point of (pa)aeD and is thereby non-empty. Proof Proposition (4.9) ensures that (pJaED has at least one cluster point p. As \ia = pay\a, Theorem (4.17) shows that pe$(y). By Proposition (5.18), p e <S,(y). a The general comments on Theorem (4.22) also apply to the preceding existence theorem. Some typical applications will be given in the examples below and, on a more concrete level, in Sections 6.2 and 6.3.

Gibbs measures with symmetries

91

(5.20) Examples. Let S be the integer lattice of any dimension d ^ 1. As in (5.17), we consider the shift-group © and the group R which is generated by all mirror reflections of S. Also, we let T° denote the group of all "pure spin transformations", i.e. of all T e T whose spatial part T + is the identity. Likewise, TA° stands for the group of all pure spin transformations which preserve a given a priori measure À. As an illustration of Proposition (5.18) and Theorem (5.19), we shall use three different types of boundary conditions to construct Gibbs measures which are invariant under some subgroup / of T° o R o @, the group of all t e T whose spatial part T + is a composition of a translation and a reflection or rotation. (1) Configurational boundary conditions. Let / be any subgroup of T° o R o @, y an /-invariant quasilocal specification, and a> e Q any configuration such that reo = œ for all x el. Consider the cubes AN = Sfl [ — N,iV]d, and suppose the sequence IANI" 1

I

)%,+i(-|û>)

(NeN)

of averaged Gibbs distributions has a cluster point ß. Then ß e ^j(y). Indeed, the /-invariance of ß follows from Proposition (5.18) by putting D = N, yN = y> VN = <5o» tN = x e I, and 02N = {AN + i: i e AN}. For, if T e / then T„, is a composition of a reflection or rotation r and a translation by some j e S. Hence {T + A: A e ^ j = {AN + i: i e AN + j} and therefore KVV: A e MN}A@N\/\<%N\ = \(AN +j)AAN\/\AN\

^ 0.

On the other hand, ß is also a cluster point of the modified sequence lAk(N)l-1

E

yA„+i(-l<ö),

' s Aie,«,

where k(N) is such that k(N)/N -> 1 and iV - /c(iV) -> oo. (For example, we can take k(N) = N - ^/N.) Comment (4.18) thus tells us that ß e <S(y). The proof is thus complete. Needless to say, the above result still holds when the /-invariant configuration co is replaced by a random boundary condition ve^(fi,/). (2) Free boundary condition. Let A e M(E, S) be finite, and suppose that / <= TA° o R. (I thus consists of compositions of reflections and A-preserving pure spin transformations.) Consider an /-invariant potential O e l Let AN be as above and let
92

Specifications with symmetries

from Proposition (5.18) and Example (4.20) (1) that each cluster point of the sequence above belongs to (SI{^>). By Comment (4.14)(1), such cluster points certainly exist when (E, S) is standard Borel. (3) Periodic boundary condition. Let X e Jt{E, S) be finite and O a 1admissible potential. Suppose that either $ has finite range or $ e 0&. We fix some integer m S: 1 and let T}m) denote the group of all A-preserving pure spin transformations x = (id; xh ie S) e T° which are m-periodic in the sense that T; = tj whenever i—j = mk for some k e S. We assume that $ is invariant with respect to a transformation group / which satisfies 0 c / c TA(m) o R o @ , For each N ^ 1 we choose a translate AN of the cube S D [1, miV]d such that A,y | S, and we let 5>N = 5>A" denote the corresponding periodic modification of $ which has been introduced in Example (4.20)(2). We let {vN)Nèi be an arbitrary sequence in ^(Q, #") and consider the sequence

to = w £

(N ^ i). s

By definition, ^jvl^ÂA i the Gibbs distribution in AN for $ with periodic boundary condition. We let %($>) denote the set of all cluster points of (A*jv)jväi- By t n e s a m e reasons as in the preceding example, the set %(Q>) does not depend on the choice of (vN)N^ 1. So we can assume that vN e ^j(Q, #") for all N. We will show that %(&) c <^7(j(Çl, #") follows from Proposition (5.18). For let x = (x*; x;, ieS)e I, N 7t 1, and x,y = (x^; xh i e S) the periodic modification of x. Here xNj|(: S -» S is defined by the conditions (i) T N*0 + W^V/) = xN*i + m ^ 7 f° r a ll U e S; and (ii) for i e AN, xNifi is the unique site in AN such that x ^ i = xj, + mNj for some j e S. We show that
Be 01(A)

K The second equality can be justified as follows. On the one hand, we have (x*B)* = xNJfB* for all BeSf, whence y(tNtlA) = {x^B: B e Sf(A)\. Since x^ preserves the relation = (for A = AN), this implies that {x^B: B e &(A)} is a complete set of ( = ^representatives of ^(xN^A\ On the other hand, ^Af, ° TN = T ° <^v For l e t i e S be arbitrary and j e AN be such that j = x"1*'. Then i = x^j = xNj(t j and thus

Gibbs measures with symmetries

= Tj o a. = T; o fft-,(. off Aj =

ffioio

93

ffA„.

The Tjy-invariance of <&N is thus proved. Now let D = N and, for each N e D, TN be as above, yN = y*", and ^jy = {Ajy}. Since / o TN = fo T eventually for all / e JS? and tNi).AN = AN for all iV, these quantities meet the hypotheses of Proposition (5.18). Thus each ß e ^oi®) is T-invariant, and the proof of the inclusion ^0(<S>) a (SI(^>) is complete, o We conclude this chapter with a basic definition. (5.21) Definition. Let y be a specification. A symmetry T of y is said to be broken if there exists some ß e &(y) such that T(/I) ^ /i. It is an immediate consequence of Remark (5.10) that \@{y)\ > 1 whenever y has a broken symmetry. This simple fact provides us with a key which opens the door to a large class of models exhibiting a phase transition. In fact, most examples of phase transitions which will be considered later on are examples of symmetry breakings. As was mentioned at the beginning of this chapter, in physical applications it is important to know if a phase transition is accompanied by a breaking of one or several symmetries, and if some other symmetries are still preserved. In the next chapter we shall encounter three basic models showing distinct kinds of symmetry breaking.

Chapter 6 Three examples of symmetry breaking

Although one of the main objectives of this book is to discuss the phenomenon of non-uniqueness of Gibbs measures, we have not shown so far that this phenomenon is indeed possible. In this chapter we will do this. We shall present three basic examples of nearest-neighbour potentials which exhibit a phase transition. In each of these examples, the phase transition is characterized by some sort of symmetry breaking. The first example is the simplest model that shows a symmetry breaking. It is a spatially inhomogeneous version of the Ising chain of Section 3.2. The corresponding potential is invariant under the spin-flip transformation, but some Gibbs measure for this potential is not. The second example is the famous Ising model on the twodimensional square lattice. This is the simplest shift-invariant model which shows a phase transition. Again, the phase transition is due to a breaking of a spin-flip symmetry. (The shift-invariance is preserved.) The third example is the two-dimensional discrete Gaussian model. Its spins take integer values and interact via a quadratic nearest-neighbour potential which is shift-invariant and shows several further symmetries. Shlosman (1983) discovered the existence of Gibbs measures with a staircase structure which breaks the shiftinvariance of the model. (The analoguous Gaussian model will be considered in Example (13.43).) The inhomogeneous Ising chain yields itself to a direct computation which will be carried out in Section 6.1. The two other models, which are twodimensional, will be treated in Sections 6.2 and 6.3 by means of a famous estimation technique which can be traced back to Peierls (1936) and combines physical and geometric ideas. The Peierls argument has turned out to be a general device for proving phase transitions. A key to this device is provided by the physical notion of a ground state. A preliminary version of this notion already appeared in Section 3.2, and a formal definition will be given at (6.18). Roughly speaking, a ground state relative to a potential <5 is a configuration at which the O-energy takes a minimum. (This minimum may be either absolute or relative. Shlosman's result will show that relative minima are to be taken into account as well.) The Peierls device applies to potentials <5 with several distinct ground states. Let us give an outline of the main steps of this device. To fix the ideas we will stick to the case where S is the square lattice and <5 is a nearestneighbour potential. (i) Pick any ground state m of <5.

Inhomogeneous Ising chains

95

(ii) Look at any configuration £ which is a local perturbation of co in the sense that {i e S: Ç; # coj is finite. Draw closed polygons along the outer boundaries of the connected components of {i e S: Ç; ^ co;}. Call each such polygon a contour. (iii) Check if co is stable in the sense that each local perturbation £ of co requires an additional amount of energy which is at least proportional to the total length of the contours. (iv) If co is stable and ß is large, a configuration £ with a long contour requires a large amount of energy relative to ß) contains a measure fi^ which is a "random perturbation" of co, in the sense that fi^ -» ôœ as ß -» oo. If this program can be carried out successfully for at least two distinct co's then clearly |^(/?)| > 1 when ß is large enough, and this will complete the proof of a phase transition. (If these co's are related to each other by means of symmetries of $ it will be sufficient to study just one of these.) Besides the Ising model and the discrete Gaussian model, there is a large number of further models which can be analyzed by means of a suitable version of the Peierls argument above. Many of these will be discussed in Chapters 18 and 19 as examples of a general theory of phase transitions that relies on Peierls' ideas. A different general theory along Peierls' lines has been developed by Pirogov and Sinai (1976); cf. the Bibliographical Notes on Chapter 19.

6.1

Inhomogeneous Ising chains

We put S = N, the set of all natural numbers, and let E = { —1,1}. The a priori measure X on E is counting measure. We choose a sequence (./„)„>! of real numbers such that Jn > 0 for all n ^ 1 and (6.1)

^ e~2J" < oo.

For instance, we might take J„ = clog(l + n) with c > 1/2. We define a nearest-neighbour potential $ by (6 2)

$

{-J"f7»f7«+i [ 0

if A =

("'" + !}> otherwise.

Under the action of $ adjacent spins tend to be aligned. That is, $ is

96

Three examples of symmetry breaking

ferromagnetic. Clearly, the spin-flip transformation (6.3)

(TCO); =

—cOi

(a> e Q, i e S)

is a symmetry of O. For n,ve ^(Q, &) we write |>,v] = {sv + (l -s)fi:0^s^

1}.

(6.4) Theorem. Consider the model above. Then

where /*_ = T(/J + ), /J+(<7;) > 0 for all i e S, and thereby n_ ^ fi+. The proof rests on the following lemma. (6.5) Lemma. Let A = {1,..., N}, n e A, x e E, and a> e Q. Then yli^n = x\co) = ( 1 + xcoN+l f l tanhJij/2. Proof. Since 2y*() - 1 = y*(on = l|ca) - tffo = -l|ca) = y>n(co)> the conclusion of the lemma is equivalent to the identity N i=n

In order to prove this we introduce a bijection ri<->fj from EA onto itself by setting N

Vi = <%+i f l nj

(i e A),

j=i

»i = fiifii+i

(i^i
iN =

fjNœN+l.

Then we can write Z%(co)yfa((o)=

I

r;e£A

^exp[-/^(fjf(a S N A )] ' N

= %+i L ( ïî»/;)exp n-1

û)Ar+12Ar r i c o s h J ; = coN+12N(

\

i=l

/ TV

msinhJ;

ncoshJ.jfntanhJ;). yie A

Inhomogeneous Ising chains

97

A similar but simpler computation shows that Z%(œ) = 2N n coshJ;. ieA

Combining these equalities we obtain the desired result, D Proof of Theorem (6.4). 1) For each <x> in the set A+ = {on = 1 for all sufficiently large n), the limit ß+ = lim yfi

N]{-\co)

N->oo

exists and does not depend on the choice of <x>. Indeed, let A be any cylinder event. We choose some n with A e ^{x,...,„-i}, and for x e E we let <x>n'x e Q be any configuration such that an(con,x) = x. For N > n we have from Lemma (6.5) (6-6)

y?u....N](A\co)

= E yTu...,n-iM\oin'x)yfi

*}K = x\a>)

xe E

= E yfi

n-1}(A\(o'-')(l

+ xœN+1 f i tanh

J^IL

In the limit N -» oo, the last expression converges to a limit which does not depend on co e A+. Thus ß+ exists. According to Theorem (4.17), ß+ e <&(<&). A similar argument shows that there exists some /i_ e ^(O) with /x_ = lim y®,...,*}(• M JV-»oo

for all CL) e A_ = xA+. In particular, [/x_,/x+] c ^(O). 2) Next we show that T(/I + ) = /x_ # /x+. Let co e A+. Then TCO e /1_ and therefore T(/I+)=

lim T(yfi

N](-\co))

JV-»oo

= lim y?i,....N}(-\*°>) JV-»oo

The second equality comes from the fact that x is a symmetry of O; cf. Corollary (5.9). Using Lemma (6.5) and hypothesis (6.1) we further obtain that ^ + ( = 2/1+ (
= 2 lim#

1

JV}(a„=

l|co)- 1

98

Three examples of symmetry breaking

= J] t a n h Ji

= n a - 2/(1 + g2/o) >0 for all n. In particular, this implies that T(/I + ) ^ /i + . 3) Finally we prove that &(<&) c [/i_,/i + ]. Let /i e ^() be given. For each n e S w e conclude from Lemma (6.5) that zeE

= I

y{*l

„}(*„= - Z | c 0 " + 1 ' z )/l(c7 n + 1 = Z )

= (1 - t a n h J „ ) / 2 = 1/(1 + e 2/ "). Hypothesis (6.1) ensures therefore that

Hence by the Borel-Cantelli lemma H(A_ U A+) = n(o„ ¥" <Jn+1 for at most finitely many n) = 1. Consequently, for each cylinder event A we obtain, using the dominated convergence theorem, /x(^)=lim

J

Mdco^^.^M

= n(A_)n_{A) +

KA+)fi+(A).

Hence \i = \i{A_)yL_ + n{A+)n+ and therefore ji e [^_,^ + ].

D

(6.7) Comments. (1) The constant configurations a>+ and a>~ minimize ^ for all A e Sf and are thus ground states of $. Let S+ and + and co~, and write ^(ßO) = [ ^ , ^ f ] when /? ^ 1. Then lim ^(
lim#(j80) = [<5_,<5+]. /)-*oo

(2) The condition (6.1) is not only sufficient but also necessary for a phase transition. For if (6.1) does not hold then

The Ising ferromagnet in two dimensions

99

CO

ntanhJ ; = 0 i=n

for all n ^ 1. Hence equation (6.6) shows that, for all co efl, lim y* ,...,#}(' \
exists and is independent of co. Applying the dominated convergence theorem as in Step 3) of the preceding proof we can conclude that |^(3>)| = 1. (3) Step 3) of the proof above has also shown that n+(A+) = p-(A_) = 1. First of all, this means that the two extreme points /*+ and p_ are supported on disjoint tail events. This is true in general, as we shall see in Chapter 7. Secondly, the event A+ U A_ can be characterized as follows. A configuration co belongs to A+ UA^ if and only if the graph G(co) with vertex set S and edge set {{i,i+ 1} c S : c o ; = co1+1} = {A = {i,j} c S: \i -j\

= l.O^co) = min
contains a (necessarily unique) infinite connected component. The relation H(A+ U A J) = 1 which has been proved to hold for all fi e &(<&) can thus be restated by saying that the random graph G(-) on the probability space (Q, 3F, fi) exhibits the phenomenon of percolation. The phase transition for can be viewed to be a consequence of this fact, together with the further fact that the percolation event splits into the two disjoint tail events A+ and A_ which are related to each other by the symmetry t. This geometrical point of view is very similar to the ideas of Peierls which will be described in the next section. In fact, the proof of Lemma (6.14) will show that the random graph G ( ) is closely related to Peierls' notion of a contour. The same random graph will reappear in Chapters 18 and 19. These chapters contain a theory of phase transitions which rests on percolation, o

6.2

The Ising ferromagnet in two dimensions

Again we choose the set E = { —1,1} for the state space (and counting measure for the a priori measure), but now the parameter set is S = Z2, the square lattice. We consider the ferromagnetic Ising potential $ with coupling constant 1 and vanishing external field. «S is given by (6.8)

<S>A = \~°i0j ( 0

if ^ = { U } , | i - 7 l = l, otherwise.

100

Three examples of symmetry breaking

again be denoted by ö+ resp. cL. As ®A takes its minimum at a>+ and eo~ for all A e £f, co+ and a>~ are ground states of <ï>. Thus <ï> exhibits a ground state degeneracy. To see whether this ground state degeneracy implies a phase transition we need to verify that the ground states eo+ and eo~ are stable in the sense that the set ~ are unstable; this was shown in Section 3.2. Here we shall prove the remarkable result that eo+ and co~ are stable in two dimensions. We let d denote any metric for the if-topology on ^"(Q, $F\ cf. Remark (4.3)(3). Recall from (5.13) that %(ßO) stands for the set of all shift-invariant Gibbs measures for /?. (6.9) Theorem. In the ferromagnetic Ising model on Z 2 as described above, lim d(%(ß®),o+) = lim d(^@(ßd>),o_) = 0. In particular, for all sufficiently large ß there exist two shift-invariant Gibbs measures pit, /x£ e ^0(/?) with t(fii) = fit and Hß-((T0) < 0 < ni(a0).

In physical terms, n+{o0) is the magnetization of the Ising spin system when Hß+ is its state. The last sentence of the theorem above can thus be rephrased as follows: At sufficiently low temperatures, the two-dimensional Ising ferromagnet admits an equilibrium state of positive magnetization, although there is no action of an external field. This phenomenon is called spontaneous magnetization. Before turning to the proof we take this opportunity to mention a number of further results on the Ising ferromagnet. Using correlation inequalities, one can show that for each ß ;> 0 the local limit ni = lim y%*(- \œ+)e %(ß<&) exists. Its magnetization is maximal, in that /4( 1 if and only if /4( 0- (See Lebowitz and MartinLöf (1972) and Ruelle (1972).) By an inequality of Griffiths (1967a), np+{a0) is a nonnegative nondecreasing function of ß. Consequently, there exists a critical inverse temperature 0. rg ßc ^ oo such that |^(jßO)| = 1 when ß < ßc and |0(/?O)| > 1 when ß > ßc. Theorem (8.7) below will imply that ßc > 0, and the second phase of Theorem (6.9) above just means that ßc < oo. Thus 0 < ßc < oo. As a matter of fact, it is even known that ß ^ s i n h ^ l ^ l o g (1 + ^/2) and /xSM = [ l - ( s i n h 2 j 8 ) - 4 ; r

(ß^ßc).

These remarkable formulas have a long history which extends over the three

The Ising ferromagnet in two dimensions

101

decades from 1941 to 1973; see the Bibliographical Notes on this section. A further remarkable result was obtained independently by Aizenman (1980a, b) and Higuchi (1981) on the basis of the pioneering work of Russo (1979): / 4 and its spin-flip image {it = t(fi+) are the only extreme Gibbs measures for ß, i.e., 9(ßQ) = lnß-,nß+} for all ß > ßc. The last fact contrasts with the behaviour of the Ising ferromagnet in three or more dimensions. To describe the latter, suppose that S = Zd for some d ^ 3. (O is still given by (6.8).) If ß is large enough, ^ 0 (/?O) again contains two random perturbations nß+ and fit of the ground states a>+ and a>~. (This can be proved by a straightforward extension of Theorem (6.9).) But in addition to nß+ and pt, ^(/?0) also contains a family of Gibbs measures which are not shift-invariant. In the limit ß -> oo, these inhomogeneous Gibbs measures converge to the Dirac measures at those configurations which are equal to + 1 on some half space of Zd, and equal to — 1 on the other. Consequently, these Gibbs measures can be considered to describe the coexistence of a plus phase and a minus phase with a sharp interface. References and further results can be found in the Bibliographical Notes. We now turn to the proof of Theorem (6.9). We follow the device of Peierls (1936) which was outlined in the introduction to this chapter. We fix some rectangle (6.10)

A = S n ( [ M 1 , N 1 ] x [M2,JV2])

(Mi < N1,M2 < N2)in S, and we consider the set (6.11)

B = {{»,;} c S: \i -j\

= 1, {i,j} D A # 0}

of all nearest-neighbour bonds which emanate from sites in A. Each bond b = {i,j} e B should be visualized as a line segment between i and j . This line segment crosses a unique "dual" line segment between two nearest-neighbour sites u, v in the dual rectangle A* = { M 1 - i , M 1 + i . . . , N 1 + | } x { M 2 - i . . . , J V 2 + ± } . The associated set b* = {u, v} is called the dual bond of b. Hence b* = {ue A*: \u - (i +j)/2\ = 1/2}. We write (6.12)

B* 4 {b*: beB} = {{u,v} c A * : | u - u | = 1}

for the set of all dual bonds. A subset c of B* is called a circuit if c = {{uik~i\ um}: 1 ^ k ^ t] for some finite sequence (u <0) ,...,u i() ) with uv) = u(0), |{u(1), ...,u(e)}\ = t, and { u ^ V ' } e B* for 1 ^ k S S. \c\ = S is called the length of c. A circuit c is said to surround a site a e A if for each "path" (i <0) ,..., i(n))

102

Three examples of symmetry breaking

in S with i<0) = a, i(n) i A, and {i*"^1', i(m)} e 5 for all 1 ^ m ^ n there is some m with {i (m - 1) ,i (m) }* e c. We let Ca denote the set of all circuits in B* which surround a. (6.13) Lemma. For each a e A and ^ 2: 1 we have \{ceCa:\c\

S}\£S3<-1.

=

Proof. Each c e Ca of length ^ contains at least one of the £ dual bonds {a + (k — 1,0), a + (/c, 0)}* (k = 1,..., ^) which cross the horizontal half-axis from a to the right. If the remaining ( — 1 dual bonds in c are successively added then at each step there are at most three possible choices for attaching the next dual bond to the preceding one. D It was the key idea of Peierls to look at circuits which occur in a configuration as follows. For each œ e Q we let B*(co)={b*:b =

{i,j}eB,col*a>j}

denote the set of all dual bonds in B* which cross a bond between spins of opposite signs. A circuit c with c c B*(a>) is called a contour for co.

1

-

+ + +

—

-

+

—

+

1

+1

+ +

+

+ + + +

Q

-

— -

-

1

+ +

-

+

+

-

+

+

+

1

+

+1

+ + + + + + + + + + + - + - - +

+ +

+ +

+ -

+ + + + + + + + + + +

Figure 6.1 A contour surrounding a, as constructed in Lemma (6.14). There are two reasons for looking at contours. First, if œ is constant on S\A but takes a different value at some a e A then a is surrounded by a contour for œ. Secondly, there is an intimate connection between the Ising Hamiltonian H®(œ) and the set B*(co). This relation implies that the occurrence of a contour c requires an energy proportional to the length of c. Consequently, if ß is large then long contours will be very improbable with respect to y^"(- \co). These two facts will be established in the next two lemmas. They will imply

The Ising ferromagnet in two dimensions

103

the theorem as follows. If we fix the co+ boundary condition outside A then with overwhelming probability the minus spins in A form small islands in an ocean of plus spins. Therefore, taking the limit A f S we will obtain some Gibbs measure ßß+ which is arbitrarily close to ö+ when ß is large enough. Similarly, the co~ boundary condition will lead to some ßt e ^(/?(D) close to cL. As ö+ and c>_ are distinct, so are ßl and ni when ß is large. Hence |^(jßO)| > 1 when ß is large. Now we turn to the details. Our first lemma ensures the existence of contours. (6.14) Lemma. Suppose co e Q is such that co; = 1 for all i e S\A and toa = — 1 for some a e A. Then there exists a contour for co which surrounds a. Proof. Thinking of S as the vertex set of a simple graph with bond set {{i,j} <= S: \i —j\ = 1}, we let D denote the largest connected subset of {i e S: oii= —1} containing a, and we also let D' be the unique infinite connected component of S\D. We define c = {{i,j}*

eB*:ieD,jeD'}.

c is the "outer boundary" of D. (See Figure 1.) We have c a B*{to) because D is a maximal connected set of spins of equal sign and D' is disjoint from D. Moreover, for each path in S from a to a site in S\A the "bond of the last exit from D" defines an element of c. Thus we have c e Ca provided c is a circuit. Since D is connected, this will follow once we have shown that nc(u)^ \{b*ec:b*3u}\

= 2

for all u e [j b*. To see this we look at the four sites in the set Nu = b'ec

{u + ( ± i , ±j)} a S. Suppose first that nc{u) = 1. Using the fact that j e D' whenever i £ D' and {i,j}* e B*\c, we conclude that Nu c D'. But Nu n D # 0. Hence nc{u) # 1. To exclude the case nc(u) = 3 it is sufficient to note that 1D(0 changes its value an even number of times when i runs clockwise through Nu. Finally, let nc(u) = 4. Then the two sites u ± (£,£) belong to one of the two sets D and D', and the remaining two sites u ± (\, —%) belong to the other one. This is impossible because D and D' are connected. (A rigorous proof of this "obvious" impossibility can be based on the Jordan curve theorem; see Theorem V. 10.2 of Newman ( 1951 ), for example.) D The following contour estimate is the heart of the Peierls argument. (6.15) Lemma. Suppose c c B* is a circuit. Then yj?{c <= B*{-)\co) S for all ß > 0 and all œ e Q.

e~2m

104

Three examples of symmetry breaking

Proof. For each ( e Q w e have -H*(0=

I

Uj

{U}eB

= \B\-

X

(1-dCj)

{iJ}eB

= =

\B\-2\{{iJ}eB:ti*i;j}\ \B*\-2\B*(C)\.

We put ^ 1 = {CeQ:CSXA = co SXA , CC J5*(0} and A2 = {C e Q: ÇSXA = coSXA, c D B*(0 = 0}. We define a mapping xc: Q -> Q by - Ci if i is surrounded by c, ' Ci otherwise.

1

Thus xc flips all spins in the "interior" of c. For all {i,j} e B we have

(For if {i,j}* e c then precisely one of the sites i,j is surrounded by c, as can be shown by means of the Jordan curve theorem.) Thus J3*(T C £)AJ3*(Ç) = c. In particular, xc is a bijection from A2 to y4l5 and for £ e ^42 w e have tf*(0 - fl^(T£0 = 2|B*(OI - 2|J5*(T C C)| = -2|c|. Thus y£*(c c B*(-)\œ) ^

£

= I

exp(-/5H*(0)/ I cxp(-ßH%(TcQ)l

Ce-42

exp(-/Mf£(0) X exp(-/5Jff*(0)

/ CsX 2

= exp(-2/S|c|), as was to be shown,

a

Proof of Theorem (6.9). For ß > 0 we define r(ß) = 1 A £ / ( 3 e - 2 " / .

The Ising ferromagnet in two dimensions

105

Then r(ß) -> 0 as ß -> oo. The preceding three lemmas yield yfK=-l|û>+) ^ X

yHccß*(')|ft)+)

and thus yrto=-l|û>+)^'(A for all a e S, /? > 0 and each rectangle A in S. For each N ^ 1 we put AN = Sfl [-iV,iV] 2 and

vj?.+= l A ^ r

£ )C<(-iß>+)-

As ^(Q, J^) is compact, the sequence (v^ + ) N à l has a cluster point /i^. From Example (5.2(D)(1) we know that \i% e ^@{ß<^). The preceding estimate implies that nß+(aa = - 1 ) ^ r{ß) for all aeS. Consequently, if A e Sf and fe &A then \iï(f)-ô+(f)\^ni(\f-f(
^ K = - l )

^ 211/111A|r(A Hence lim ßl = ô+ and therefore /S-oo

limd(^@(0O>),<S+) = O. /J->oo

Finally, we put /i^ = x{ßß+). Since T preserves $ and commutes with each translation, ixt e ^@(/?<ï>); cf. Remark (5.10). As x maps local functions to local functions, lim ni = x ( lim fi{\

= T(C5+) = c5_

and thus limd(%(ß 0. Theorem (6.9) is thus proved,

a

106

Three examples of symmetry breaking

We conclude this section with a few remarks on the two-dimensional Ising antiferromagnet without external field. The corresponding potential 4* differs from the ferromagnetic Ising potential <1> by its sign: *F = — <1>. In the notation (5.3) we have *? = f(), where f: co -> ((— l)m+Bco(m>II))(m n ) e S is the spin flip on the odd sublattice of S. Let fi{ e y@(ß(b) be as above. By Proposition (5.6) and Remark (5.10), the measure v£ = r(ni) belongs to ^(ß^V). If ß is so large that ni{a0) > 0 then vÇ(a0) > 0 > VÇ( 1. Using the above-mentioned Aizenman-Higuchi identity
M+i0 ( O > 1 ) (v$) = f ( M + ^ ) .

Whilst all Gibbs measures for the Ising antiferromagnet on Z2 are still periodic, the example below exhibits a complete breaking of shift-invariance: There exist Gibbs measures which differ from all its translates.

6.3

Shlosman's random staircases

As in the preceding section, we let S = Z2 be the square lattice. But now we choose a different state space: We put E = Z, and we let k be counting measure. We consider the potential (6.16)

(ot-oj)2 0

HA = {i,j},\i-j\ otherwise.

= l,

(We note that 0 is equivalent to the Ising potential (6.8) when E = { — 1,1}. On the other hand, if E = R and X is Lebesgue measure then the potential
exp(-jSfl*(0) ^ Z I I exp(-/?(Ç ; - C.--(i.o))2)

^(Z £ e x p(-^ 2 )) I A ! < C 0 whenever ß > 0, co e Q, and A e y . Next we observe that has several symmetries. (6.17) Remark. Each of the following transformations of Q preserves Q). (i) The lattice translations 6h i e S; cf. Example (5.2)(1).

Shlosman's random staircases

107

(ii) The lattice reflections r^ and r2. These act on Q as {rlco)i = co(_iijJ2), (r2(o)i = co(ii _,-2). Here i = (i*i, i2) e S and coeQ. (iii) The lattice rotation r0 which is defined by (r0co); = co(i2 ^ ii); i = (iiJj) e S, co e Q. (iv) The spin reflection T: CO -> -co; cf. Example (5.2)(2). (v) The spin translation t which acts as (to) ; = co,- — 1, J e S, co e Q. o Knowing the results on the two-dimensional Ising model we might expect a breaking of the symmetries T and t. However, far more than this is true. We will show that at sufficiently low temperatures each of the O-symmetries listed above is broken. Let us consider the ground states of <£ first. So far we have not provided a general definition of a ground state. In some particular cases we used the term ground state for each configuration co which minimizes the potential. In the case under discussion, the class of all configurations which minimize each ®A consists of all constant configurations. However, there are also some non-constant configurations which will be seen to play an important rôle in the low temperature limit. These non-constant configurations are ground states of <£ in the following, more general, sense. (6.18) Definition. Let ¥ be a potential. A configuration co e Q is called a ground state of ¥ if

Hl(0 ^ Hl(co) whenever A e y and £ e Q are such that CS\A = œs\\Thus co is a ground state if the energy of each finite perturbation of co exceeds that of co. Clearly, if co minimizes each of the functions T x for some potential T ~ ¥ then co is a ground state of T. Let us return to the potential O at (6.16). For each z e Z we define a configuration coz by (6.19) cof = z Î! if i = (f J, i2) e S. (For z # 0, coz reminds of an infinite staircase.) Clearly, TCOZ = co~z. The configurations coz, z ^ 0, belong to distinct levels of the specific energy. Indeed, if AN = S fl [ - N, N] 2 then lim \AN\~lHl(œz)

= z2.

JV->oo

Nevertheless, all coz's are ground states of <£. (6.20) Remark. The configurations coz, z eZ, (and also their images under the symmetries r0 and t" (n e Z) of <£) are ground states of <£. Proof. We fix some z e Z and a rectangle A of the form (6.10). Let £ e Q be such that CS\A = œs\A but Ç # coz. Then

108

Three examples of symmetry breaking

H*(t)-H*(œ*)=

£

£

t=M, e=M-L-\

+ Z

(C(t.m) - C(k,o)2

Z

[(C(k+i,o - C(k,0)2 - z 2 ] .

The first term is strictly positive. To see that the second term is nonnegative we use the inequality s2 - z2 ^ 2z(s - z)

(se [R)

and observe that for each / ZJ

(C(k+iy> ~~ C(k,o ~

=

Z

k= M 1 - l =

z

)

[(C(k + l,0 ~~ ^fk + l/)) ~~ (C(k,0 — tt'fk/))]

(C(Afi+l,0 ~ ^ ( f y + l . o ) ~~ U>(M,-l,
= 0.

D

There are further ground states of 0, for example the configuration co with CO; = 1 for il ^ 0 and co; = 0 otherwise. The ground states t"œz and r0 o fco z (z, n e Z), however, are stable: In the low temperature limit ß -> oo, the set ^(ßO) is attracted by each of the Dirac measures at these staircase-shaped ground states. In other words, if ß is large then ^{ß(5>) contains a collection of random fields which are random perturbations of these staircase configurations. It is these Gibbs measures to which the title of this section refers. They were discovered by Shlosman (1983). (6.21) Theorem. Let E = Z, S = Z 2 , and suppose 0 is given by (6.16). / / ß is sufficiently large then there exists a family (/if)z6z of Gibbs measures in ^(/?) with the following properties. (i) For all i e S and z e Z, /if (CT,) = œz and ß^i = œï) > 1/2- In the low temperature limit ß -> oo, /if tends to the Dirac measure bz at a>z. (ii) For each z e Z, /if is invariant under the (^-symmetries ö ( 0 1 ) , t~z o ö ( 1 0 ) , rl o T, and r2 which preserve a>z. Moreover, i(/if) = ^(/if) = \itz and r0(ßo) = A«o(iii) The subset {t"(/if ), t" o r0(/if ): n, z e Z} of$(ß) is linearly independent. We emphasize the following consequence of the theorem above: If 0 ^ w ^ z # 0 and 0 < s < 1 then the Gibbs measure s^ + (1 — s)r0(n^) for /?$ is different from all its images under the O-symmetries 0,- (0 # i e S), t" (0 # n e Z), x, r 0 , r l5 and r2. The proof rests on a modification of the Peierls contour argument which has been used in the preceding section. We fix again a rectangle A of the form

Shlosman's random staircases

109

(6.10), and we consider the sets B and B* of bonds, resp. dual bonds, in A. These have been defined in (6.11) and (6.12). Let Ç e Q, z eZ, / c ^ l . Let us agree to call a circuit c in B* a (z, k)-contour for £ if £,- — cof ^ /c and £,- — co? < k whenever i,j e S are such that {i,j}* e c and i is surrounded by c whilst) is not surrounded by c. (6.22) Lemma. Let z e Z, /c ^ 1, and suppose Ç e Q and a e A are suc/i f/iaf Ca ~ œa = k and CS\A = Ö | \ A - ^ e n there ex/sfs some (z, k)-contour c for £ w/iic/i surrounds a. Proof. Same as Lemma (6.14).

D

Next we need an estimate for the probability of contours. For each circuit c in B* we consider the mapping tc: Q ->• Q defined by {Ci - 1 if i is surrounded by c, {tcÇ)i = < [Ci otherwise; i e S, C e Q. Thus fc equals the O-symmetry t in the interior of c and leaves all spins in the exterior of c unchanged.

(6.23)

(6.24) Lemma. Let z e Z, It ^ 1, £ e Q, and c be a (z, k)-contour for Ç. TTien

ifftO - fl*(te0 ^ |c|. Proo/. For each fceß with fc* ^ c we have <5fc(Ç) = <J>6(£CC j - TTius

H*(C) - H*{tc0

=

X

<5(&),

beB:b*ec

where <5(fe) = <5fc(Ç) — Q>b(tcÇ). For each fe e B with fe* e c we write fe = where i6 is surrounded by c and j f c is not. Then

{ib,jb},

ô(b) = (C„ - C J 2 - (Cib - Ch - l ) 2 = 2(Cib - CJb) - l. We split c into the three disjoint subsets T = {b*ec:jb

= ib + (1,0)},

-c = {b* e c:jb = i b - (1,0)}, and J = {b*ec:jb

= ib±

(0,1)}.

2

Each horizontal line in U through sites of S meets the polygon in IR2 through the sites in c an even number of times. This shows that \~c\ = |"c"|. If b* e c1 then

k-Ch

= (Clh-K)-(^-K)^k-(k-i)

=i

110

Three examples of symmetry breaking

and thus 0(b) ^ 1. If b* e ? then

and therefore ô(b) 2; 1 + 2z. Finally, for b* e ~c we have Ci. - CA = (k - O - (0. - < ) - 2 è 1 - 2 and thereby 0(b) ^ 1 — 2z. Hence V 5(6) ^ |c>| + |c"|(l + 2z) + \r\(l - 2z)

= |c'i + \r\ + \r\ = 14 This proves the lemma,

a

For /? > 0 we set

r(0) = 1 A 2 X A3e"YNote that r(/?) -> 0 as ß -> oo. (6.25) Lemma. For all ß > 0,z eZ,k^.l,

and aeAwe

have

y H k - - ««I = *l<»*) = r(j8)k. Proof. In the case r(/?) = 1 there is nothing to show. Thus we assume r(ß) < 1. From the t-invariance of O we obtain y f V „ - otf = -*!<»*) = yl*(-oa

- to~a*

=

/c|œz)

= yrK-cor^fe|co-z). Therefore we only need to prove that

yfV„ - col = fe|«z) ^ K/?)72 for all z e Z, /c ^ 1, and a e A. Lemma (6.22) asserts that the probability on the left is dominated by Z ys*(aa — œa ^ K c is a (z,/c)-contour|coz). ceC a

(Recall that Ca stands for the set of all circuits surrounding a.) For fixed c e Ca, tc is an injection from Al = {Ceil: Ca~co^k,c

is a (z, fe)-contour for Ç, CS\A = ^S\A}

into ^ 2 = {C e Q: Ca-co*^kThus Lemma (6.24) gives

1, CS\A = Û>|\A}.

Shlosman's random staircases

ZP(coz)yP(oa

- vza ^ fc, c is a (z, /c)-contour|coz)

= £

exp(-M(0)

^ £

exp(-#**(tcO-0|c|)

^e-W

111

£

exp(-ßH*(Q)

CeA2

Using Lemma (6.13) we arrive at the estimate tf>«

- (ol £ k\œ*)

g £ ^ 3 ' e - " r ô > . - « z ^ /c - l|coz) /gl

=

r{ß)yZ*(aa-*a>k-l\co*)/2.

Iterating this inequality we obtain the result. D Proof of Theorem (6.21). Let ß be so large that r(ß) < 1/2. We consider the squares AN = S D [ — N, AT]2, JV ^ 1. For each z e Z we define

vLHAjvr1 £ yC+;(-l«z). ieA„

Lemma (6.25) shows that

vkAWa-<\^k)Sr{ßf for all a e S and k ^ 1. In particular, lim s u p v £ z ( K | ^ / c ) = 0 fc->oo N i l

for all a e S. Thus Corollary (4.13) implies that the sequence (v$ z ) N è l has a cluster point /zf. Comment (4.18) shows that /zf e 0(/?O). Assertion (ii) follows immediately from Example (5.20) (1). To prove (i) we observe that HßA\oa-o)l\^k)^r{ßf forallaeSandfc ^ 1.Thus/zffloj) < oo and lim npz{f) = 5 z (/)forall/e JSf. /)->00

Since /xf is preserved by rx o T, /xf(
/lf((7ao(r"'o0(ai>O))o0(Oia2))

= ßt(Go + for all a e S.

za

l)

112

Three examples of symmetry breaking

Finally we turn to the proof of (iii). We shall prove the slightly stronger statement that it is impossible to find real numbers an z, bn z(n, z e Z) such that 0 < I ( K z | + |fc„, z |)
I [ < W V ) + Kj" o r 0 ( ^ ) ] = 0, and b„i0 = 0 for all neZ. (The last condition takes account of the fact that r o(^o) = ^o-) Suppose such numbers existed. Then n,z

because /zf (Q) = 1. Without loss we may assume that n,z

n,z

+

here a = a v 0 and a ~ = ( —a) + . We choose a number ô > 0 with 2(5 < 1 — 2r(/?) and an integer N > 1 such that

I

( K z | + |b„,z|)<«5.

|n|v|z|âlV

We also define the sets I± = {2Nz - n: \n\ v \z\ < N, ±a„ iZ > 0} U {4N2z - n: \n\ v \z\ < N, ±b„, z > 0}. First of all, we claim that I+ D /_ = 0. For suppose the contrary is true. Then there exist integers — N < m, w, n, z < N such that either 2N(z — w) = n — m and (m,w) ^ (n,z); or 4N2(z — w) = n — m and (m, w) ^ (n,z); or 4N2z = 2Nw — m + n and z # 0. Looking at absolute values we see that each of these three equation is impossible. We consider the sites i = (2N,4N2) and j = (4N2, — 2N), and we let — N < n, z < N be given. If an z > 0 then t\nl)(ai

e I+) £ fitfKa,

= 2Nz - n)

= n'M = cof) £ 1 - r(ß). Similarly, if b„z > 0 then f o r0(ßl)(ai e I+) £ t" o r 0 (^)((7, = 4N2z - n) = pfaj = co])^lOn the other hand, if a„ z < 0 then t"(nl)(at e I+) ^ f(ßf)(at = fi^i and for bn z < 0 we have

* 2Nz - n)

± oif) S r(ß),

r(ß).

Shlosman's random staircases

f o rMWi

113

e h) ^ tf(/) ^ r(ß).

Thus we obtain the contradiction 0 = 1 K^OxfXff,- e / + ) + fen,zf ° r0(/i£)(<7, e / + )]

^

Z

[ « z + C ) ( l - riß)) - (a~z + b-Mß)~] - «5

|n|v|z|<JV

^ (1 - <5)(1 - r(j8)) - r{ß) - Ô

^ 1 - 2r(ß) - 20 > 0. This completes the proof of Theorem (6.21).

a

To conclude this section we mention that Lemma (6.24) and therefore Theorem (6.21) can easily be extended to the class of potentials which arise if we replace the function x -> x2 in (6.16) by a strictly convex, even function on Z. Of course, the results of this section can also be extended to higher dimensions (by looking at closed hypersurfaces instead of circuits). Finally, we note that some further aspects of this model will be discussed below in (9.17) and (9.25).

Chapter 7 Extreme Gibbs measures

In Chapter 6 we have seen that ^(y) is not necessarily a singleton. This fact leads us to examine the question of the structure of ^(y) in general. By the very definition of ^(y), this problem is a special case of a more general question: What is the structure of the family of all probability measures that are preserved by a family of probability kernels? This question is a, by now, classical topic of ergodic theory, especially the ergodic theory of Markov processes. Therefore we may (and will) take advantage of some results and ideas that were developed in that context. The basic observation is that ^(y) is a convex set. Therefore our interest concentrates on the extreme elements of &(y), and the question above will be specified as follows. I) What are the special properties of the extreme elements of 'Siyp. II) Is ^(y) uniquely determined by its extreme elements? As for question I, we shall see that the extreme elements of ^(y) are characterized by a zero-one law for all y-invariant events. The latter are nothing but the tail events. Therefore the extreme elements of ^(y) may be interpreted as the possible thermodynamic phases of a physical system that is modelled by y. Each such phase can be approximately rediscovered in large but finite volumes with suitable external boundary conditions. All of this will be the subject of Section 7.1. Section 7.2 is devoted to four applications. In particular, we shall deal with exchangeable distributions and with the product of two specifications. In Section 7.3 we shall turn to question II. Under fairly general conditions we shall show that ^(y) is a simplex: Each element \i of &(y) can be uniquely decomposed into extreme elements of ^(y), in that ß is the barycenter of a unique probability measure wfl on the extreme boundary of ^(y). The construction of w^ is very similar to the construction of an entrance boundary of a Markov process. In the final Section 7.4 we shall apply this result to pairs y, y of specifications that are microscopic perturbations of each other. We shall show that then &(y) and ^(y) are isomorphic and macroscopically identical in some sense. From a logical point of view, Sections 7.2 and 7.4 will not be needed for the understanding of later chapters. On a first reading one should thus concentrate on Sections 7.1 and 7.3. A few results of this chapter will depend on the backward martingale convergence theorem. A reader who is not familiar with this theorem is referred to Bauer (1981), Corollary 60.9, or Breiman (1968), Theorem 5.24, or any other (non-elementary) introduction to probability theory.

Tail triviality and approximation

7.1

115

Tail triviality and approximation

We consider the general setting of Chapter 1. We are thus given a countably infinite product space (Q, &) = (£, ê f, and we look at the set ^(y) of all Gibbs measures for a given specification y. We start from the observation that ^(y) is convex, i.e., if \i, v £ <&(y) and 0 < s < 1 then s/z + (1 — s)v e ^(y). The most interesting elements of a convex set are its extreme points. (7.1) Definition. An element ß of a convex subset ^ of any real vector space is said to be extreme in <é if \i # sv + (1 — s)v' for all 0 < s < 1 and v, v' e <€ with v # v'. The set of all extreme elements of # will be denoted by ex # and is called the extreme boundary of (€. First of all we note that the extreme boundary of &(y) remains invariant under symmetries of y: This fact is a complement to Remark (5.10). (7.2) Remark. Let y be a specification and x e T be a transformation of Q. If ß e ex ^(y) then T(/I) £ ex (S(x(y)). In particular, each symmetry of y maps ex ^(y) onto itself. Proof. Suppose T(/I) = sv + (1 — s)v', where 0 < s < 1, v, v' e ^(f(y)), and v # v'. Applying t" 1 to this equation and using Remark (5.10) to conclude that t~1(v), T - 1 (V') e ^(y), we obtain a contradiction to the extremality of \i. a

The main objective of this section is to obtain a characterization of the extreme elements of ^(y) by a zero-one law for the tail events. This characterization rests on a general proposition which is well-known in ergodic theory. (7.3) Proposition. Let (Q, J^) be a measurable space, n a probability kernel from iF to 3F, and \i e ^(O, J^) with \m = \i. Then the system Sn(ß) =

{Ae
of all ß-almost surely n-invariant sets is a o-algebra, and for all measurable functions f: Q -*• [0, oo [ we have (fß)n = f\i if and only if fis J^n(ß)-measurable. Proof. 1) Jn(ß) obviously contains Q and is stable under the formation of complements and countable disjoint unions. In order to show that Jn{\i) is a cr-algebra we therefore only need to check that J^(/i) is stable under intersections. We take any two sets A, B e J^n(ß). Then n{Ar\B\-)£n{A\-)ATt{B\-)=lAA

1B = I ^ B

and MUrm -n(Ar\B\-))

= fi(Ar\B)-

ßn(A fl B) = 0.

/WI.S.

116

Extreme Gibbs measures

Thus A PI B e Jn(fi). 2) Suppose {f/j)n = f/j,, and let c > 0. We will show that {/ ^ c} e «/„(//). Writing g = l{ / ä c j, we have J dfifng = fi(fng) - fi(fg ng) {/
= (fuMg) - Kfg ng) = (M(g) - n(fg ng) = KfgiX - ng))Since fg ^ cgf and ng ^ 1, the last expression is at least cn(g(l - ng)) = cfin(g) - cfi(g ng) = c j

d/i 7r#.

{/
Thus j

d/x(/-c)»rg^0

and therefore \^f
/i-a.s.

As /i(g — 7tg) = 0,ng = g /i-almost surely. Hence {/ ^ c} e J^(/x). This proves that / is j^(/i)-measurable. 3) Conversely, suppose / is j^(/i)-measurable. We will show that (fn)n = fix. / i s the limit of an increasing sequence of ./„(/^-measurable step functions. Therefore it is sufficient to prove that (lAn)n = lAjx for all A e JK(n). We choose any B e 2F. Then (lAfi)n(B) = (lAfi)n(A f) B) + £fin(AnB)

+

= fi{ArtB) =

(lAn)n(B\A)

fi(lAn(Q\A\-)) +n(lAlau)

VAH)(B).

Similarly, (lAn)n(Q\B) ^ (l^)(Q\ß). This implies (\Ap)n(B) = {lAfi)(B) because (lAfi)n{B) + (lAfi)n(Q\B) = n(A) = (lAfi)(B) + The proof is thus complete.

(lAnM\B).

•

We will say a probability measure fi is trivial on a c-algebra se, or se is trivial relative to \i, if \i satisfies the zero-one law JX(A) = 0 or 1 for all Ae se. (7.4) Corollary. Let (Q,J^) be a measurable space, U a non-empty set of probability kernels from 3F to 3F, and 0>u = {n e &>(Q, &): fin = n for all % e n }

Tail triviality and approximation

117

the convex set of all H-invariant probability measures on (Q, #"). Let \i e 0>n be given and Ju{n) = (~) J*n{n) be the a-algebra of all ^-almost surely 7ten

U-invariant sets. Then fi is extreme in 0>n if and only if /i is trivial on

Proof. 1) Suppose there exists a set A e J'ni/J.) such that 0 < /i{A) < 1. Then we may look at the conditional probabilities V = H(-\A) = M

v = fi(-\a\A) = fn,

where / = lA/n(A) and / ' = ln\A/n(Q\A). Clearly, v # v' and /x = n(A)v + (1 — n(A))V. But Proposition (7.3) asserts that v, v' e ^ n because / and / ' are ./„(/^-measurable for all n e Yl. Thus /i is not extreme. 2) Conversely, suppose /i is trivial on J"n{n) and has a representation /i = sv + (1 — s)v' with 0 < s < 1 and v, v' e â?n. Then v is absolutely continuous with respect to \i. Thus v = fn for some measurable function / j ^ 0. From Proposition (7.3) we conclude that / is ^ n (/i)-measurable, and the /i-triviality of */n(/z) implies/ = n(f) = 1 /x-a.s.. Hence v = /i and also v' = \i. This shows that ß is extreme, D The reader will wonder if Corollary (7.4) remains true when Jn{n) is replaced by the c-algebra Sn = {Ae&:n(A\-)

= lA for all ne 11}

of all strictly Il-invariant sets. (The proof that Jn is a cr-algebra is similar to the argument below (1.19).) However, simple examples show that this is not necessarily the case. We include such an example as an aside. (7.5) Example. Let Q = {1,2,3} and n be the stochastic matrix / 1 1/2 \ 0

0 0 0

0 \ 1/2 1 /.

Then ^ Jt = { ( s , 0 , l - s ) : 0 ^ s g l } is the set of all rc-invariant probability vectors, and ex^„ = {(1,0,0), (0,0,1)}. On the other hand, a function / on Q satisfies nf = f if and only if 2/(2) = /(I) + /(3). Thus the cr-algebra of all strictly rc-invariant events is JK = {£1,0}. Consequently, each \i e ^ is trivial on JK. Thus ./„-triviality does not imply extremality. But if JX e S?n then Jn(n) is the power set of Q. Thus ./„(/^-triviality clearly implies extremality. o

118

Extreme Gibbs measures

Fortunately, in all cases in which we are interested J^uiß) may indeed be replaced by Jn because J*n(ß) turns out to be the /^-completion of Jn. For example, this holds when n is a specification. Recall definition (2.19) of the tail (j-field ST. (7.6) Remark. Suppose y is a specification. Then J>y = ST. If ß e &(y) = 0>y then ^y(n) is the /i-completion of ST. Proof. 1) If A G ST then yA(A | • ) = \A for all A e Sf because all yA's are proper. Conversely, i(AeSy then A = {yA(.4 | • ) — 1} e ^ for all A e ^ and therefore 2) Let ß G <%) and ,4 e J^(/i). Then

5= 0 Ae£f

U

{yAA\-)=l}

AcAey

belongs to ST, and /i(ylA5) = 0 because l„ = limsupl { y A W . ) = 1 } = limsupl x = l x

/i-a.s.. D

The preceding remark implies in particular that Jy{ß) is /i-trivial if and only if 3~ is /i-trivial. Consequently, Corollary (7.4) implies assertion (a) of the following theorem. (7.7) Theorem. Let y be a specification. Then the following conclusions hold. (a) A Gibbs measure ß e @(y) is extreme in &(y) if and only if ß is trivial on the tail a-field ST. (b) If /te @(y) and v e ^(Ci, 3?) is absolutely continuous with respect to ß then v e 1S(y) if and only if v = f\i for some ^-measurable function / 2ï 0. (c) Each ß G 'S (y) is uniquely determined (within ^(y)) by its restriction to the tail a-field ST. (d) Distinct extreme elements ß, v of @(y) are mutually singular on 3T, in that there is some A e ST with ß(A) = 1, v(A) = 0. Proof, (a) This has already been proved. (b) Since J*y{ß) is the /i-completion of ST, each ^.(/immeasurable function is /i-almost surely equal to a ^-measurable function. Thus v = f/j. for a 2Tmeasurable function if and only if v = fß for an Jy{/immeasurable function. Therefore the assertion follows from Proposition (7.3) and the Radon-Nikodym theorem. (c) Let fi, v e &(y) be such that ß = v on ST. Then ß = (/t + v)/2 e &(y), and assertion (b) implies that ß = fß and v = gß for ^-measurable functions /, g ^ 0. But ß = v = p on ST. Thus / = g = 1 jû-a.s. and therefore ß = v. (d) This is an immediate consequence of (a) and (c). D

Tail triviality and approximation

119

At this stage we should comment on the physical meaning of extreme Gibbs measures. (7.8) Comment. Suppose we are observing a well-defined state of a real physical system in equilibrium. We will find that the microscopic quantities are subject to rapid fluctuations, whereas the macroscopic quantities remain constant (on a "human" time scale and within the given bounds of accuracy). Suppose we want to describe the observed state by a mathematical object. Because of the microscopic fluctuations we are led to a probabilistic point of view: We shall try to describe the state by a probability measure \i. Of course, \x should be consistent with the observed empirical distributions of the microscopic variables. According to the basic principles of Statistical Mechanics, this can be achieved by assuming that /i is a Gibbs measure for a suitably chosen Gibbs specification y; cf. the Introduction. There is, however, a further requirement on \v. fi should be such that the macroscopic quantities are non-random. In Section 2.2 we argued that the macroscopic quantities are just the tail measurable functions. Consequently, the tail measurable functions should be constant /i-almost surely, and this means that \i should be trivial on ST. Theorem (7.7)(a) thus tells us that the system's state will be described by a suitable extreme element of ^(y). (According to assertion (7.7) (c), there exists but one extreme element of ^(y) which exhibits the right macroscopic behaviour). From this we conclude that only extreme Gibbs measures are suitable to describe an equilibrium state of a real system. In more catching terms we may say that a physical system will always pick an extreme Gibbs measure for its equilibrium state. For this reason, an extreme Gibbs measure is often called a phase. This somewhat vague term should not be confused with the physical concept of a pure phase. In fact, the stable coexistence of distinct pure phases in separated regions of space (such as in Figure la below) will also be represented by an extreme Gibbs measure. This can be seen quite nicely in the three-dimensional Ising model; cf. Dobrushin (1973a) and the survey below Theorem (6.9). It is a tempting misunderstanding to believe that the coexistence of two pure phases (which are considered to be described by two extreme Gibbs measures /i, and /i2) was described by a mixture like (/ix + /i 2 )/2. Such a mixture rather corresponds to an uncertainty about the true phase of the system, as will be explained in more detail at (7.27) below. The following pictures which are taken from Aizenman (1980b) may serve as illustration.

120

Extreme Gibbs measures

Figure 7.1a

An extreme fi,coexistence Figure 7.1b

(/zwater + fiice)/2

Let us return to mathematics. The next proposition gives a fairly standard characterization of probability measures with a trivial tail cr-field. Because of this characterization, such measures are sometimes said to have short-range correlations. (7.9) Proposition. For each /i e ^(Q, 3F} the following statements are equivalent. (i) p. is trivial on ST. (ii) For all cylinder events A (or, equivalently, for all A e 3F\ lim sup \n(A D B) - n(A)n(B)\ = 0. Proof. 1) Suppose \i is trivial on &~. Let A e #" be given. The backward martingale convergence theorem asserts that for each cofinal increasing sequence (A„)Bfel in £f H(A\
in the L (/i)-sense. As \i is trivial on 3~, ii(A\ST) = \i(A) /i-a.s.. Thus for each e > 0 there is some A e y such that fi(\fi(A\jrA) - fi(A)\) < e. For all A <= A e Sf we have sup \n(Af)B)-

fi(A)fi(B)\

BeiT.

^ sup \ dn(n(Am < H(\n(Am

- fi{A))

- n(A)\) < e.

This proves (ii). 2) Suppose assertion (ii) holds for all cylinder events A. Then /x(A D B) = H(A)n{B) whenever B e ST and A is a cylinder event. For fixed B e J , the

Tail triviality and approximation

121

system 3> of all A e 3F with \i{A fl B) = n(A)n(B) is a Dynkin system. That is, <2> satisfies the conditions (i) Q e 2; (ii) A2\A1 e 2 when Ax, A2e 3 and A1 cz A2; and (iii) (J„ ä i A„ e 3) when An e 2 (n ^ 1) are pairwise disjoint. Since 3) contains the cylinder events, 3 = #"; see Theorem 2.3 of Bauer (1981), for example. Hence B e § , and this implies that /i(B) = /i(5) 2 and thereby fi(B) = 0 or 1. Thus /i is trivial on 2T. D It is natural to ask if the extreme elements \i of ^(y) even satisfy the stronger regularity property (7.10)

lim &

sup

\n(A\B) - n{A)\ = §

Be^:ß(B)>0

for all cylinder events A. For example, if y is the non-quasilocal specification of Example (2.27) then (7.10) holds for all p. e ex^(y) because these are the Bernoulli measures. However, if y is Gibbsian for a uniformly convergent potential then (7.10) can only be satisfied for some \i e ^(y) when |^(y)| = 1. This follows from the next proposition. (7.11) Proposition. Let X e Ji(E, S) and y = pX_ be a quasilocal ^-specification. (a) If p is positive and (7.10) holds for some \i e &(y) then <&{y) — {//}. (b) Conversely, suppose {E,ê) is standard Borel and, for simplicity, all pA's are bounded and X is finite. If ^(y) = {/i} then ÏA(' \œ) -£ ^ uniformly in œ e Q, and p. satisfies (7.10). Proof, (a) Suppose \i e @(y) satisfies (7.10) and v e &(y) is arbitrary. We show that [i = v. Let A be a cylinder event and e > 0. Then there is some A e y such that H(A) -eSn(A\B)

= niB)-1 J dnyA(A\ •) S ft(A) + e B

for all B e STA with /i(B) > 0. This means that H(A) - e S JA(A I • ) ^ KA) + e

/i-a.s..

By hypothesis, yA(A\ •) e Sf. Thus there are some A e y and fe£fA with ||yA(yl|-) — f\\ S £• Consequently, \f — fi(A)\ ^ 2e /i-a.s.. But /i and v are equivalent on #"A because of Remark (1.28)(2). This implies \f — fi(A)\ ^ 2e v-a.s. and therefore \yA(A\ •) — fi{A)\ fï 3e v-a.s. and |vU)-/x(^)| = |vyA(^)-/x(^)|^3e.

We thus conclude that \i — v. (b) We need to show that lim sup |yA(yl|cü) - \i{A)\ = 0 Sf

med

122

Extreme Gibbs measures

for all cylinder events A. This will also imply that p satisfies (7.10) because n(A\B) = n(By1

idnyA{A\-) B

for all ß e ^ with p(B) > 0. Suppose the converse. Then there exist a cylinder set A, an e > 0, a cofinal sequence (A„) nèl in £f, and a sequence (co"^^ in Q such that \yAn(A\co")-p(A)\^e for all n ^ 1. In view of Theorem (4.12), the sequence (yA„(- la)"))näi has a cluster point v. Theorem (4.17) asserts that v e &(y). But v # /x because \v(A) — ^04) | ^ e. This contradicts our hypothesis &(y) = {p}. a As a by-product of the preceding discussion of property (7.10) we have obtained the following result: For some specifications y the uniqueness relation &(y) = {/"} implies that p is the local limit of the yA(- |eo)'s as A runs through y . We ask if a similar approximation theorem holds when |^(y)| > 1. In this case only the extreme elements of ^(y) can be shown to be limits of suitable yAn(-\co)% and it is impossible in general to specify the approximating boundary conditions a>. (7.12) Theorem. Let y be a specification, p e ex &(y), and (A„)„ >i an increasing cofinal sequence in £?. The following conclusions hold as n -» oo. (fl)

7AJ -» Kf)

A*-a.s.

/or a// bounded measurable functions f on Q. (b) / / £ is a compact metric space with Borel a-algebra S then 7A„( - |O>)-»^

weafc/y

for p-almost all œ e Q. (c) / / y is a ^-specification for some X e Jt(E, S) then for p-almost all œ e Q we have sup

|y A „(/M - p(f)\ -> 0

/ e i ? 4 : 11/11 g l

/or a// A e £f, and therefore

in the topology of local convergence. Proof, (a) By the very definition of a Gibbs measure, VAJ = Kf\#kn)

A*-a.s..

The backward martingale convergence theorem thus implies

Tail triviality and approximation

IKJ^MW)

123

A*-a.s..

Since 9~ is ^.-trivial, p.{f\3~) = p.(f) /i-a.s.. (b) As £ is a compact metric space, so is Q in the product topology, and J^ is the Borel cr-algebra. The space of all bounded continuous functions on D. contains a countable subset C0 which is dense with respect to sup-norm. Thus, for each <x>, yAn(- |co) converges weakly to p. if yAn(/|co) -> p{f) for all fe C0. But (a) asserts that the latter holds for /i-almost all co. (c) Because of Remark (1.28) (3) we may assume that X e 0>{E, $). Let p be such that y = pX_, and let A e y be given. For each n ^ 1 with A c A„ we consider the function pA = AAn\ApAn. We have, writing v = pJ.A, Pi = V(PA\^A„\A)

v-a.s..

This is because pA is ,^^-measurable, and V(PA/) = P-(/) = P A „ ( P A „ / )

= P-^A„\A(PAJ)

= v(p A /)

for each J^^-measurable function / . The backward martingale convergence theorem thus implies v a s

"--

PÂW^PAJI-^AA) We claim that V

(PA

Q ^\„\AJ = P A - J^(df/)/9AKf/s\A)

v-a.s..

For suppose / is measurable with respect to (~) 2TAn\A. Then for each Ç e EA «ai

the function co -> /(ÇcoS\A) is 5~-measurable and therefore /i-a.s. constant. Thus Fubini's theorem gives v(P A /) = J p(deo) J AA(dC)pA(CcoSXA)/(CcoSXA) = J AA(dC) J A*(df/)pA(C^s\A) I p(dco)/(CcoSXA) = jAA(dC)Jp(df/)pA(Cf/SXA)/(C^\A) = v(p A /). So far we have shown that pA -> pA v-a.s.. Now we recall that v = pXA, and we use the identities ^APA = ^A„PA„ = 1»

^APA = P^A(PA) = 1

to obtain that ^AGPA-PAD^O

P-a.s..

124

Extreme Gibbs measures

Indeed, Fatou's lemma implies 2 - J d/x lim sup XA(\p"A - pA\) n->oo

= J d/x liminf AA(pA + PA - IPÂ - PAI) «-•co

^ J d / U j l i m i n f (pA + pA - |pA - p A |)) = MA(2PA) = 2.

Assertion (c) now follows immediately because sup

|y A „(/M - /x(/)|

feä>A:\\f\\il

sup

I4^A„\A(PA„/|W)-MA(PA/)|

sup

|AA(PÂ/-PA/|W)|

^A(IPA-PAIM

for all to e Q. (By the way, choosing / = sign(pA(o-Acus\A) — pA) we see that the last inequality is an identity, and all four terms are nothing other than the total variation distance of the restrictions yAn(- \co)\^A and n\!FA to #"A.) a Let us note that Theorem (7.12) (a) provides us with a more explicit version of Theorem (7.7) (d). For distinct measures /x, v e ex % ) we choose any bounded measurable function / with /x(/) ^ v(/), and we consider the event A = { 7 A „ / " ^ ^ > /•*(/)} f° r some cofinal increasing sequence in S. Then Ae ST, H(A) = l,andv(^4) = 0. We note further that Theorem (7.12)(a) has a converse: If /i e g(y) is such that yAnf -» /x(/) /x-almost surely for all / e i£ then /x is extreme. For, in this case we have fi(f\&~) = /x(/) /x-almost surely for a l l / e =£?, and this means that /x is trivial on ST. Recalling the contents of this section the reader will notice that the product structure of (Q, #") was only used at a few places, namely Proposition (7.11) and Theorem (7.12)(c). Its use in Proposition (7.9) and Theorem (7.12)(b) was unessential. So we may state the following remark. (7.13) Remark. Theorem (7.7) and Theorem (7.12) (a) remain valid without change in the more general setting below. (Q, #") may be an arbitrary measurable space, y any countable index set which is directed upwards by a partial ordering "<=", 3~ the intersection of any decreasing family (2TA)A^y> ofsub-calgebras of 8F, y any family of proper probability kernels yA from STA to #", and &(y) = 0>1 the set of all y-invariant probabilities on (Q, #"). This framework also includes some other cases of probabilistic interest. One such case will be considered in Example (7.16). o

Some applications

7.2

125

Some applications

Before we continue with the general theory we will apply the preceding results to four examples. The first three of these are classical topics of probability theory. (7.14) Example. Kolmogorov's zero-one law. Let (E, S) be a measurable space, S a countably infinite set, (Q, &) = (E,é)s, and X e 0>{E,S\ Kolmogorov's zero-one law asserts that the product measure Xs is trivial on ST. This is a special case of Theorem (7.7) (a) because Xs is the unique element oi^(X_) and thus extreme in ^{X); cf. Remark (1.25). More generally, if /x = Lj <X; is the ieS

product of not necessarily identical probability measures a; on (E, S) then \i is trivial on ST. For \i is the unique element of "&(y), where y is defined by

yA/M=i(nA«i)(dc)/(ctoSXA). Here œ e Q, A e if, and / : Q -> IR is bounded and measurable,

o

(7.15) Example. Stationary Markov chains. Let £ be a finite set, S its power set, P a positive stochastic matrix on E, and pP the distribution of the stationary Markov chain with transition matrix P and parameter set Z. pP was defined at (3.3). From Theorem (3.5) we know, and we shall see again in (8.41) (3), that ßp is the unique element of "&(y), where y is defined in terms of P by (3.6) and Comment (3.8)(a). Thus ßP is trivial on 2T. In particular, \iP is trivial on the right-sided tail f] ^znin.œv This is a particular case of neZ

a well-known zero-one law of Blackwell and Freedman (1964). Moreover, each of the propositions (7.9) and (7.11)(b) implies the ergodic theorem (3.A3) for P. For let x,y e E and a be the unique probability vector on E with ccP = a. Then |P"(x,j;)-aO0| = « M " 1 IM^O = y->a-n = X)~ ßp{(T0 = y)ßp{0„n = X)| -> 0

as n -> oo

because {a^„ = x} e ^zn]-«,«[- For the case of a general state space E we refer to Theorem (10.34). o (7.16) Example. Exchangeability and the zero-one law of Hewitt and Savage. This example is a digression from the theory of Gibbs measures but fits into the more general setting of Remark (7.13). Let {E,S) be a measurable space and (Q, &) = (E,S)N. For each n :> 1 we let /„ c T denote the group of all transformations x of Q of the form x: œ -> (co t4 ;) ièl , where x^: M -> M is

126

Extreme Gibbs measures

a bijection with xj = i for all i > n. Clearly, \I„\ = n\, and I„ increases with n. The union I = \J I„ is the group of all permutations of finitely many coordinates. "- 1 We consider the set 0>l = ^ ( Q , &) of all /-invariant probability measures on (Q, &). Each \i e 0>l is called an exchangeable or symmetric distribution. &J admits a description in the spirit of Gibbs measures. For let JH = {Ae &: z~xA = A for all t e I„} denote the a-algebra of all /„-invariant events. The intersection J = f] J„ is nil

called the a-algebra of all symmetric events. We define a family (y„)näl of proper probability kernels from Jn to 3F by yn(A\a)) = -

£

ni

i£j„

lA(tœ)

(Ae^,œeQ,n^l).

It is then easily checked that 0>1 = {lie 0>(C1,&): nyn = \i for all n ^ 1}.

We are thus in the framework of Remark (7.13). In particular, ex ^ consists of all / j e ^ which are trivial on J. A well-known theorem on exchangeable distributions states that (7.17)

{XN:Xe0>{E,£)}.

ex^f =

Let us give a proof that fits into the ideas of Section 7.1. The key observation is that lim n-*ao

Mn/'off«)-fu(/'°ffi)

= lim

k

(n — kY »•

k

q>eM(k,n) i = l

= 0 for all k ^ 1 and bounded measurable functions fx,..., fk on E. Here M(k, n) is the set of all mappings from {l,...,/c} to {1,..., n} and Mt(k,n) the set of all injective mappings in M(k, n), and the last identity follows from the fact that \M,{k,n)\l\M{k,n)\ = (nl/(n - k)\)/nk -> 1 as n —> oo. Now if/i e ex ^ then Theorem (7.12)(a) in the extended version of Remark (7.13) implies

n[Y\fio(7i)

= limyn

(Ufio(ji

k

= n

iim

y n (yi° ff i)

i —1 n-»oo

Some applications ff

i) ^_a-s-

= if=ll P(fi ° for all k and fy,..., / j e ^ , w e have

fk and therefore \i = a^fi^.

»[lift0*

S) =

127

On the other hand, for each

tirnytt[Y\fioai Hm

= ifl= l

n-*oo

k

=n

Hm

Vnifi ° al) {

n

^-a-s->

- z /i ° ^

i = l n-^co ft j = l

and the last expression is ^"-measurable. This implies AM

Y\fioai

i= l

Mn/« o a, ^

/i-a.s.

i= l

because 2T a J. Consequently, the system 3 = {A e &: n(A\f)

= n(A\F)

/i-a.s.}

contains all cylinder sets of the form A = Ay x ••• x Ak x £ Nnik,co[ with/c ^ 1 and yl t , . . . , Ak e ê. As 3} is a Dynkin system, ^ = ÜF and therefore J = 2T /i-a.s.. Now let /i = AN for some X e 0>(E, ê). Then ji e ^ j . By Kolmogorov's zeroone law, ß is trivial on ST. In view of the above, \i is trivial on J. This is the zero-one law of Hewitt and Savage (1955). We conclude that \i e e x ^ j , thus completing the proof of (7.17). We shall return to exchangeable distributions at (7.31). o (7.18) Example. Product specifications. Let (£, S) be an arbitrary state space, S a countably infinite parameter set, Sy U S2 = S a partition of S into two nonempty disjoint subsets, (Qk, &k) = (£, k) is the Dirac measure at eok (k = 1,2). Routine arguments show that y is a specification, indeed. We have {n1 x pi1: nk e g(yk),k = 1,2} c 0(y) with strict inclusion when ^(y 1 )! > 1 and |^(y 2 )| > 1. For if//* e &(yk) then (A*1 x

A*2)?A

= A^Àns,

x

ß2ylns2

= M1 x fi2

128

Extreme Gibbs measures

for all A e y , and if v* e 9{yk)\{nk} then (fi1 x ft2 + v1 x v2)/2 belongs to ^(y) but is not a product measure. The precise relation between g(y) and the 0(y*)'s is (7.19)

ex^(y) = {/i1 x /i 2 : //* e ex»(y*),fc = 1,2}.

In particular, |ex»(y)| = |ex^(y 1 )||ex^(y 2 )|. The latter fact may be used to construct specifications with a prescribed number of phases. For instance, an iterated product of the Gibbsian specifications for the potentials in Section 6.1 has a power of two as its number of phases. We turn to the proof of (7.19). Let fxk e ex^(y k ) be given and n = /i 1 x /x2. To show that fi is extreme in <&{y) we check that fi is trivial on ST = D ^ S A A x ^S2\A- We take an arbitrary A e ST, and for co1 e Q.1 we let ^(co1) = {co2 e Ü.2: œ1 œ2 e A} denote the co1-section of A. It follows from the definition of product a-algebras that A{œl) e 2/~2 = f] ^i2\\ a n d the Aey

function co1 -> fj.2(A(co1)) is measurable with respect to ST1 = f] ^ A A Acy

As nk is trivial on 2Tk, /i2(yl(co1)) = 0 for /^-a.a. co1, or /^(^(co 1 )) = 1 for /^-a.a. co1. Hence H(A) = J /i1(doj1)/i2(yl(co1)) = 0 or 1. Conversely, let fi e ex^(y) be given. Then Theorem (7.12) (a) asserts that for all Ak e &k (k = 1,2) and /i-a.a. co = co1co2eQ HiA1 x A2) = lim yAn(Ax x yl2|co) n-*oo

= limyAnnSlM1|co1)y2„nS2M2|co2) n-*oo

= lim yAn(Ax x Q ^ t u ^ Q 1 x ^ 2 |co) «-•co

= ^(/l 1 x n 2 ) / ^ 1 x A2); here (A„) is any increasing cofinal sequence in y. Hence fi = fi1 x fi2 with /x* e ^(Q*,.^*), fc = 1, 2. We have /i 1 e ^(y 1 ), for if Ai <= S1 is finite and A1 e F 1 then Moreover, fi1 is extreme in ^(y1). For suppose fi1 = sv + (1 — s)v' with distinct v, v' e ^(y 1 ) and 0 < s < 1. Then fi = sv x fi2 + (I ~ s)v' x fi2 and v x fi2, v' x ^i2 E @(y) in contradiction to the extremality of fj.. Similarly, fi2 e ex ^(y2). The proof of (7.19) is thus complete. As a complement to equation (7.19) we note that 2T = 2Tl x 2T2 /^-almost surely for all n e &(y). Indeed, for any Ak e 3Fk it follows from the backward

Extreme decomposition

129

martingale convergence theorem and the product structure of y that ^(A1 x A2\£T) is /i-almost surely equal to a &~l x .^-measurable function. Since ^" 1 x 3~2 <=. 3~, we can conclude that \x(A\9~) = \x(A\3~^ x ST1) ^-almost surely for all A e !F, and the result follows, o

7.3

Extreme decomposition

In this section we shall deal with non-extreme Gibbs measures. We will show that each Gibbs measure admits a unique representation as a mixture of extreme Gibbs measures, at least when the state space (E, S) is standard Borel. Extreme decompositions are the subject of a well-known branch of convex analysis, the Choquet theory. Under suitable topological assumptions on E, y, and &(y) this theory also applies to the convex set &(y) of all Gibbs measures for a specification y. However, we prefer to follow a measuretheoretic approach due to Dynkin. This approach works without any topological assumption on fx(A) on 3P. We let #(0>) denote the smallest a-algebra on 3? such that all eAs are measurable. In other words, £(&") is generated by the sets {eA ^ c] with i e # and 0 g c ^ 1. e(âP) is called the evaluation a-algebra on 3P. We note three simple facts. (i) For each bounded measurable function / o n ( l the map ef: ix -> ix(f) on 3P is ^(^)-measurable. This is because / is the uniform limit of step functions. (ii) Suppose # <=. 3F generates 3* and is stable under finite intersections. Then s(0P) is already generated by the maps eA with A e cß. For let £
for all fieSP. We are going to represent probability measures /x e & in the form (7.20)

n = J v w(dv),

where w e 3P{3P0,£{%)) is a probability measure on a specified subset ^ 0 of

130

Extreme Gibbs measures

&. Equation (7.20) means that

Kf) = J v(/)w(dv) or, equivalently, ef(p) = w(ef) for all bounded measurable functions / on Q. Standard extension arguments show that this already holds whenever /i(A) = J v04)w(dv) for all A in a generator of F that is stable under finite intersections. Proposition (7.22) below will show that representations of the form (7.20) can be established by means of probability kernels with some particular properties. Let us introduce these kernels. (7.21) Definition. Let (Q, F) be a measurable space, sf a sub-er-algebra of &, and & a non-empty subset of &(Sl, F). A probability kernel it from (Q, s/) to (Q, F) is called a (&, j^)-kernel if it enjoys the following properties: (i) For all f j e ^ and A e # , (JL(A\S/) = JI(A\-)

/x-a.s..

(ii) Q.gs = {&> G Q : Ji(-\co) e ^ } 6 ^ . (iii) / x ( ß ^ ) = 1 for all /x G ^ . It will be convenient to write itw = TT(-\CO), w Ê fl. In fact, we can and will assume that Q.@> = Q, which strengthens (ii) and (iii). Otherwise, we take any v G & and define itw = v for co £ Çl&>. The usefulness of ( ^ , jz/)-kernels is demonstrated by the next proposition. (7.22) Proposition. Let (fl, F) be a measurable space, and suppose IF is countably generated. Consider a non-empty subset & of SPÏÇl, êF), a a-algebra i c ^ , and the subset 0>^ = {p e 0>: p(A) = 0 or 1 for all

Aesé]

of all measures in 0* that are trivial on se. Suppose there exists a (0>, sé)-kernel n. Then &^ # 0, and for each p. e 0> there is a unique w e ^(SP^, £{^^)) such that p = j" v vv(dv). w is given by w(M) = fi(n' e M), M e

s(^).

Proof. 1) We express SP^ in terms of n. Let <& be a countable generator of F that is stable under finite intersections. For A e <& and / j e ^ w e let vA{p) =

\{n-{A)-p{A)Yàp

= p(n-(A)2) - p(A)2 denote the variance of n'(A) with respect to p. (Note that p(n'(A)) = p{p(A\sé)) = p(A).) The mapping vA: 0> -» [0,1] is ^(^)-measurable because

Extreme decomposition

131

vA = eK{A)2 — eA. We will show that (7.23)

^/={Aie^:^(7r- =

Ai)=l}

= {ne&: vA{n) = 0 for all A e <#}. In particular, this will imply that 3P^ e s(SP). Let \i e SP be given. Then \i(A\s4) = 7i"(v4) /i-a.s. for allAeïF. Thus \i e ^ if and only if 7i'(i4) = fi(A) /i-a.s. for all ,4 e # \ As {,4 e #": 7t'(y4) = n(A) ^-a.s.} is a Dynkin system, the latter holds if and only if (7.24)

n-(A) = p(A)

ii-a.s.

foralMetf.

(7.24) is satisfied if and only if vA(fi) = 0 for all A e <%. On the other hand, (7.24) is equivalent to the statement /x{w = /x) = 1 because {71' = /1} = {7i"(^) = /i(,4) for all A e
= 0}

because of (7.23). Thus {it e SP^} e se, and we only need to prove that fi(vA(n') = 0) = 1 for all A e # and ^ e ^ . Since vA ^ 0, this will follow once we have shown that /i(vA(iï)) = 0. But fi(vA(n-)) = $ n(dœ)ln»(n:(A)2) - n»(A)2^ = \ d^{n{A)2\sé)

-

n^A)^

= |d/i[7r-(^)2-7r-(^)2]

= 0. 3) Let n e SP be given and w be defined as in the last sentence of the proposition. We first need to show that w is well-defined, i.e., that {71' e M} e se for all M e e(SP^). It is sufficient to check this in the special case when M = {v e SP^. v(A) ^ c} with A e #" and 0 ^ c ^ 1. But then we have {TT e M} = {TC- e ^ } 0 {%\A)

^c}estf.

Thus w is well-defined. Step 2) above ensures that w is a probability measure. By definition, j" (pdw = J (p(n')dfi whenever (? = 1M for some M e e{SP^). By standard arguments this identity extends to all bounded ^(^^-measurable functions cp on SP^. In particular, choosing

132

Extreme Gibbs measures

4) To show the uniqueness of w we take an arbitrary w e ^(^, ei^)) with p = J v w(dv). Let M e s{^) be given. From (7.23) we conclude that V(TT- e M) = lM(v) for all v e 0>^. Thus w(M) = \ lM(v)w(dv) = \ v(7i" e M)w(dv) = /x(7T* e M ) = w(M).

This completes the proof of the proposition,

n

We intend to apply the preceding proposition to the sets 0> = ^(y), y being a specification. In this case Theorem (7.7) (a) asserts that ex^(y) = ^ r , where 3~ is the tail a-field. Thus Proposition (7.22) will give us the extreme decomposition of Gibbs measures provided we are able to construct a (&(y), ST)kernel. This can be done when (E, S) is a standard Borel space. The crucial property of standard Borel spaces is the existence of countable cores, cf. (4. A9) and(4.All). From now on we assume again that (Q, #") = (E, . {_

n-co

"

J

Clearly, Q 0 e y. By the definition of a core, for each œ e Q 0 there is a unique na = n{-\co)eâP{Q.,^) such that n(A\m) = lim yAn(A\m) for all A e oo

œ e Q\Q 0 w e P u t Tcm = n(- \co) = v0 for some fixed v0 e ^(O, #"). The function n : !F x Q -> [0,1] thus defined is a probability kernel from 3~ to J^. This is because {A e 2F: n(A\ •) is ^-measurable} is a Dynkin system containing (€. 2) Next we let p. e &(y) be given. The backward martingale convergence theorem implies that p(A\9~) = lim p{A\^n) «-•oo

= lim yAn(A\ •)

p-a.s.

n->cc

for all A e (ê. Hence p{£l0) = 1 and p{A\2T) = n{A\ •) ^i-a.s. for all A e ^ and thus for all A e J^. 3) Finally we consider the set Q t = {n' e ^(y)}. Since Q

I = n AeSf

n OOAW=n{A)i Ae
Qj e 5T We need to prove that p{Qx ) = 1 for all p e &(y). This will follow once

Extreme decomposition

133

we have shown that n'yA(A) = n'(A) /i-almost surely for all A e Sf, A e (€, and jx e <&(y). But the last statement follows from Step 2) above because 7fyA(A) = fi(yA(A\ •)]
n(n(A\^)\^) /i-a.s..

The proof is thus complete, D In Chapter 14 we shall construct (0, ^-kernels for pairs (0, se) different from {@{y), 0~)- Here we shall apply the preceding two propositions to show that the set $(y) of Gibbs measures is a simplex, in that each [i e &(y) can be uniquely decomposed into extreme components. More precisely, ^(y) is isomorphic to the simplex of all probability measures on ex ^(y). (7.26) Theorem. Let (£,
The mapping a>: \i -> w„ is an affine bijection from @(y) onto 0(ex &(y), e{ex &(y))). In fact, u> is induced by a (&(y), 0~)-kernel n, in that wß is the image of n under the mapping œ -> n™. Proof We put 0 = <S(y) and se = 0~. Then Theorem (7.7)(a) asserts that 0^ = ex &(y). Proposition (7.25) ensures the existence of a (0, ^ - k e r n e l n. By Proposition (7.22), the mapping w. \i -> wu = n(co: nm e •) is an injection from <S(y) into 0(ex<$(y), ^(ex^(y))) such that wß is the unique probability measure on ex <&(y) that represents \i. It remains to show that «* is surjective. Let w e 0(ex <&{y), #(ex <S(y))) be given. Then \i = \ v w(dv) is a well-defined probability measure on (Q, #"). /z e @(y) because WA = JvyAw(dv) = /x for all A e y . By the uniqueness of the representing measure, wß = w. The proof is thus complete, D (7.27) Comment. Suppose we are given a Gibbsian specification y that serves as a model of a physical system. The state space (E, S) will then clearly be standard Borel. In (7.8) we have argued that a physical system will always choose an extreme element of <&(y) for its equilibrium state. However, an experimenter will not always be able to determine the chosen extreme element. It may be impossible to observe all relevant macroscopic quantities sufficiently accurately, and the observed data may suggest a macroscopic picture that varies in the course of a long experiment (which possibly includes repetitions). In this situation the experimenter will use his partial knowledge of the system's

134

Extreme Gibbs measures

macroscopic behaviour to guess the system's state. This guess will be represented by a (subjective or empirical) weight w on the set ex ^(y) of all possible phases, w will be chosen in such a way that the tail behaviour of the mixture ji =

j

v vv(dv)

ex <S(y)

fits the macroscopic data. The experimenter will then agree to say that /x describes the state of the system. Now Theorem (7.26) asserts that each fi e ^(y) is obtained in this way from a unique weight w. Consequently, each /x e <&(y) may be viewed as an experimenter's subjective description of the system's state. This description is characterized by a certain amount of information on the true macroscopic behaviour of the system. If no information is available, the associated weight w will be some sort of equidistribution on ex ^(y); cf. Figure lb above. In the opposite case of complete information, \x will be extreme and w = 6ß. o Theorem (7.26) will be complemented by three corollaries. The first of these will state that the bijection «* commutes with the action of each symmetry x of y. Let T e T be a symmetry of y. According to Remark (7.2), x may also be considered as a transformation of ex &(y) that maps each /xeex &(y) to x(fj.) = /io T" 1 e ex^(y). This transformation is easily seen to be measurable with respect to ^(ex^(y)). Thus for each w e ^(ex ^(y), ^(ex ^(y))) we may define its T-image T(w)

= w(v: T(V) e •) e ^(ex<S(y),^(ex9(y))).

(7.28) Corollary. Let (E, S) be standard Borel and y a specification with &(y) ^ 0. If x e T is a symmetry of y then wt(rt = x(wß) for all /x e ^(y). In particular, \x e @(y) is x-invariant if and only if wß is x-invariant. Proof For each n e ^(y) we have T{H) = j T(v)w„(dv) = j vT(w„)(dv). Thus x(wß) is equal to the unique wt(At) that represents T(^). D It is a further immediate consequence of Theorem (7.26) that @(y) has the same linear dimension as ^(ex^(y), ^(ex^(y))). In particular, we have the following corollary. (7.29) Corollary. Let (E, S) be a standard Borel space, y a specification, and N ^ 1 an integer. Then \ex@(y)\ ^ N if and only if ^(y) is at least (N — 1)dimensional, in that @(y) contains N linearly independent measures. Proof. 1) Suppose ex <&(y) contains N distinct elements fi1,..., fiN. By Theorem (7.7)(d), n1} ..., /xN are pairwise mutually singular. The latter implies the

Extreme decomposition

135

existence of sets Au...,ANe3F such that /J.„(A„) = 1 and Hs(A„) = 0 if t # n. This fact immediately shows that nu ..., nN are linearly independent. 2) Conversely, suppose |ex^(y)| = m < N. Then ^(ex^(y), ^(ex^y))) is isomorphic to the simplex spanned by the basis vectors of Mm. The bijection ^ in Theorem (7.26) is affine. Thus if ^(y) contains N linearly independent elements then so does Rm. The latter is impossible, D The next and final corollary will show in which sense the set ^(y) of all Gibbs measures may be considered as the infinite volume limit of the finite volume Gibbs distributions yA(-1 w), A e y, w e Q. We consider the if-topology on ^*(Q, IF). For a quasilocal specification y we let ^Um(y) denote the set of all /x e ^(Q, J^) that enjoy the following property: There exist nets (or, if you prefer, sequences) (AX)XED in y and (ö)J)iej) in Q such that Ka -$ S and yK(- |coa) -$ /x. Theorem (4.17) ensures that ^iim(y) c y(y)- If y is Gibbsian, each /x e ^Iim(y) is called a limiting Gibbs measure. (7.30) Corollary. Let (E, S) be standard Bor el, X e M(E, S), and suppose y is a quasilocal X-specification. Then &(y) is the closed convex hull of %m(y). Proof. 1) <&(y) is convex and, by Theorem (4.17), closed. Thus &(y) surely contains the closed convex hull of its subset %m(y). On the other hand, Theorem (7.12)(c) asserts that ex^(y) a %m(y). Hence we only need to show that ^(y) is equal to the closed convex hull of ex
consists of all probability measures of the form £ ajVj with v} e ex ^(y), a} ^ 0, Y, Oj = 1, m ^ 1. We will show that ex ex ^(y) is dense in ^(y). Let \i e ^(y) be given and U = jv e <S(y): max |v(^ k ) - fi(Ak)\ < el a neighbourhood of /x. Here Ax, ..., A„ are any cylinder events and e > 0. We need to show that U D ex ex ^(y) # 0. (In fact, we shall prove this when Ax, ..., A„ are arbitrary sets in #".) We choose an integer r > 1/e and split the range [0,1] of the evaluation maps eAi, ..., eAn into r disjoint intervals 1(1), ..., I(r) of length less than e. For each g = {gy,...,Q e {l,...,r}" we consider the set M , = H {veex^(y):v(X k )e/(4)}. k= l

The sets M, constitute a measurable partition of ex ^(y). If v, v' belong to

136

Extreme Gibbs measures

the same M( then \v(Ak) — V(Ak)\ < s for all k. For each { with Me ^ 0 we pick some ve e Me, and we define

Then v e ex ex ^(y). But also v e U because max \v(Ak) - n(Ak)\ = max £

j (e^ - ^k(v,))dwM

< Z w»(Me)£ = £e

This completes the proof of the corollary,

o

We conclude this section with the observation that Proposition (7.25) and thereby Theorem (7.26) carry over without change to the more general setting that was described in Remark (7.13), provided (Q, #") is standard Borel. As was shown in (7.16), this framework also includes the exchangeable distributions. We thus arrive at the following example. (7.31) Example. Let (E, S) be a standard Borel space, (Q, #") = (E, é)N and 0>t be the set of all exchangeable distributions on (Q,#"), cf. Example (7.16). We already know that 0>I consists of all probability measures that are preserved by a family of probability kernels satisfying the conditions of Remark (7.13). The argument for Proposition (7.25) therefore provides us with a (^7, Jf)-kernel, where J is the cr-algebra of all symmetric events. Thus Proposition (7.22) implies that each n e 0>1 is uniquely represented by a probability measure w^ on ex^j. By (7.17), the marginal projection v -> at(v) is a bijection from ex 0>1 to SP{E, ê). It is easily seen to be measurable. Therefore, letting mfl denote the image of wfi under this marginal projection, we end up with the following result: For each n e ^ there is a unique m^ e 0>(0>(E, S), e(0>(E, S))) such that H = Jl N m^(dl). This is the well-known theorem of de Finetti (1931) in the version of Dynkin (1953). o

7.4

Macroscopic equivalence of Gibbs simplices

In this section we shall consider specifications y and y which, in some sense, differ from each other only microscopically. We will show that in this case the simplices <&(y) and ^(y) are macroscopically identical, in that there exists an isomorphism between &(y) and ^(y) that preserves the probability of tail events. Let y, y be two specifications with the same parameter set S and state space (£, £). We will say y and y are equivalent, and write y « y, if there exists

Macroscopic equivalence of Gibbs simplices

137

a number 1 ^ c < oo such that c-lyA(A\-)^UA\-)^cyA(A\-) for all A e 3F and all A in a cofinal subset SP0 of SP. Clearly, we have y % y whenever for each A e SP there is a positive measurable function fA on Q such that yA = fAyA and sup ||log/ A || < GO. In Aey particular, we have the following example. (7.32) Example. Let X e Jt(E, ê) be an a priori measure. Suppose y = pk, and y = pi. are two ^-specifications which are such that c"1 pA :g pA ^ cpA for all A e SP and some 1 ^ c < oo. Then y % y. In particular, let $ and Ô be two A-admissible potentials. Then y® % y* provided that sup ||H* - H%\\ < oo. More generally, y* % y* whenever 5> is equivalent to a potential ¥ with sup \\H* - fl*|| < oo. This condition certainly holds when

Z 11^ - OJ < oo. (The last condition simply means that ¥ is a local perturbation of Q>. IfS = Zd and $ is shift-invariant then there is no shift-invariant ¥ # O that satisfies this condition.) o (7.33) Theorem. Suppose (E,S) is a standard Bor el space. Let y and y be two equivalent specifications. Then &(y) # 0if and only if &(y) # 0, and in this case there is an affine bijection p. «-> fi between @(y) and &(y) having the properties below: (i) p = fi on &~. (ii) fi = pn, where ft is any Ç&(y), 3T\kernel. (Hi) x(fi) = x(p) whenever T e T is a common symmetry of y and y. In particular, |ex^(y)| = |ex^(y)|. Proof. 1) Suppose there exists some p e &{y). Consider the net {pyA)Aey0. For each A e SP0 and A e ^ we have pyA(A) ^ cpyA(A) = cp(A). Thus the net {pyA)A£^0 is equicontinuous, in that lim lim sup pyA{Am) = 0 n->oo

A 6 y0

for every sequence {Am)mèl in !F with Am[0. A standard argument similar to

138

Extreme Gibbs measures

that in the proof of Proposition (4.9) thus shows that there exists a subnet (\)aeD a n d a measure p e 0>(ÇI, 3F) such that WA„(^) 1

ß(A)

for all A e !F, and therefore

mtSf) 1 ß(f) for all bounded measurable functions / on Q; cf. the argument in Remark (4.3)(2). From this we conclude that p e @(y). Indeed, let A e y and A e J*\ Then the consistency of y yields ßMA) = Hm nyAJA(A) D

= um iiy^(A) = p(A). D

In particular, we have ^(y) ^ 0, and Proposition (7.24) ensures the existence of a (^(y), ^ k e r n e l Ä. 2) Let \i e ex <&(y) be given and let p e &(y) be constructed as above. Then p(A) = lim nyK(A) ^ cn(A) D

for all A e 2F. In particular, if A e ST then either p(A) ^ cn(A) = 0 or /ï(Q\yl) S c[i(Q\A) = 0. Therefore p = \i on 2T and, by Theorem (7.7.)(a), peex @(y). As p is represented by 5^ and {it' = /i} e ^", ^(TJ- = p) = £(£• = £) = ^({/i}) = 1. Consequently, /i = /in. 3) Let ^ e ^(y) be arbitrary, and define p = \in. Then p e ^(y). This is because Step 2) implies that H(n- e 9(y)) = J v(£" e 3(y))w„(dv) = 1 and therefore WA = J V-{àoj)nmyA = J /i(d p = fift is injective. For if /^ = p2 then /i t = p2 o n ^" ar>d therefore fj.l = \i2 because of Theorem (7.7)(c). Finally, let T e T be a symmetry of y and y. Then the tail-measurability of nz('\ together with (i), implies that t ( / l ) = T(/X)7T = /I7T 1 ''' = / Ü r 1 ' ' ' = T(-JÛ)7T = t ( / ï ) .

This proves (iii).

Macroscopic equivalence of Gibbs simplices

139

4) It remains to show that the injection ß -> ßn is surjective. This is done by interchanging the rôles of y and y. Let n be a (@(y), ^ - k e r n e l and fi e ^(y). Then 3) shows that fin e ^(y) and fin = fi on ST. Thus (fin)ft = fin — fi. This completes the proof of the theorem. • The following example shows how the theorem above can be applied to one-dimensional systems. (7.34) Example. Let S = Z, E = { — 1,1}, X be counting measure, and (J„)„eZ a two-sided sequence of positive real numbers such that £ e~2Jn < oo. We consider the nearest-neighbour potential $

= {~J»a"a"+i { 0

üA = {n,n+\},neZ, otherwise.

We will show that |ex^(). In fact, this result still holds when the state space is arbitrary and Q> is any potential such that

I A e £?'• min A<^n<max A

for some n e Z.

o

11®J < °°

Chapter 8 Uniqueness

From Chapter 6 we know of some examples of specifications that admit several distinct Gibbs measures. We are thus led to the question of whether there is a class of specifications each of which admits only one Gibbs measure. In other words, we ask for conditions that imply the absence of phase transitions. We shall investigate two such conditions. The first condition (which is due to Dobrushin) can roughly be stated as follows: The total interaction of a given spin with all other spins should be so small that some crucial quantity is less than one. Obviously, such a condition is tailored for some kind of contraction argument. This contraction argument, together with a discussion of the circumstances under which Dobrushin's condition holds, will be the subject of Section 8.1. In fact, Dobrushin's contraction technique will yield more than uniqueness of the Gibbs measure. It will also allow us to estimate the distance, in some sense, of the unique Gibbs measure from other probability measures. In Section 8.2 we shall use these estimates to obtain further information on the unique Gibbs measure fi: We shall construct consistent regular versions of the conditional probabilities \i{-\3Ty) for arbitrary, not necessarily finite regions V, and we shall also derive some upper bounds on the mixing coefficients of /i which describe the decay of correlations between distant regions. A second uniqueness condition will be studied in Section 8.3. This condition only applies to one-dimensional systems. It states that the interaction between very distant spins should decrease so rapidly that the total interaction energy of the spins on any two complementary half-lines is finite. As a matter of fact, we shall see that any two Gibbs measures for such an interaction are necessarily absolutely continuous with respect to each other. By the zero-one law for extreme Gibbs measures, this property already implies the uniqueness of the Gibbs measure.

8.1

Dobrushin's condition of weak dependence

In this and the subsequent section we shall exploit a technique that was invented by Dobrushin (1968a). We shall analyse a specification y by looking at an 5 x S-matrix C(y) = (Cij(y))ljES that describes the interdependence between the spins at distinct sites. More specifically, Cy(y) will be an estimate

Dobrushin's condition of weak dependence

141

of how much the conditional distribution yt(- \co) of ai depends on the value oij of the spin at j . To define C{y) we first need to introduce a distance between probability measures on the state space E. We shall simply take the uniform distance which is nothing other than one half of the total variation distance. This choice will be convenient for our purposes. Let (E, S) be a measurable space and a t , a 2 e ^{E, ê). We define a distance ||œ1 — a2|| of al and a 2 by three identical expressions, namely (8.1)

K-aall^maxM^-MA)! = max|ai(/)-a2(/)|/<5(/) s

= a(l0i -02D/2. Here max extends over all bounded (non-constant) measurable functions / on E. f (8.2)

0(f) 4 sup |/(x) - /OOI = sup /(x) - inf f(x) x,y€E

xe E

XG E

is the oscillation of such an /, a e Jt{E, S) is an arbitrary measure with a.l « a. and cc2 « a (such as a = a t + a 2 , e.g.), and gx and g2 are the a-densities of ax resp. a 2 . It may be worth mentioning a special case. If E is countable (and
(8.3)

K-a2|=Jl|ai({x})-«2(W)|.

(Just let a be counting measure in (8.1).) Let us check that the expressions on the right of (8.1) are equal. For each bounded measurable / we have, putting m = I sup f(x) + inf f(x) I / 2, \XG E

\*1(f)-«2{f)\

=

x eE

J j

\a((g1-g2)(f-m))\

^«(101-021)11/-«Il = a(l0i-02IW)/2, with equality when f — m = sign(ö'1 — g2) \\f — m\\ on {gt # g2}. In particular, equality holds w h e n / = l{ 9l>92 }- The equality of the three expressions on the right of (8.1) now follows immediately. After these preliminaries we turn to the definition of Dobrushin's interaction matrix C(y). Let y be a specification with state space E and an arbitrary parameter set S. In fact we shall only look at the kernels yt = y^ (i e S) that correspond to the singletons in SP. To some extent this procedure is suggested by Theorem (1.33). Let ie S be fixed. As yt is proper, yt is uniquely determined by its ar projection y?(- \co) = <7;(y;(- \co)), coeQ.. More explicitely, y° is defined by

142

(8.4)

Uniqueness

y?(A\œ) = yi({oieA}\œ)

(Ae^œeQ).

Thus y? is a probability kernel from (Q, ^ l } ) to (E, S). We are interested in the cOj-dependence of y°(- \co) for each j e S. This co^-dependence may be estimated by the quantity

(8.5)

cu(y)=

sup

\\y?(-\0-y?(-m\.

The matrix C(y) = (Q/y)),- J s S is called Dobrushin's interdependence matrix for y. It is the basic object of this section. We intend to introduce a condition on a specification y which implies that ^(y) contains but one element. For intuitive reasons, <$(y) should be a singleton whenever y is close to an independent specification. In other words, the interdependencies described by y should be weak, in that yf (• |co) depends on co only weakly. We therefore look for a quantity which estimates the dependence of yf(- |co) on all of co. Having defined C(y) it is natural to think of the sum YJ Qj(y) a s s u c n a quantity. There is, however, an important fact which must be remembered: This sum tells us nothing about the dependence of y°(• |co) on the behaviour of co at infinity. For example, if y is the specification of Example (2.27) then C^y) = 0 for all i, j e S, but yf (• |co) does depend on co and ^(y) is not a singleton. Therefore, if £ C^y) is taken as an estimate JsS

of the co-dependence of y?(- |co), a tail dependence of yf must be excluded. This can be done by assuming that y is quasilocal. We are thus led to introduce the following condition of weak dependence. (8.6) Definition. Let y be a specification, y will be said to satisfy Dobrushin's condition if y is quasilocal and c(y) * sup X Cy(y) < 1. ieS

jeS

We are now ready to state Dobrushin's uniqueness theorem. (8.7) Theorem. Suppose y satisfies Dobrushin's condition. Then \^(y)\ ^ 1. If (E, S) is a standard Bor el space then |^(y)| = 1. The above theorem is an immediate consequence of a more general comparison theorem which will be stated and proved at (8.20) below. Before going on to that subject we will provide a sufficient condition for Dobrushin's condition to hold, and we will also look at some examples. In analogy with (8.2), we let

s(f) = sup i/(o - m\ denote the oscillation of a real function / on Q.

Dobrushin's condition of weak dependence

143

(8.8) Proposition. Let X e <M(E, S) and Q> be a À-admissible potential such that SUPYJ(\A\-1)S(<S>A)<2ieS

A3Î

Then y* satisfies Dobrushin's condition. (Note that, up to ^-admissibility, there are no restrictions on the selfpotential part of <£.) Proof. First of all, each H* is quasilocal. For if A e y then

sup

\H*(0~H%(r,)\^Y<

I

<W,

and the expression on the right tends to zero as A runs through y . Thus Proposition (2.24)(b) guarantees that y*" is quasilocal. To check that c(y*) < 1 we need to estimate Cy(y*) for i, j e S, i ^ j . Let Ç, n e Q with Cs\{j} = *ls\{j} b e given. We put "oW = -Hfi}(xCs\{i}l

«iW = -^}(^s\{i})

(* e £)

and v = u1 — u 0 . v is bounded. More specifically, we have <5(«) ^ sup X l«M*Cs\{i}) - ^(Xf7s\{;}) x,}> e £ A3 i

- <MKs\{i}) + <M)»fe\{i})l

^2

X ,5(a>j.

Next, for each 0 ^ t ^ 1 we write ut = tUi + (1 — t)u0 = u0 + t«;, /i, = exp ut/A(exp ut), Xt = Ji,A. Then X0 = ot(yf(-\Ç)) and At = afyf(• |rç)). We thus need to estimate ||A0 — Aj_||. By (8.1) and Fubini's theorem, 2||A0 — Ax|| = A ( | Ä 1 - Ä 0 | ) ^ J d t A 0

at K

On the other hand, the dominated convergence theorem implies that — A(exput) = X(v expu t ). at d Thus —ht = (v — Xt(v))ht and therefore 2||A0 — Ax|| ^ J d t A . d « - ^ « ) ! ) o

144

Uniqueness

s)àtXt{(v-Xt(v)fY12 o

^ J dÜ,((D - m)2)112 o

for each me U. Choosing m = I sup v(x) + inf v(x) J / 2 \xeE

xeE

) \

we obtain 2||A0 — At|| ^ ||v — m\\ = S(v)/2. Inserting our previous estimate of ô(v) and taking the sup over (, r\ we find that Cu{y*)^

£

ô(d>A).

Consequently,

jeS

£ j*i

A=>{i,j)

Our hypothesis on O thus implies that cfy*) < 1.

•

It is interesting to note that the constant 2 in Proposition (8.8) is sharp. Indeed, we shall see at (16.27) that for every e > 0 there is a pair potential O with state space { — 1,1} such that sup £ (\A\ - 1)0(®A) < 2 + e ieS

A3i

and |^(0)| > 1 and thus c(y9) ^ 1. In some special cases, however, the estimates in the preceding proof can be improved. We consider two examples. (8.9) Examples. (1) Lattice gas models. Let E = {0,1} and X be counting measure. Suppose
[0

otherwise.

Clearly, ô(
Asi

Dobrushin's condition of weak dependence

145

Thus, compared with the condition in Proposition (8.8) we have an additional factor of 2. This factor can be picked up by improving the estimate of ô(v) in the preceding proof. Let us assume that £,- = 1, rjj = 0. Then v(0) = 0 and

.(1) =

X

K(A).

Thus <5(»)g

I

\K(A)\.

Inserting this estimate into the preceding proof we obtain the result. (2) Spin models. Let £ = { — 1,1} and suppose $ is a spin potential in the sense of Example (2.38). Thus there is a function J: y -> M with 0>A = — J(A)<jA for all AeST. Here aA = \\ at. Plainly, ô{d>A) = 2\J{A)\. We will show that y"> satisfies Dobrushin's condition whenever (8.10)

sup X (|^| - l)tanh|J(^)| < 1. ieS

A3i

This condition improves Proposition (8.8) in that \J(A)\ is replaced by the smaller quantity tanh \J{A)\. This can be achieved by an ad hoc estimate of Cij{y
2\\x0-x,\\

= X IMx)-Äo(*)l = 2 | M D - M D I x=±l

= Itanhu^l) — tanhu 0 (l)| because ut(— 1) = — u,(l) and thus 2/i((l) = expu t (l)/coshu t (l) = tanhu t (l) + 1. Next we use the inequality Itanhb - tanha| < 2tanh(|b - a\/2). (For its proof it is sufficient to note that tanh is increasing, odd, and concave on [0, oo [. Indeed, let a < b. If a > 0 then tanh b — tanh a ^ tanh (b — a) — tanh 0. Thus we only need to consider the case a ^ 0. For similar reasons we can assume that b ^ 0. But then the concavity yields tanhb + tanh ( - a ) ^ 2 tanh (b - a)/2, and the claimed inequality follows.) We obtain P o - l J gtanh|ü(l)/2| ^tanh

£

^

tanh|J04)|

X A^{i,j]

\J(A)\

146

Uniqueness

and thus I

C y (y*)^ X

jeS

(\A\-l)Uinh\J(A)\.

A3i

Thus (8.10) implies c(y*) < 1. It is worth looking at an even more specific case. Let S = Zd and suppose [0, oo[ with J(0) = 0 such that <&A = J(i —j)^ when A = {i,j} and <&A = 0 otherwise. Then (8.10) reduces to the condition (8.11)

£tanhJ(i)
This in turn holds when (8.12)

£ J(i) S 1.

We note that (8.12) is fairly close to (8.11) if J takes many positive but small values. Two further comments are in order. (i) In general, the number 1 that appears in condition (8.12) cannot be replaced by a larger constant. This will follow from Theorem (16.27) below. On the other hand, it is obvious that (8.11) (and all the more (8.12)) may be far from being optimal in specific cases. This holds in particular when d = 1 and J{i) = J if |i\ = 1 and J(i) = 0 otherwise. Then (8.11) reads tanh J < 1/2, but we know from Section 3.2 that the Gibbs measure is unique for every J. (ii) Condition (8.12) coincides with the condition for the absence of spontaneous magnetization that is obtained in the so-called mean field approximation. In this approximation all spins except one, say
(The factor of
\

i#0

//

xeE

\

i#0

/

= tanh I s YJ J(i) ) • By consistency the effective spin s should be chosen in such a way that m{s) = s. In other words, in the mean field approximation the absence of spontaneous magnetization just means that the equation m(s) = s does not admit a non-zero solution. Of course, s = 0 is always a solution. As tanh is concave on [0, oo [, 0 is the only solution if and only if

Dobrushin's condition of weak dependence

147

_ d X J(i) = -rm(s) s = 0 ^ 1 ,

ieS

OS

in agreement with (8.12).

o

Proposition (8.8) and the examples above can be summarized by saying that Dobrushin's condition holds whenever the interaction is small enough. However, the validity of Dobrushin's condition is not limited to this case, as we will show now in three further examples. (8.13) Examples. (1) Large external fields. Let E = { — 1,1} and O be any potential such that sup exp f i

£

ieS

«5(0^)] ^ (\A\ - 1)0{<S>A).

Asi:\A\>l

A3i

The Gibbs specification y® then satisfies Dobrushin's condition. To check this we can assume that h > 0. (Otherwise we interchange the rôles of 1 and — 1.) We then proceed just as in the proof of Proposition (8.8). In the notation of this proof we have, putting m = v(l), 2U0-^\\^)dtXt((v-m)2yi2 o g sup U(v - v(l))2)112 Ogtgl

supht{-l)1>2\v(-l)-v(l)\

= t

= sup(l + exp[u,(l) - u f (-l)])- 1 ' 2 5(i;). t

Moreover, ut(l)-ut(-l)^2h-

£

Ô(OA).

A=>i:\A\>l

Combining these inequalities with our earlier estimate of ô(v) we obtain the result. (2) Low temperatures. Suppose O is any potential such that a^supX ieS

(\A\-l)ô(®A)

A3i

is finite, and consider the Gibbsian specification ypo for $ at inverse temperature ß. By Proposition (8.8), yßo satisfies Dobrushins's condition whenever ß is small enough, i.e., when the temperature is high. We will now show that Dobrushin's condition can also hold at low temperatures, provided O admits but one ground state. Let E be finite and X counting measure. In addition to the hypothesis a0 < oo, we assume that, for some (necessarily unique) co e Q,

148

Uniqueness

b9 4 inf

inf

lHfi}(0 - Hf^œ^K

> 0.

In particular, there exists at most one ground state for O, namely a>. (The assumption b® > 0 certainly holds when E and O are as in the preceding example, with an external field h which satisfies the condition |fc|>supi

X

<5(OJ.)

Under these hypotheses, yß
ilill ^

«K-)) 2 ) 1/2

SUp J,((l> 0<(<1

exp[u t (x)-u ( ((U;)]) 1/2

z

^ <5(«)(

xe.E\{eOij

^ô(v)(\E\

-l^expC-j?^]

and thus c ( 0 ^ « | £ | - l^expE-/^]. Hence c(y^">) < 1 when ß is large enough. (3) Potts antiferromagnet for large JV. Let E = {1,..., JV} and X be counting measure. Also, let (S,B) be a simple graph as defined in (2.17). Consider the potential \J5{oi,oj) if A = {i,j}eB, ®A = 10 otherwise. Here J > 0 is any constant, and S{x,y) =

'1 if x = y, 0 otherwise

is Kronecker's function. (In the case JV = 2, the above <5 equivalent to the antiferromagnetic Ising potential.) For each i e S w e let di = {j e S: {i,j} e B} be the set of all neighbours of i. Suppose that JV > 3|<3i| for all i e S. Then c(y°) < 1, independently of the choice of J. Indeed, let i e S and j e di be given. Using the notation of the proof of Proposition (8.8) we can write

Uo - IJ = Wi

- h\)/2

^/l(|e"' — e"°|)/2(eUl)

zUnV Z xe E

exp -J

Z <*(*.£*) ke3i

— exp - J £ <5(x,rçk) tea«

Dobrushin's condition of weak dependence

S Zftfa)- 1 ^ 2(N -

£

l ex P l-JSixXj)-]

- exp

149

1-mx^yW

\di\y\

In the last step we have used that Z^(fy) ^ | {x e E: x # nk for all k e di} | ^ N - |<9i|. We thus obtain that Cy(y*) ^ 2(iV - [<9ff)"1 whenever) e di. Since Cy(y*) = 0 when) <£ 5i, we conclude that c(y*) ^ sup 2\di\(N - |ôi|) _1 < 1. As a final remark, we note that the preceding estimate of c(y°) can be improved by a factor of 1/2 if J = oo. This shows that c(y*) < 1 whenever N > 2\di\ for all i e S and J is sufficiently large, o After these examples we turn to the proof of Theorem (8.7). Without additional effort we may consider a more general problem, the comparison of Gibbs measures for two (possibly distinct) specifications. It is worthwhile doing so, as will turn out in the next section. We need to develop a concept of comparison of two probability measures H,fion (fi, &). In contrast to the macroscopic equivalence which was considered in Section 7.4 we will now look at the microscopic (that is, local) behaviour of ß andfi.To state our criterion of local comparison we first need to introduce the oscillations at single sites. Let / e <£ and j e S be given. The oscillation off at j is defined by

(8.14)

00)=

sup

|/(0 -Ml

Clearly, 00) = sup

S(fjia),

caefi

where fJsX{j}) {x e E, œ efi).We also note that (8.15)

ô(f) S £ ôj{f)

for all fe&. This follows from (2.22) because

i/(o - m \ ^ i/(o - /(CvffcvOi + z ôj(f) for all C,)|eQ and A e y . We are now ready to introduce the concept of local comparison. Let /i, fi e ^>(fi, 3F) be given. We shall say a vector a = (a,-),-eS e [0, oo[ s is an estimate for fi andfiif

(8.16)

\ß(f)-fi{f)\=Y.^¥f) je S

150

Uniqueness

for all / G !£. Note that in this case a;- is an upper bound of o)(M - Oj(p)\\ More generally, if A e 5^ then £ a;- is an upper bound of jeA

\\ß-fi\\A±

sup

\ß(f)-fi(f)\/o(f).

Let us state some further simple facts. (8.17) Remarks. (1) The constant vector a = 1 is always an estimate. This follows from (8.15) because the left-hand side of (8.16) is at most 5(f). (2) Suppse a is such that (8.16) holds for all f e ££ only. Then a is an estimate. For l e t / e J ? . Define fA e <£ by fA = /(
IM/) - ß(f)\ = Hm IMA) - ß(A)\ ^ lim sup £ Aey

ajôj(fA)

jeS

je S

(3) A coordinatewise limit of a sequence of estimates for ß and /i is also an estimate for ß and fi. This follows from the preceding remark. For if / e S£ then the sum in (8.16) contains only finitely many non-zero terms, o Now we fix two specifications y and y, and we let \i e <£(y) and fi e ^(y) be given. We look for an estimate for ß and fi in terms of y and y. To this end we assume that y is quasilocal, and for each i e S we let bt: Q -> [0, oo[ be a measurable function such that \\y?(-\)\\£bt(œ) for all co G Q. Then the following key result holds. (8.18) Lemma. Consider the situation described above, and suppose a is an estimate for ß and fi. Define a vector a by ä = C(y)a + fi(b). That is, the i'th coordinate of a is «; = Z cij(y)aj + £(&;)• je S

Then ä is also an estimate for ß and fi. Proof. For each finite A c S w e define a vector aA by . [at A a, a-1 = s [üi

if i e A otherwise.

Dobrushin's condition of weak dependence

151

We show by induction on |A| that each aA is an estimate. The lemma then follows from Remark (8.17)(3). By hypothesis, a9 is an estimate. So we assume that aA is an estimate, and we take any i e S\A and put A = A U {i}. We will show that aA is an estimate. Let / e J*? be given. Then IM/) - ß(f)\ S \K7if) - ß(ytf)\ + \ß(ytf -

yJ)\

because n e &(y) and ß e ^(y). We estimate each term on the right separately. The second term is at most /ï(b ; )^(/). This is because for each a> \7ifW - yJH\

= \y?{fUa\a>) -

tft/UMI

£\\y?{-\o>)-y?{-\a>)\Wiia) S bt(œ)ôt(f); the notation fi(0 was introduced after (8.14). To estimate the first term we use the hypotheses that yj e !£ and aA is an estimate. These hypotheses imply jeS

We are thus led to look at àjiyif). Clearly, àfyif) = 0. So we fix any j # i. For C i e f i with Cs\{j} = Vs\{j) we have

\yt(f\0-yt(m\ S \y?{f,,i-fiJQ\ ^Wfi,,-fiJ ^ W)

+

+ \y?(fi,,\Q +

y?(fi,,m

\\y?(-\0-yX\ri)\\Hfi,n)

CijiyMf).

Consequently, W)

S ôj(f) +

C^yMf).

Combining all preceding inequalities we obtain \fi(f) - ß(f)\ S I afWf)

+ CyftM/)] +

fifaMf)

because aA ^ a,- and Cu(y) = 0. Since aA is an estimate, we may replace at by ïït A a;. This shows that aA is an estimate, and the proof is complete. • The preceding lemma readily implies the comparison theorem belows For a given specification y and any « ^ O w e let cn(y) = (q(y)) i J e S denote the n'th power of the interaction matrix C(y). In particular, C°(y) is the identity matrix. We put

152

(8.19)

Uniqueness

D(y) = (Dil(y))lJeS = £

C(y).

(8.20) Theorem. Let y and y be two specifications. Suppose y satisfies Dobrushin's condition. For each i e S we let b{ be a measurable function on Q such that

lly,°(-M-y?(-MH^M for all co e Q. If jx e &(y) and fi e &(y) then \n(f)-fi(f)\^

Z

aiinDuMßibj)

ij'e S

for all

fe&.

Proof. We write C = C(y), D = D(y\ and b = (fi(b,))leS. Replacing bt by 1 A bt if necessary we may assume that bt ^ 1 for all i e S. We need to show that the vector Db is an estimate for n and fi. According to Remark (8.17)(1) the constant vector a = 1 is an estimate. An iterated application of Lemma (8.18) shows that for each n ^ 1 the vector fl(")

= Ca + £ Ckb k= 0

is an estimate. By Remark (8.17)(3) we therefore only need to prove that a(n) tends to Db coordinatewise as n -> oo. Dobrushin's condition implies that

Z q ^ c(y)" for all n ^ 0 and i e S. Thus the row sums of D are at most 1/(1 — c(y)). In particular, D has finite entries and Db exists. Also, Ca = Z Q -» 0 coordinatewise as n -> oo. Thus a(n) -> Db, and the proof is complete, D Theorem (8.20) was stated for specifications y and y and Gibbs measures H e @(y),fie @(y). But in fact we have only worked with the "single-spin parts" (}>;) ; e S and (y~i)ieS oî y resp. y, and we have only used the fact that nyt = n and fi% = fi for all i e S. We have thus proved a more general result than stated. In many cases, however, this generalization is only formal. This follows from Theorem (1.33). We still need to give the proof of Theorem (8.7). Its first assertion follows immediately from Theorem (8.20): Putting y = y and bt = 0 for all ieSwe see that n(f) = fi{f) for all / e J5f and thus n = fi whenever ß, fie &(y). The statement of existence will also follow from Theorem (8.20). It will be restated and proved in (8.23) below.

Further consequences of Dobrushin's condition

8.2

153

Further consequences of Dobrushin's condition

In this section we shall exploit the comparison theorem (8.20). We suppose y satisfies Dobrushin's condition. First we shall show that &(y) contains a (necessarily unique) element /u. Then we shall derive some properties of this fi. We shall construct consistent regular versions yv of the conditional probabilities n(-\&y) for each, not necessarily finite, subset V of S, and obtain explicit estimates on their local variational distances. In particular, we shall sharpen the conclusions of Proposition (7.11)(b) by estimating the speed of convergence of yA(- |co) to /i as A runs through Sf, as well as the speed of convergence in the uniform mixing property (7.10) that describes the decay of correlations of /i. We start with a simple lemma that deals with the conditioning of a specification relative to a fixed configuration on some part of S. It will be convenient to add to each specification y a Dirac kernel y0. y0 is a probability kernel from ^"0 = $F to & and is defined by (8.21)

y$(A\œ) = lA(œ)

(Ae^,œeQ).

(8.22) Lemma. Let y be a specification, and let V c S and œ eQ.be given. We define probability kernels yA by y/XA\Q = y\nv(A\a>f&s\K); here A e Sf, A e # \ £ e Q. Then y(V'a) ^ (y A ) A e ^ is a specification. Moreover, the following conclusions hold. (0 IfV = Stheny(V'm) = y. (ii) If V is finite then yv(- \œ) e
10 = 1 7AnF(d'7|toACS\A)yAnF(- |coA/jSXA).

As

n\\v = <JiK\v for y AnF ( - |coACSXA)-almost all n, the expression on the right equals 7AnK)W(- I^ACSXA) = TWO \Ü)ACS\A) = M' 10(V œ)

Thus y ' is a specification. Assertion (i) is obvious. To prove (ii) we assume that V e ^. Then for each A e ^ w e have

154

Uniqueness 7v(- \co)yA = J yv(dC\a>)yAnv('

|Û> A CS\A)

d

= hv( CHyAnv(-\0 = yv(-\œ). Thusy K (-|co)e^(y (K - m) )Next, if A e Sf and / e i f then 7 A / = yAf\vf(wA<7s\A) e ^ provided that y is quasilocal. This proves (iii). For the proof of (iv) we let i, j e S be two sites. If i e V then % = y; and therefore Cij(yiV,). In particular, ys(- \a>) = /J.. (ii) For any infinite V c S and co eQ., yv(-\o))=

lim yA(-\a>) AeSfy

in the J5?'-topology. In fact, if A e y and A e Sfv then sup

\yA(A\œ) - yv(A\œ)\ S

Ae^.ioEil

I

Dtj(y).

ieA,jeV\A

The expression on the right tends to zero as A runs through £fv. In particular, sup Ae^.toeCl

\yA(A\co) - (i(A)\ S

Z ie

A/y)-

A,jeS\A

(iii) If V <^W c S then y^yK = yw. In particular, nyv = n; that is, yv is a regular version of n(- \3~y). (iv) (yv)VŒS is quasilocal, in that yvf e S for all f e S and V c S. Proof. 1) Let F c S b e infinite, coeQ, and A, A' e Sfv with A c A' be given.

Further consequences of Dobrushin's condition

155

We consider the specifications y — y(A'm) and y = y(A,m). We know from (8.22) that y satisfies Dobrushin's condition. Also, yA(-\co)e ^(y) and yA.(-\co)e ^{y). Finally, if j e A then % = % = yp and if j e S\A' then $>(• | 0 = #>(• 10 = <5ra, for all £ e Q. Hence llf(-|0-f,°(-|OII^^(0=lA-\A(j) for all Ç e Q and j e S. Consequently, we have all ingredients that are necessary to apply Theorem (8.20). We obtain \yA(A\<°) - yAA\a>)\ z

Z

<5,(UA/y)

ieS,j'eA'\A

for each cylinder event A. Applying (8.22)(iv) we conclude that for each A e ^ IITA

— TA-IIA —

sup

\yA(A\oi) - yA,(A\œ)\

AE^COEQ

^

Z

DM

ieA,jeV\A

As Z A/y) < °° f° r all U the last expression tends to zero as A runs through J'ES

Sfy. Thus the preceding estimate shows that the net (yA)Ae <,v is a Cauchy-net with respect to || • ||A for each A e y . Consequently, the limit yr(A\co)= lim yA(A\co) AeSTy

exists for all cylinder events A and toefl, and HTÀ-TVIIA^

Z

A/y)

ieA,jeK\A

for all A e Sf and A e £fv. We also conclude that yv(- \co) is a probability measure on each J \ . As (E,S) is standard Borel, yK(-1co) can be uniquely extended to a probability measure on J^; cf. the proof of Proposition (4.9). The resulting yv is a probability kernel from & to #" (hopefully from ^"K to #"), and assertion (ii) holds. 2) We still do not know if yv{A\-) is ^y-measurable for each Ae&. Nevertheless, we proceed showing that yv is proper. Let A e 3~v be a cylinder event. Then yy(A\-)=

lim

yA(A\-)=lA

because each yA is proper and A e 3~A for all A e y K . By a monotone class argument the identity yv(A\ •) = lA extends to all A e 9~Y. 3) Next we prove (iv). Let V a S be infinite and / e &, say / e <£A with A e Sf. By hypothesis, yJ e <£ for all A e £fv. From (8.1) and Step 1) we conclude that lim \\yAf - yvf\\ ^ 0(f) lim \\yA - yv\\A = 0. AEtfy

At/,

156

Uniqueness

Thus yvf e !£. 4) To prove (iii) we let V c W and / e if. As yvf e ^ and y^(- |co) = lim yA( • | co) for all co, ywyvf=

lim yAyF/.

On the other hand, from Step 3) we know that lim \\yAnvf - 7vf\\ = 0 . Thus lim y A y F / = Hm yAyAn^/

= lim y A / = ywf. 4ey„

As / was arbitrary, y^yF = yw. The last part of (iii) follows by (1.20). 5) Turning to the proof of (i) we let V c S and co e Q be given. As a consequence of 2) and 4) above, the argument proving Lemma (8.22)(ii) also applies when Fis infinite. Thus yF(- \co) e ^(y (F ' m) )- But |^(y(F-£u))l ^ 1 because of Lemma (8.22)(iv) and the first part of Theorem (8.7). 6) It remains to show that yv(A\ •) is ^-measurable for all A e 3F and V cz S. (This does not follow from (ii) because 9~v is strictly smaller than 0 9~A if V is infinite.) By construction, yv(A\ •) is ^-measurable. Therefore AEJ;

we only need to prove that yv(- |Q = yv(' \rj) whenever Ç, rj e Q are such that Cs\v = Vs\v- B u t f° r s u c n C, V we have yiV'° = y(V^. Thus the result follows from assertion (i). D The preceding theorem has several aspects. One of these is illustrated by the following example. (8.24) Example. Let (E, S) be a standard Borel space and (S, B) a simple graph as defined in (2.17). Suppose y is a Markovian specification. Here Markovian means that, for all A e 5^ and A e !FA, yA{A\ •) is measurable with respect to #a A , where ÔA 4 {i e S\A: {ij} e B for some j e A} is the boundary of A. If (S, B) is locally finite (in that dA e ^ for all A e ^ ) then y is quasilocal. We assume that y even satisfies Dobrushin's condition. Then the extension (y F ) F c S of y is still Markovian. That is, for each (not necessarily finite) V c S and any A e J'y, the function yFC4|-) is êFdvmeasurable. To see this we may assume that A is a cylinder event. Let Ç e Ev be a fixed reference configuration. Then for all co e Q we have yv(A|co) = yv(A|Ccos\v) =

lim Ae&>y

yA(A\Çcos\v)

Further consequences of Dobrushin's condition

157

because yv{A\ •) is ^y-measurable. As dA\V c dV for all A e £fv, the Markov property of the yA's implies that the last expression is an J^-measurable function of co. Thus we have proved that yv is also Markovian. Consider now the unique element ß of &(y). If V c S and A e J'y then by (8.23)(iii) fi{A\^dr) = M y ^ l O I ^ K ) = yr(A\ •) /x-a.s. and therefore M^4|^K) = M ^ I ^ 3 K )

/i-a.s..

This property of ß is called the global Markov property, as opposed to the local Markov property which states that ß{A\2TA) = /x(A\^dA) /i-a.s. whenever i e , f A and A c S i s finite. We have thus shown that under Dobrushin's condition the local Markov property of ß (which holds by assumption) extends to the global Markov property. (Incidentally, a random field with the local Markov property is called a Markov field.) o A second aspect of Theorem (8.23) is the final estimate in assertion (ii). It shows that ß = lim yA( • | co) uniformly in co. (Under different hypotheses, the same conclusion was obtained in Proposition (7.11)(b).) In addition, it provides an estimate of the speed of this convergence by means of the quantities (8.25)

D(A,A,y)4

£

Dij(y).

ieA,jeS\A

We are thus led to ask how fast D(A, A, y) decays to zero when A is fixed and A runs through £f. We will show how the rate of this decay can be estimated in terms of the decay properties of the interdependence matrix C(y). (8.26) Remark. Let y be a specification. Suppose there is a semimetric s on S such that cs(y) A sup X QGOe*-» < 1. ieS

jeS

Then for all A, A e if we have D(A,A,y)^ |A|(1 -c s (y))- 1 exp[-s(A,S\A)], where s(A, S\ A) =

min

s(i, j) is the s-distance of A and S\ A. The condition

ie A,j'eS\A

cs(y) < 1 is satisfied whenever y is Gibbsian for some potential Q> such that sup £ e«A)(\A\ - 1)Ö(0A) < 2. ieS

Asi

Here s(A) — max s(ij) is the s-diameter of A e if. iyje

A

Proof. We start noting that (8.27)

sup £ e'<'-»D0.(y) S 1/(1 - cs(y)). ieS

jeS

158

Uniqueness

Indeed, using the definition (8.19) of D(y) and the triangle inequality for s we see that for each i0 e S

X e^-x/y) ^ i + I jeS

n ï l it

£ I

fleS
I

i„eS Jt = l

cjiyf = 1/(1 - cs(y))-

Thus for each A, A e if we have e s(A - s \ A) D(A,A,y)^

£

e^'D^y)

ieA,jeS\A

^ |A|/(1 - cs(y)). The second statement is obtained by a slight modification of Proposition (8.8). There we have shown that

for all i 7^ j . Thus

X Cy(y*)e*-» g \ X e^»(|^| - 1)<5((DJ for all i e S.

D

(8.28) Examples. Let S = Zd for some d ^ 1, and let y be a shift-invariant specification which satisfies Dobrushin's condition. (1) Exponential decay. Suppose, in addition, that (8.29)

X e'|JlC0;(y) < oo

for some t > 0. For example, this holds when y has finite range in that C0j(y) = 0 for all but finitely many j . Condition (8.29) also holds when y = y* for some shift-invariant potential $ which satisfies £ etdiamA(\A\

- 1)5(
here diam A is the Euclidean diameter of A. Under (8.29), D(A, A, y) Aey,> 0 exponentially. More precisely, there are constants 0 < c, C < oo such that D(A, A, y) ^ C | A | exp [ - cd(A, S\A)] for all A, A e ^ ; d(A, S\A) is the Euclidean distance of A and S\A. Indeed, combining (8.29) and Dobrushin's condition we may find a constant 0 < c < t such that C - i A l - £ e<WC0j.(y) > 0 .

Further consequences of Dobrushin's condition

159

Putting s(i,j) = c\i — j | in Remark (8.26) we obtain the result. (2) Power decay. Suppose, instead of (8.29), that there is a power p > 0 such that (8.30) £ \j\pCoj(y) < oo. JeS

This holds, for instance, if y = y * for some shift-invariant potential $ with £

(diamA)"(|^|-l)5(Oit)
By the dominated convergence theorem, (8.30) and Dobrushin's condition imply that £ ec|j'|Aplog(1+|Jl)C0j-(y)< 1 JeS

for some c > 0. Thus cs(y) < 1 for the metric s{i,j) = c\i - j \ A plog(l + |i - ./I). As s(i, _/') — p log 11" — j \ is bounded from below, we conclude from Remark (8.26) that there is a constant 0 < C < oo such that D(A,A,y) £ C|A|d(A,S\A)- p for all A, A e Sf with A c A . Thus D(A, •, y) has a power decay with the same power p. o As a third consequence of Theorem (8.23) we obtain detailed information on the decay of correlations of the unique p. e $(y). We know from Proposition (7.11)(b) that, under some hypotheses, p exhibits the uniform mixing property (7.10). Roughly speaking, condition (7.10) states that an event in a finite region A of S is almost independent of all events that occur sufficiently far apart from A. The degree of dependence may be described by the so-called uniform mixing coefficients (8.31)

cp(A,A,p)=

sup

\p(A\B) - p(A)\

of p.; here A, A e £f. Now we are able to improve (7.10) by finding upper bounds on these coefficients: Under Dobrushin's condition, cp(A, A, p) S D(A, A, y) for all A, A e £f. In particular, the preceding Remark (8.26) and Examples (8.28) provide us with estimates on the decay of cp(A, A, p) as A tends to infinity. Such estimates are especially useful for proving central limit theorems; see the Bibliographical Notes. (8.32) Corollary. Suppose y satisfies Dobrushin's condition, and let $(y) = {p}. Then (p(A,A,p)SD(A,A,y) for all A, A e Sf.

160

Uniqueness

Proof. Let A, A e Sf. If A e &A and B e 3~A with n(B) > 0 then B

Thus
n(A)\\,

and the result follows from Theorem (8.23)(ii). (In Theorem (8.23) {E,ê) was assumed to be standard Borel. But this hypothesis was only used to prove the existence of ft and the yF's. Here we may dispense with the standard Borel property because n exists by assumption.) o We conclude this section looking at the covariances fi(fg) — n(f)n(g) of bounded measurable functions / and g. We first note that these covariances may be estimated in terms of the uniform mixing coefficients provided / is local and g only depends on all spins outside some finite region, or vice versa. Indeed, if A, A e 5^ and / is J^-measurable and g is ^-measurable then (8.33)

\n(fg) - n(f)n(g)\ £

q>{A,A,n)ô(f)ô(g)/2.

To see this we may assume that 0 :g / ^ 1 and 0 ^ g :g 1. This is because (8.33) is invariant under affine transformations of / and g. We can also assume that fi(g) ^ 1/2; otherwise we replace g by 1 — g. Then Fubini's theorem gives Mfg) -

ii(fMg)\

rg } dt } ds | M / ^ t,g ^ s) - n(f ^ t)fi(g ^ s)\ o o l

^(p{A,A,/i)jdsfi(g^s) o = (p{A,A,(j)(j.(g) ^cp{A,A,li)à{f)ô{g)l2. Combining (8.33), Corollary (8.32), and Remark (8.26) we obtain covariance estimates for certain pairs of functions /, g. The next proposition deals with pairs of quasilocal functions. (8.34) Proposition. Suppose y satisfies Dobrushin's condition, and let <§(y) = {ft}. Then for all fge&'we have

\Kfg) - Kf)p(9)\ ^ Z

WWyWg).

Proof Rescaling g if necessary we may assume that g > 0 and fi(g) = 1. We define a specification y by yA = hAyA, where A e ^ and hA = g/yAg. y is easily

Further consequences of Dobrushin's condition

161

seen to be a specification. (In fact we may dispense with checking the consistency of y because we shall only need its single-spin part Cy,);,= s .) We put fi = g\i. Then fi e <&(y). For if A e £f and / e 3? then

ßn(f) = Kg yÀKPi) = WA(» yÀfg)h\g) = W A ( / S ) = Kfg) = £(/)•

Thus Theorem (8.20) applies, giving the estimate

\Kfg) - Kf)Kg)\ = W) - Kf)\ S I ôi(f)Du(y)fl(bj, iJeS

where bj is any measurable upper bound of the function co -> ||y?(- |ct>) — y/°(- |<x»)||. We claim that bi = ôj(g)/4y}g is such an upper bound. This will complete the proof because fi(bj) = ôj(g)fi(g/4yjg) = Sj(g)nyj(g/4yjg) = ôj(g)/4. Wefixany j e S and co e Q, and we write a = yf(- \co) and y,°(- \co) = t/a, where u{x) = g{xcos\y})/yjg(co), x e E. We also put m = (supu + inf«)/2. Then we obtain, using (8.1), ||«a - a|| = a(|u - l|)/2 ^ a((u - a(u))2)1/2/2 ^ a((u - m)2)1/2/2 g ||u - m||/2 ^ (5(u)/4 g bj((o). The proposition is thus proved,

a

As a first application of the preceding proposition we state a counterpart of Remark (8.26). (8.35) Remark. Let S = Zd for some d ^ 1 and s a shift-invariant semimetric on S. Suppose y is a quasilocal specification such that cs(y) A SU p £ CyOOe*-* < 1, and let <S(y) = {n}. Then for all /, g e S we have

I IM/^-M/^MJIe«0'0 ieS

^ <5S(/)<5S(0)/4(1 - c,(y)). Here

Proof. The properties of s imply that s(0,/c) = s(0, -k)^

s(0,i) + s(i,j) + s(0,./ + fc)

162

Uniqueness

for all i, j , ke S. Proposition (8.34) thus shows that \ß(fg°0k)-ß(f)ß(g°0k)\es(O'k)

Z SeS

Summing over k, j , and i in that order and using (8.27) we obtain the result, a Needless to say, the considerations of Example (8.28) apply to the preceding remark as well. The details are left to the reader. As a second application of Proposition (8.34), we establish a smoothness property of the unique Gibbs measure for a potential ® which satisfies Dobrushin's condition. For simplicity we confine ourselves to a homogeneous setting. Suppose that S = Zd. We consider the Banach space M& of all shiftinvariant potentials ® which are such that (8.36)

|||®|||4 £

\A\\\®A\\

A30

is finite. We put @ = {® e $B: \\\®\\\ < 1}. Clearly, each ®e 3 satisfies the hypothesis of Proposition (8.8). (8.37) Corollary. Suppose that S = Zd and I e J((E, ê) is finite. For each ® e 3> let ßgf be the unique Gibbs measure for ® and X. Also, let g e ^ be such that Z ôi(g) < oo. Then the function ® -> /%(#) on 3) is continuously differentiable. ieS

More precisely, for each ® e 3) and *F e 3l& we have p.

dtn*+w(g)

Here fv = £

t=0

= - Z [>*(# fv°0i) - ß{g)ß*{fv)l isS

\A\-^A.

A3Ü

Proof. We fix any O e ®, 4* e l 0 , and œ e Q. For each t e R we let fA = yr^ifg^)

- yt+tv{f\
and

+t+tv(g) be the covariances of g and any bounded measurable / relative to the measures y^+tV(-|eo) and /i(I>+,4,. Let t0 > 0 be so small that |||0||| + tolll^lll < 1- A glance at the proof of Proposition (8.8) then shows that

sup Z sup q/y™) < l. ieS

j e S \t\ S t 0

Therefore, if Dtj = sup Dii(y9+ev) then sup Z A; < °o|r|£r0

ieS

jeS

Further consequences of Dobrushin's condition

163

Turning to the main part of the proof, we conclude from Theorem (8.23)(ii) that /VH4-(0)

= lim yA +,4, (0M

for all t with \t\ ^ t0. On the other hand, it is easily checked that

whenever \t\ ^ t0. It is therefore sufficient to show that

supK/tf,^-! A - s\> and T3 = 2sup

sup

\k\-

£

|»|£»o AeyU{S},A=>A

keA\A

In view of Lemma (8.22)(ii) we can apply Proposition (8.34) to estimate T±. Since *J(HJ\

I fv°e-k) = ôj( £ keA

/

\A\A\\A\-^A

V^nA^O

S2

£

TOI,

A3J:A\A*Q

we obtain that

The inner sum tends to zero as A runs through y . The dominated convergence theorem thus shows that TY -> 0 as A -> S. To estimate T2 we observe that fv°6_k e S for all keA. Theorem (8.23)(ii) (together with a version of equation (8.1)) thus ensures that T2 -> 0 as A -> S. As for the term T3, we can again use Lemma (8.22) and Proposition (8.34). We obtain the estimate £•

Since

i,jeS,keS\A

164

Uniqueness

Z ^ ) ^ Z keS

keS

I

i^r1ii^ii = 2ii^ii0<œ,

A=>{0,k}

we conclude that T3 can be made arbitrarily small when A is chosen sufficiently large. The proof is thus complete, D

8.3

Uniqueness in one dimension

A main advantage of Dobrushin's uniqueness condition (8.6) is its universality: It is not supposed that the parameter set S has any specific structure. •Here we shall prove a more particular uniqueness result. We shall assume that S has a chain structure. That is, we take S = Z, the integers, or S = N, the natural numbers. We shall also confine ourselves to the case of Gibbsian specifications. But then we shall see that Dobrushin's condition of small total interaction can be replaced by a condition on the decay of the interaction. In contrast to the former the latter condition remains unaffected when the potential is multiplied by a scalar factor ß. (Note that a uniqueness condition for general S can never have this property. This is exemplified by the phase transition in the two-dimensional Ising model.) Whilst Dobrushin's uniqueness theorem (8.7) relied on a general contraction argument we shall now use a special feature of one-dimensional systems. Z and M may be exhausted by intervals, and intervals have but a bounded number of boundary sites. This fact will allow us to find a uniform bound on the oscillation of energy in arbitrarily large regions when the boundary condition varies. The usefulness of such a uniform bound is demonstrated by the following proposition. The hypothesis of this proposition is similar to the concept of macroscopic equivalence which has been studied in Section 7.4. In fact, these conditions are both designed for implying a (uniform) mutual absolute continuity of Gibbs measures. A further condition of this type will appear in (9.1). (8.38) Proposition. Let y be any specification. Suppose there exists a constant c > 0 with the following property. For every cylinder event A e 3F there is a set A e y such that yA(A\0^cyA(A\n) for all Ç,rçe Q. Then \<${y)\ S 1. Proof. Suppose @(y) contains two distinct elements ji and v. Then (^u + v)/2 e @(y)\ex @(y). Therefore it is sufficient to show that ^(y) c ex ^(y). Let n e <&(y), Be ST, and suppose \x{B) > 0. We put v = //(• \B). By Theorem (7.7)(b), v e &(y). Our hypothesis implies that v ^ c/z. Indeed, let A be any

Uniqueness in one dimension

165

cylinder event. Then we have for some suitable A e ^ v(A) = vyA(A) = I v(d{) J n(dri)yA(A\Q >c[v(àQ[ii{àri)yM\ri)

= cii(A).

Using the monotone class theorem we may conclude that v ^ c/i on all of SF. In particular, 0 = v(Q\B) 2; c/x(Q\B) and therefore fi(B) = 1. Consequently, fi is trivial on &~. Hence Theorem (7.7)(a) ensures that ß is extreme, a We now state the uniqueness theorem for Gibbs specifications in one dimension. As before, Ô(OA) denotes the oscillation of <&A. (8.39) Theorem. Let S = Z or N, (£, ê) be any state space, X e JÎ(E,S) an a priori measure, and $ a A-admissible potential such that (8.40)

sup ieS

Z A G£f,min A^i<ma\

ô(%) < oo. A

Then |#(0)| ^ 1. / / (£, S) is standard Borel then |#(0)| = 1. Proof. 1) To prove the first statement we shall check that y* satisfies the hypothesis of Proposition (8.38). We let s denote the finite supremum in (8.40). We consider an interval A in S of the form A = ] — n, n] D S, n ^ 1. For all C, n e Q and œ e EA we have

l*tf Ksw) - #A(™/S\A)I S

Z

Ô(OA)

/inA#0,/l\A#0

s Z

Z

w)

fc= + n min ^4^fc<max A

S 2s. (The term with k = —n does not appear when S = N.) Thus e-2s/tf(a»ySXA) ^ A*(Û)CS\A) ^ e ^ ^ s v O Integrating over co with respect to AA we obtain Z*(C) ^ e 2 s Z » and for all Ae tFA. Consequently, putting c = e~4s we have

whenever i e ^ A and £ rç e Q. But for each cylinder event A there is an interval A with A e 3FK. The hypothesis of Proposition (8.38) is thus verified. Consequently, |^(0)| S 12) To prove the second statement we assume that (£, S) is standard Borel.

166

Uniqueness

Passing to an equivalent potential if necessary we can assume that \\<J>A\\ = ô(%)/2 for all ,4 e 5^ with \A\ > 1. Condition (8.40) then implies that |H% — YJ ®{i}\\ < °° for all A e 5^. Therefore the /l-admissibility of
means that >l(exp( —O^)) < oo for all i. Suppose for a moment that O^ does not depend on i, in that 0{j} = cp o at for all i e S and some measurable cp: E -* U. y® then coincides with the Gibbs specification for the potential Ô = (^{\A\>i}(^A)Ae^ ar*d the finite a priori measure I = e_<"/l; cf. (1.28)(3) and (2.18). By (8.40), $ e f . Thus Theorem (4.23) guarantees that ^()| ^ 1. D (8.41) Comments. (1) Suppose that S = Z and
£

diamA 0(®A) < oo.

Here diam /I = max A — min A is the diameter of A. Indeed, the sum in (8.40) takes separate account of diam A translates of each set A, whereas in (8.42) one of these translates is singled out by the requirement min A = 0. By similar reasoning, (8.42) is equivalent to the condition „ diam A AsO

I-A |

(2) Let S = Z, and suppose O is a shift-invariant pair potential of the form (ai>aj) A (0

ttA = {i,j},i¥=j, otherwise,

where q>: E x E -> M is measurable, symmetric, and bounded. Then the expression in (8.42) takes the form ô(q>) £ n1"". nil

Thus 2. (Note that 1.) In fact, the condition p > 2 is not only sufficient but also necessary for uniqueness: for 1 < p ^ 2 and a suitable choice of E and (p a phase transition occurs. The case 1 satisfies (8.42) if and only if O{0 1} is bounded. The latter clearly holds when E is finite. Thus

Uniqueness in one dimension

167

Theorem (8.39) provides a second proof of the uniqueness statement in Theorem (3.5). On the other hand, if ${0,1} is unbounded then $ may exhibit a phase transition; see Chapters 11 and 13. (4) The uniqueness result in Comment (6.7)(2) is not covered by Theorem (8.39). This does not detract from Proposition (8.38). In fact, looking at (6.6) we easily see that the hypothesis of Proposition (8.38) is satisfied when
Chapter 9 Absence of symmetry breaking. Non-existence

As we have pointed out in Chapter 5, the phenomenon of symmetry breaking is an important special type of phase transition. In fact, the discussion of this phenomenon is one of the main themes of this book. Three examples were already presented in Chapter 6, and quite a number of further examples will be provided later on. Here we shall deal with the converse problem, the conservation of symmetries. More precisely, we shall ask for conditions under which a given symmetry T of a specification y preserves each element [i of ^(y). By (5.11), this is trivially true when \@{y)\ = 1. Therefore we shall only be interested in conditions which are weak enough to cover at least some cases of phase transitions. For example, we want to consider specifications y with two (or more) symmetries, one of which is broken whilst we will show that the other remains unbroken. We shall distinguish between two kinds of symmetry: discrete symmetry and continuous symmetry. A symmetry T of y is said to be continuous if x belongs to the image of a homomorphism from the additive group R into the symmetry group of y. Otherwise x is said to be discrete. (For example, if the symmetry group of y is isomorphic to (Z/2Z) x (IR/Z) then the transformations that correspond to an element of {0} x U/Z are continuous symmetries whereas all other symmetries are discrete.) It is not surprising that it is much more difficult to break a continuous symmetry than to break a discrete symmetry. The former requires a slower decay of interaction or a higher spatial dimension than the latter. Suppose, for example, that S = Zd and y is Gibbsian for some shift-invariant bounded nearest-neighbour potential. If d = 1 then \@(y)\ = 1, by Theorem (8.39). In particular, there is no symmetry breaking in one dimension. Two dimensions are sufficient for a discrete symmetry breaking in bounded nearest-neighbour systems. This is exemplified by the two-dimensional Ising model of Section 6.2. A continuous symmetry breaking is possible in three or more dimensions, as will be shown in Chapter 20. What about the breaking of continuous symmetries in two dimensions? One of the main results of this chapter states that this is impossible, at least when we impose some natural conditions. We shall confine ourselves to Gibbsian specifications. The underlying parameter space S will be assumed to be either one-dimensional or the square lattice Z 2 . In the first case we shall look at single symmetries x such as spatial translations and pure spin transformations, regardless of whether the latter are discrete or continuous, x will be shown to remain unbroken, provided the potential satisfies a mild decay condition in terms of x. The case S = Z2 will

Discrete symmetries in one dimension

169

be treated in Section 9.2. In this case a discrete symmetry breaking occurs even in simple models. Therefore we shall only consider continuous symmetries. Accordingly, we shall look at (possibly periodic) one-parameter families (T')teR °f P u r e s P m transformations. A breaking of such a family of symmetries will turn out to be impossible for any potential that depends smoothly on (T')teR and satisfies a decay condition. As a by-product, the preceding results on the absence of symmetry breaking also imply the non-existence of Gibbs measures for certain potentials <E>. Indeed, suppose T e T is dissipative. Roughly speaking, this means that under the action of T each point eventually tends towards some sort of infinity. Such a T does not admit an invariant probability measure. Consequently, if z can be shown to preserve each p e ^(<E>) then ^(<E>) must be empty.

9.1

Discrete symmetries in one dimension

As a starting point for all the results of this chapter we shall prove a general proposition which is closely related to Theorem (7.33) and Proposition (8.38). (9.1) Proposition. Let (E, <S) be a standard Borel space, y a specification for E and S, and T e T a symmetry of y. Suppose there exist two numbers a, b 5: 0 such that for each cylinder event A e êF there is a set A e y with (9.2)

ayA(T-'A\-)

+

byA(xA\-)^yA(A\-).

Then T preserves each p e ^(y). Proof. By the extreme decomposition theorem (7.26) it is sufficient to show that each /jeex ^(y) is r-invariant. So we let /jeex ^(y) be given. Let A e êF be a cylinder event and A e £f be such that (9.2) holds. Then a H(T~1A) + b H(TA) = a pyA(x~1A) + b pyA(rA) ^ pyA(A) = p(A). This means that a x(p) + b x"l(p) ^ / j o n the algebra of all cylinder events. The monotone class theorem then implies that this inequality holds even on all of &. Now suppose that T(P) =£ p. Then also T _ 1 (/I) =£ p. By Remark (7.2), r(p) and T_1(JU) are extreme in ^(y). Thus Theorem (7.7)(d) ensures that there are sets Ax, A2 e ^"such that piA^ = p(A2) = 1 andr( J u)(^ 1 ) = %~i(n)(A2) = 0. Putting A = A1 H A2 we conclude that 0 = a z(p)(A) + b T-1(/*)(,4) ^ p(A) = 1. This contradiction proves the proposition, D

170

Absence of symmetry breaking. Non-existence

(We note that a misunderstanding of the notion of a T-invariant specification might lead to the erroneous conclusion that hypothesis (9.2) was a trivial consequence of the assumed t-invariance of y. We can avoid this confusion by again looking at (5.4) and (5.7)(b).) We intend applying the preceding proposition to Gibbsian specifications. To this end we need to replace (9.2) by suitable conditions on potentials. Our next proposition is a first step in this direction. (9.3) Proposition. Let (E, i) be a standard Borel space, X e Jt{E, ê) an a priori measure and <S>bea À-admissible potential. Also, let xbea X-preserving symmetry of Q>. Suppose we can find two constants 0 ^ c ^ 1 and 0 ^ C < oo with the property below. For each A e y there are a set A e y with A c A and a X-preserving transformation x e T such that (i) x is a localized version of T, in that (xœ)A = (xœ)A, ( T - 1 ^ = ( T - 1 « ^ , and (fa»)s\A = a>s\A for all œ e Q; and (ii) c H* o f + (1 - c)H* o r 1 -H*^C. Under this hypothesis, each p. e $((&) is T-invariant. Proof. Let a = (1 — c)ec, b = c e c and A e J^ be any cylinder event. We choose A e y so large that A e ^A and Ae y with A c A as in the hypothesis. We will show that y = y* satisfies (9.2). Condition (ii) implies that

cfc*o f + (i - c)hX ° r 1 ^ (A* o f)c(/j* o r 1 ) 1 - ' ^ e" c /i*.

On the other hand, as f equals the identity outside A we have Z*of = Z*of-i=Z*.

Thus ap*0x-'+bPZox^P*. Further, we have f M = x~lA and xA = xA because Ae JFA and f = x on A and T™1 = T - 1 on A. From this we conclude that a tf(T-U| •) + b y * M | •) = XA{pZ(alA ° f + b\A o f" 1 )) = XA{{aP*ox-'

+

bp%ox)\A)

= AA(p*li4) = y*(X|-). The second equality follows from the hypotheses that f is A-preserving and preserves all spins outside of A. Thus (9.2) holds, and an application of Proposition (9.1) completes the proof. • After these general preliminaries we turn to a study of one-dimensional systems. We shall look at two kinds of transformation: spatial translations and transformations without spatial part. We start with translations.

Discrete symmetries in one dimension

171

Let S = Z. We seek conditions on a potential such that given these conditions each Gibbs measure is shift-periodic with a fixed period p. For simplicity we shall look only at shift-invariant pair potentials. (An extension to more general potentials is possible. We leave this to the interested reader.) Without loss we shall assume that the self-potential part is incorporated into the a priori measure; see (1.28)(3) and (2.18). Consequently, we shall confine ourselves to potentials O of the form
[0

i f A. = {i, i + k} a n d

k^l,

otherwise,

for any choice of measurable real functions q>k on E x E. Such a potential O is absolutely summable if and only if £ \\cpk\\ < oo. (9.5) Theorem. Let S = Z, (£, ê) be any standard Borel space, X e Jt(E, S) a finite a priori measure, and e ^® a potential of the form (9.4). Let p ^ 1 be given and suppose that (9.6)

£ k\\
(pk\\ < co.

Then each fi e &(
6p-invariant.

Proof We shall verify the hypothesis of Proposition (9.3) with the constants c = 1 and

C = 4p X ll%ll +2 X *ll%+p-%llLet A e 9* be given. We choose two integers m, n with A c [m + p, n — p], and we put A = [m, n] fl Z. We also define a localized version 0 of 0p by a>;_p

if m + p ^ i ^ n,

(Op«); = < (Oi-p+„-m+1 a»,-

if m ^ i < m + p, if i e Z\A.

Condition (i) of Proposition (9.3) then evidently holds. We are thus left with proving condition (ii). We write Ht = Zi + E 2 + E 3 + E with E

i =

Z

o))»

m ^ i <j ^ n — p m ^ i ^ n — p,j>n E

3 =

Z

«Pj-I^J» 0))»

172

Absence of symmetry breaking. Non-existence

and £4 =

Z

®A-

An[n-p+l,n]#0

Similarly, with £l =

Z

^2 =

Z

^3=

(Pj-iiOi-p'Oj-p)' °}\

Z

Oj-p\

i<m,m+pi^j^n

and

£4 =

Z

^ ° 0,.

An[m,m + p - l ] # 0

Clearly, we have Z t = t,1. Moreover, lS2-^2l^

Z

l^-ifa.^-^-i-pfa»«^)

s Z ( k -p)ll%-%- P ll k>p

= Z

k

\\
k>l

and IZ3 - £ s l ^

Z

l<ü>f-.-(ffi»^-)-<ü!/+p-£(^»<»))l

i < m ^j ^ n — p

^ Z

feii%+p-%ll-

Finally, | Z 4 | v | £ 4 | ^ p | | ( D | | 0 = 2p Z ll%llfc>i

Combining these estimates we conclude that \\H*oÖp-H*\\ZC.

The theorem thus follows from Proposition (9.3).

a

We now make some simple remarks on hypothesis (9.6) of Theorem (9.5). (9.7) Comments. (1) The set of all integers p that satisfy (9.6) is a subgroup of Z. Thus condition (9.6) is consistent with the fact that {p e Z: 6p(n) = n for all p. e <&(&)}

is also a subgroup of Z.

Discrete symmetries in one dimension

(2) Condition (9.6) is closely related to the condition

173

Z \\(pk\\ < oo of

absolute summability. On the one hand, if lim ||
Z ll%ll ^ Z kll%+p-%ll for all p ^ 1. This is because -
li%li ^ Z \\• [0, oo[ such that cpk = J{k)q> for allfc^ 1. Then

Z &H%+i - %H = II9II Z Z VW - J{k + 1)) til

nil tin

^ Z li%iifcfel

Equality holds whenever the last sum is finite. (3) Condition (9.6) is weaker than the uniqueness condition (8.42). More precisely, suppose O is of the form (9.4) and satisfies (8.42). Let q>k = q>k — inf ks by means of (9.4). Then Ö is equivalent to O and satisfies (9.6) for all p ^ 1. o We proceed with some examples of Theorem (9.5). (9.8) Examples. (1) Long range Ising ferromagnets. Let S = Z, E = { — 1,1}, and X be counting measure. For given parameters a > 1 and /? > 0 we let
oA = \~ßv ~jr(7i(7j

iïA = { u }

' '' *j'

} 0 otherwise. By Comment (9.7)(2), (9.6) holds for p = 1. Consequently, 0(O) = %(0). This result is trivial when a > 2 or ß is small since |^(0)| = 1 in these cases. But it is non-trivial when 1 < a ^ 2 and /? is large because then the spin-flip symmetry of <5 is broken; see (20.21) and Comment (8.41)(2). (2) Long range Ising antiferromagnets. Let S, E, X, a, ß be as above and define a potential Ô by

Ô, = j - ^ - 1 ) ^ - Jl""^ \

0

if

^ = {U}, i * J,

otherwise.

Ô satisfies (9.6) for p = 2. Consequently, each p. e $(<&) is periodic with period at most two. As a matter of fact, the present model is isomorphic to the

174

Absence of symmetry breaking. Non-existence

preceding example. For let x e T be defined by (TCO), = (— l)'cOj, co e Q, i e S. Then <ï> = x(O). Hence Proposition (5.6) and Remark (5.10) imply that the involution fi -> x(fi) is a bijection between ^(O) and (S{^>). As x commutes with 62, the 62-invariance of the Gibbs measures in ^(O) implies that of those in ^(0), and vice versa. The fact that ^(O) and ^(<ï>) are isomorphic also shows that (S{^>) exhibits a breaking of 61 whenever 1 < a ^ 2 and ß is large enough. Thus it is no accident that Ô fails to satisfy (9.6) for p = 1. o We now turn to the second subject of this section, the conservation of pure spin transformations in one-dimensional systems. Here we say that a transformation x e T is a pure spin transformation if x can be written in the form (9.9)

T.œ^iZiœùçs,

with a family (x,-, i e S) of invertible transformations of E. In other words, x is a pure spin transformation if its spatial part x,,, (cf. (5.1)) is the identity. As in (5.20), we let T° denote the set of all pure spin transformations which preserve the underlying a priori measure X e Ji{E, S). We put S = Z or M. As before, we shall confine ourselves to the consideration of pair potentials, and the extension of our results to more general potentials is left as an exercise. But now it is not appropriate to assume shift-invariance. So we shall look at all potentials O of the form (9 10)

G i )

1ÎA

= & j} otherwise,

and

'

<

j

'

with arbitrary measurable functions %/. E x E -> R. A pure spin transformation x = (x;, i e S) is a symmetry of such a O if and only if ) = (Pij(x, y) for all i < j and x, y e E. We shall write z+ = z v 0 for the positive part of a real number z. (9.11) Theorem. Let S = Z or N, (E, S) be any standard Bor el space, X e M{E, S) an a priori measure, O a X-admissible pair potential of the form (9.10), and x = (t.-, i e S) e T° a symmetry of O. Define J(i,j)=

sup l
+

x,yeE

for i < j , and suppose that (9.12)

C(0,x)^sup X

J(hJ)
ne S i^n<j

Then x preserves each /J. e ^(O). Proof. We shall again verify the hypothesis of Proposition (9.3). This time we choose the constants c = 1/2 and C = C(0, x). Let A e y be arbitrary and m < n be two sites in S such that A ^ [m,n]nS3A.

Discrete symmetries in one dimension

175

We define a localized version £ e T° of x by xœ = (TCO)ACOS\A (CO e Q). £ clearly satisfies condition (i) of Proposition (9.3). To prove condition (ii) we will show that Hi o £ + H* o r 1 - 2Ht ^ 2C(d>, T). The expression on the left equals the sum of the three terms Zi =

l>ufo Tjrlff>)

Z

_ 2(

PiMh o/)].

m ^ i<j ^ n

and

As O is T-invariant, Z x = 0. The summands of 2 2 and Z 3 are dominated by J(i,j) because (p^xj1 a}) = (p^x^aj) and (ft/rf1 , T) and Z 3 ^ C(, T). This proves condition (ii) of Proposition (9.3) and thereby the theorem, D (9.13) Comments. (1) Condition (9.12) is implied by the stronger condition (9.14)

sup Z

sup

$pij{Tix,y)-(pij(x,y)\
(provided O is i-invariant). This follows from the identity \^xïkx,y) - 2(Pij(x,y) = Z (\k\-W$L(pij(Tt+1x,y)

+

(pij(xf-'x,y)-2
V\<\k\

shows that C(
176

Absence of symmetry breaking. Non-existence

Conditions (9.12) and (9.14) are spatially uniform. This does not exclude inhomogeneities, as is shown in the example below. (9.15) Example. Let S = N, £ = {-1,1} 2 , and X be equidistribution. We think of E as the four vertices of a square in U2. Let us write x = (x1,x2), œ i = (œn> œn\ a n d Oi = {on,oi2) whenever x e E, œ e Q, and i e S. We define a nearest-neighbour potential $ of the form (9.10) by , ij

v '

=

f - ^ i y i - Kxlx2yly2 \ 0

if j = i + 1, otherwise.

Here 1 ^ i < j < co, K ^ 0, and (J„)„ èl is a sequence in ]0, oo[ such that £ exp(-2J„) < oo. näl

3> is invariant under the two reflections T (1) , T (2) e T° which are given by (T^'CO); = ( - C O ; ! , ^ ) , ^

2

^ ) ; = ((On, ~

(Ot2),

coeQieS. For this $ and T = T<2) the expression at (9.14) equals 2K. Thus Theorem (9.11) implies that each p e <$(<&) is T(2)-invariant. On the other hand, the symmetry T (1) is broken. This is easy to see when K — 0. For let Sj and S2 be two disjoint copies of N. By the natural identification of EN and {— 1, l} s i us 2 ; y® can be thought of as a specification with state space { — 1,1} and parameter set S t U S2. In the case K = 0, y® is a product specification. Its factors are the Gibbs specification for the potential (6.2) and the independent specification with single spin distribution (ôl + 3.^/2. Consequently, (7.19) and (the proof of) Theorem (6.4) show that ^($) contains a measure /i + which satisfies n+{an) > 0 for all i e N and is given by n+ = lim ?*,...,#}(• |co+). Here a>+ is the constant configuration with value + 1 . Such a /i + also exists in the nontrivial case K > 0. This can be deduced from a famous inequality of Griffiths (1967a). This inequality implies that each y*,...,jv}(oo

we refer the reader to section VI of Griffiths (1972), for example.) Consequently, the inequality fi+((rn) > 0 also holds when K > 0. In particular, t (1) (/i + ) # fi+. Thus ^($) exhibits a breaking of the symmetry T (1) . O As we have said in the introduction to this chapter, the absence of symmetry breaking entails the non-existence of Gibbs measures for potentials with dissipative symmetries. This is made precise by the next corollary to Theorem (9.11). (9.16) Corollary. Let S = Z or N, {E,S) be a standard Bor el space, Xe Ji(E, S), and t e T° a pure spin transformation. Suppose T is dissipative, in that

Discrete symmetries in one dimension

177

there is a bounded measurable function / ^ 0 on E such that X(f) > 0 and l i m / O T ' = 0 A-a.s. for some i e S. Also, let O be a x-invariant X-admissible pair potential of the form (9.10) with the property (9.12). Then <${<$>) = 0. Proof. Suppose that ^( 0 and lim / o x\ o at = 0 /i-a.s.. The dominated convergence theorem thus gives the contradiction 0 < \x{f o o\) = lim / i ( / o cr; o T ) = /i I lim / o tf o er; = 0. fc->00

In applications of Corollary (9.16) it will often be natural to choose f{x) = 1 A h*i}(xœs\{i})

(x e E)

for suitable i e S and t a e Q . For instance, we may do so in the following example. (9.17) Example. Let S = Z, E = Z or IR, and X be counting resp. Lebesgue measure. Also, let u: E -> [0, oo[ be a measurable function. We assume that u diverges at infinity in such a way that X(e~") < oo, but satisfies the quadratic growth condition c(u) = sup [u(x + 1) + u(x — 1) — 2u(x)] + < oo. xe E

For example, we may take u{x) = ß\x\p (x e E) with ß > 0 and 0 i+l.

Here i, j e S and x j e £ . We will show that $ is 1-admissible but <S(^>) = 0. On the one hand, for each A e y and œ e f i w e have | AA(dC|
|A|

= l(e~") . Hence 0 is ^-admissible. On the other hand,
= e~2uM

(x e E),

where œ = 0.) Moreover, C(, T) = c(u) < oo. We thus conclude from Corollary (9.16) that (S(^>) is empty. (This example shows the advantage of

178

Absence of symmetry breaking. Non-existence

condition (9.12) over (9.14). The latter only holds when the growth of u is at most linear.) An additional remark is in order. If E = Z and u(x) = ßx2 for some ß > 0 then $ is nothing other than the one-dimensional version of the potential (6.16) which was investigated in Section 6.3. In that section, the underlying parameter set was Z2, and we have shown that, for sufficiently large ß, ^(ß<5)) is infinite dimensional (and thereby non-empty). We thus have an example of a potential which admits no Gibbs measure in one dimension but infinitely many extreme Gibbs measures in two dimensions, o

9.2

Continuous symmetries in two dimensions

Throughout this section we set S = Z2, the square lattice. We assume we are given a family (xt)teU of A-preserving pure spin transformations x* = (x\, i e S)e T° which enjoy the property (9.18)

T S O T ' = TS+'

for all s, te U.

That is, the mapping t -> x' is a homomorphism from the additive group U. into T°. Note that t -> x* is not assumed to be an isomorphism. In other words, p (T') ( S R may be periodic in that some x is the identity. Thus our setting includes an action of R as well as an action of the compact circle group U./Z. (We do not need to require that T'CÜ be jointly measurable in t and a>, although this will generally hold in applications.) We now turn to the problem of showing that it is impossible to break the continuous symmetry (T ( ) (eR . For simplicity we shall again only consider pair potentials. So we shall look at potentials $ of the form

(9 19)

ai) A

(0

XA =

{i.j},i±j,

otherwise.

The (pijS are assumed to be measurable real functions on E x E such that (Pij{x, y) = cpjiiy, x) for all i # j and x, y e E. The theorem below is the main result of this section. Its proof is postponed until the end. (9.20) Theorem. Let S = Z 2 , (E, S) be a standard Borel state space, X e Ji(E, S) an a priori measure, and (x')teU a family of X-preserving pure spin transformations that satisfy (9.18). Also, let <5) be a (x')teU-invariant k-admissible pair potential of the form (9.19). Suppose that for <S> and {x')teR the following conditions of smoothness and decay hold. (i) For all i # j and x, y e E, the function t -> (ptj{x, x]y) is twice continuously differ entiable.

Continuous symmetries in two dimensions

179

(ii) There exists a symmetric function J: S x S -> [0, oo[ and a constant K < oo such that d2 sup sup --y ç>y(x, t]y) ^ J(i, j) x.yeE

teU

tu

for all i 7^ j , and (9.21)

£

|i-j|2J(U)^Klogn

>ES:0<|i-j|gn

/or a// i e S and n ^ 2. Under these hypotheses, each fi e @(
mod 2n

(i e S,œ

e Q).

r

We define a (T )(eR-invariant pair potential $ of the form (9.19) by
- x).

Here x, y e E, i, j e S with i ^ j , and ß > 0 and p > 2 are fixed parameters. (The condition p > 2 implies that is absolutely summable.) Proposition (8.8) and Dobrushin's uniqueness theorem show that |^(0)| = 1 whenever ß is small enough. What about the case of large ßl We shall see in (20.21) that the continuous symmetry (Tr),eK of $ is broken whenever 2 < p < 4 and ß is sufficiently large. Here we obtain an intermediate result: For p ^ 4 and arbitrary ß there is no breaking of (t') ( E R . Indeed, in the latter case Theorem (9.20) applies because d2 foïViM^jy)

_ d2 = ~ß\i - J\ "-^cosiy

+

t-x)

Sß\i-j\-p and

z \j\2\j\-p^i 0
fc

z /e2-"=s z F-* = l |j"i|v|j 2 | = k

k=l

180

Absence of symmetry breaking. Non-existence

^ 8 £ fc"1 ^241ogn k= \

for all n ^ 2. (2) Shlosman's rotator. Let E, X, and (T') I S R be as above. Define <5 by (9.19) and (Pij(x,y)=

— ßcos(x — y) iî\i — j \ = y/2, < ßcos2(x — y) if\i — j \ = l, 0 otherwise;

here i, j e S,x,y e E, and ß > 0. It follows immediately from Theorem (9.20) that each ß e ^(<5) is preserved by the symmetries T' of $. The point of this example is that O exhibits a phase transition when ß is large enough. The phase transition depends on the fact that O has a further (discrete) symmetry, namely the rotation by n of all spins on the even sublattice of S. For large ß this additional symmetry is broken, as will be proved in Subsection 18.3.8. o Theorem (9.20) may also be used to prove the absence of symmetry breaking for symmetry groups / which are larger than K o r a factor group of U, provided each element of / can be imbedded into a one-parameter subgroup of L The latter is possible whenever / is a connected Lie group. We will not discuss these generalities, however. Instead we shall illustrate the basic principle by a further example. (9.23) Example. Classical Heisenberg models. Let JV ^ 2, E be the unit sphere in UN, and X be normalized surface measure on E. Let <5 be defined via (9.19) by putting
{x,yeE,i^

j).

Here K(-, •) is any symmetric real function on S x S, and the dot denotes the inner product on UN. We assume that \K(i,j)\ ^ K\i — j \ ~ p for some K > 0 and p ^ 4 and all i ^ ;. (Note that the present example includes a reformulation of Example (9.22)(1) as a special case.) Let r be any rotation of UN and T e TA° be defined by TCO = (rco;)ieS. We will show that every /i e ^(<ï>) is T-invariant. In a suitable orthonormal basis of UN, r is represented by a matrix of the form 1 M(r 1 ,...,r„) =

M(r x ) M(rn),

Here

Continuous symmetries in two dimensions (COSt

— Sin£\

M(t) = [

)

,

181

^

(te M\

\sint cost / and r1,..., rn e ] 0,2% [ are suitable rotation angles. The mapping (t1,...,tn)-> M(t 1 ; ..., tn) establishes a homomorphism (with kernel 27iZ") from the additive group W onto a (necessarily abelian) subgroup of the rotation group SO(N). For each t e R we let T' e TA° denote the simultaneous rotation of all spins by means of the rotation that is determined by M(tr1,...,trn). Then f T 1 = T, and (T'XSR satisfies (9.18). (The group (T )teR is isomorphic to IR if and only if there is no s ^ 0 such that (sr1,...,sr„)eZn. Otherwise (T'), E R is isomorphic to IR/Z.) O and (T'), E R satisfy the assumptions (i) and (ii) of Theorem (9.20). Indeed, a straightforward calculation shows that

A 22(x-M(tr1,...,tr„)y)

< max r} dt for all x, y e E. Thus (9.21) follows in the same way as in Example (9.22)(1). Consequently, Theorem (9.20) implies that each p e %(<&) is r-invariant. We have thus proved that the S0(./V)-symmetry of O is not broken, o As a final application of Theorem (9.20) we state a result on the non-existence of Gibbs measures for potentials with dissipative continuous symmetries. This result is a two-dimensional counterpart of Corollary (9.16) and is proved in the very same way. (9.24) Corollary. Let S = Z 2 be the square lattice, (£, S) a standard Borel space, XeJt(E, ê) and (Tf)reR a one-parameter family in T° that satisfies (9.18). Suppose (T')ISR iS dissipative, in that there exists a site ie S and a bounded measurable function f ^ 0 on E such that 1(f) > 0 and lim / o Tfk) = 0 X-a.s. k->oo

for some sequence (t(k))keN in IR. Also, let <5?be a (r')teR-invariant pair potential of the form (9.19) which meets the hypotheses (i) and (ii) of Theorem (9.20). Then <&(<&) is empty. (9.25) Example. Let E = IR and X be Lebesgue measure. Also, let u: E -» [0, co[ be an even twice continuously differentiable function such that X(e~") < co and supu"(x) < co. (For example, we may takew(x) = ß\x\p/(l + xe E

\x\9) with ß > 0 and 0 < p - q ^ 2 ^ p.) We define a potential $ via (9.19) by setting \u(y - x) i f | i - ; | = l, 11 ' (0 otherwise. Here x,yeE and i, j c S. The same argument as in Example (9.17) shows that O is A-admissible. On the other hand, O is invariant under the dissipative one-parameter group (T') [E R of spin translations that are given by T'CO =

182

Absence of symmetry breaking. Non-existence

(coi + t)ieS, t e U, œ e Q. The hypotheses (i) and (ii) of Theorem (9.20) evidently hold. Thus Corollary (9.24) shows that ^(O) = 0. This result is particularly interesting when compared with the behaviour of the discrete Gaussian model which was studied in Section 6.3. Suppose that u(x) = ßx2 for some ß > 0. (This choice of u implies that y° is Gaussian; cf. (13.13). It gives us a model for a system of harmonic oscillators with nearest neighbour coupling.) The resulting potential O is then identical with the potential (6.16) of Section 6.3. But in that section the underlying a priori measure X was counting measure on Z instead of Lebesgue measure. The choice of X has dramatic consequences. Whilst ^(O) = 0 when X is Lebesgue measure, we have |ex^(fl>)| = oo when X is counting measure on Z and ß is large enough; cf. Theorem (6.21) (and Corollary (7.29)). The former result sheds some light on the latter. The restriction to integer valued spins prevents y° from being invariant under the full continuous translation group (T') ( 6 R which cannot be broken in two dimensions. The remaining discrete symmetry group (T')rez °f th e discrete Gaussian model is small enough to admit existence, and large enough to imply (together with the spatial symmetry group 0 ) an interesting phase structure of ^(O). The picture changes when S = Zd for d ^ 3. Then ^(O) ^ 0 even when X is Lebesgue measure. This will be seen in (13.43). o In the rest of this section we shall prove Theorem (9.20). So we assume that its hypotheses hold. We fix a parameter f e R and look at the symmetry T' of $. We will show that O and T' satisfy the conditions of Proposition (9.3). That is, we shall construct, for each square A e Sf, a transformation f e TA° which coincides with T' on A, preserves all spins outside some larger square A, and is such that (9.26)

^ o i + ^ o f

1

-

2Ht

admits an upper bound that does not depend on A and A. It is fairly clear that we shall take a transformation f of the form (9.27)

f = (T 1 '«,ieS),

where t( • ) is a real function on S that equals t on A, gradually decreases outside A, and is identically zero outside some large square A. The point will be to find a proper rate of decrease. The main part of the proof will consist of two lemmas. In the first lemma we shall exploit the smoothness hypothesis (i) of Theorem (9.20) to obtain an upper bound for (9.26). Then we shall use the decay condition (9.21) to estimate that bound when t(-) is suitably chosen. (9.28) Lemma. Consider the setting of Theorem (9.20). Let t(-) be any real function on S, and let f e TA° be given by (9.27). For any A e if, the expression in (9.26) then admits the upper bound

Continuous symmetries in two dimensions

(9.29)

183

J(i,j)(t(i)-t(j))2.

X {ij}r\A*Q

Proof. By definition, the expression in (9.26) is equal to the sum

Here Mx>y) = Vvtt^X'^y)

+
- 2<^(x,j,)

for all x, y e E and i ^ j . Thus, we need only show that

for all i ^ j . This follows from the (T'), e R-invariance of O and Taylor's formula. Indeed, let i ^ j be given and write s = t(j) — t(i). Then ^iM,y) = (Pij(x,T?y) + (p^x,ij'y)

- 2<prfx,y)

2

d \u\)-^(pij{x,x]y)du

= 1 (s S

< J(iJ)

J (s-

\u\)du

= J(i,j)s2 for all x,yeE.

This proves the lemma,

o

Lemma (9.28) leaves us with the problem of defining t(-) in such a way that t(-) = t on a given A, t(-) = 0 outside some A, and (9.29) remains bounded. This will be achieved by means of the following key quantities. We define

(9 30)

-

««-{JHO,*)-

Uli,

and for each L ^ 1 (9.31)

Q(L) = £ q(/c)

and fl (9.32)

r(«,i)

ifn<0, X

<ï(fc)/Q(L)

ifO
n
0

if n > L.

For each L ^ 1, (r(n,L))„ ai is a decreasing sequence. Its rate of decrease is precisely adjusted to the decay condition (9.21), as will become evident from the next lemma.

184

Absence of symmetry breaking. Non-existence

We shall work with the maximum norm on S. So we put ||i|| = 1^1 v \i2\ when i = (i1,i2) e S. (9.33) Lemma. Let N ^ 1 and C > 0 be given. Under the decay condition (9.21) on J there exists an integer L ^ 1 such that the following conclusion holds. If A = {Il il ^ N + L) and t(-) = r(|| • || - N, L) then the sum at (9.29) is at most C. Proof. 1) We start with an estimate for \t(i) - t(j)\. Clearly, t(i) = t(j) whenever ||i|| = \\j\\ or ||i|| ^ N and \\j\\ ^ N. So we may assume that \\j\\ > || i|| v N. In this case we have 0 S t(i) - t(j) =

X

q(k-

N)/Q(L).

Afv||«||
As q(-) is decreasing and \\j\\ — \\i\\ ^ \\j — i||, the last sum is at most \\j - i\\q(\\i\\ - N)/Q(L). Also for the same reasons it is not larger than h

fiV)/Q(L)

=

Q(\\j-i\\)/Q(L).

(=1

2) Next we observe that log log L < Q(L) < 1 + logL for all L ^ 2. Indeed, L

dx Q(L) > f — = log log L - log log 2 > log log L 2 xlogx and L

dx

2(L)<1 + J — = l + l o g L . 1

X

3) For every 0 < p < 2 there is a constant K(p) < oo such that

I j(u)ii;-;ir^K(p) for all i e S. This follows from (9.21). To prove this we write the preceding sum in the form I

J(i,j)\\j-i\\2+(p-2).

I k

fcgO jeS:2 g|U'-i||<2

k +1

As ||j|| ^ | j | ^ 2|| j | | for all j , the above expression is less than

X 2^"2> kgO

J(i,;)l;-»l 2

X |j-i|g2

k +2

^ £ 2 , t ( ^ 2 ) Klog2 , t + 2 = Klog2 X (fc + 2)2- ( 2 - p , k kgO

£ K(p) < oo.

Continuous symmetries in two dimensions

185

4) After the preceding preparations we are ready to consider the sum (9.29). We split it into three partial sums, each of which will be shown to converge to zero as L tends to infinity. The first partial sum (which will be denoted by Z J extends over all i, j e S with ||i|| ^ JV < \\j\\. The second sum, Z 2 , comprises all pairs {i,j} which satisfy JV < ||i|| ^ N + L, \\j\\ > \\i\\, and \\j — 'II ^ (ll'll ~ N) 2 . The third one, E 3 , runs over all remaining pairs {i,j} with £(0 # t(j). These are characterized by the properties JV < ||/|| ^ N + L, yil>lli||,andy-i||>(||i||-iv)2Applying the estimates in the preceding steps 1), 2), and 3) successively we see that Z ^

J(UJM\\J-i\\)2/Q(L)2

£ \\i\\SN<\\j\\

^Q{LT2

I

J(i,j)(l+log||;-«||)2

I

^2ß(L)" 2 £

£

J(Uj)\\j-i\\

PUSN j ' e S 2

^ 2(2JV + l) K(l)ß(L)^ 2 . Because of 2), the last expression tends to zero as L tends to infinity. Next, 1) and (9.21) yield £2^

I

^

J(i,j)\\j-i\\2q(\\i\\-N)2/Q(L)2

I

N<\\i\\^N+L |lJ-it|S(||i||-^) 2 Q(L)-2 £ 9 (||i|| N<||i||SN+L

= KQ(L)-2

- JV) 2 Klog[2(||;|| - JV)2]

£ 8(JV + fe)g(/c)2log(2/c2) k=l

^ 8(JV + l)Kß(L)- 2 t

kq(k)2log(2k2)

k= l

We have used the fact that |{||-|| = l}\ = 8*f for all { ^ 1. As fclog(2fc2) ^ 3/q(k) for all k ^ 1, the last sum is less than 3ß(L). Thus L 2 ^ 24(JV + 1)K/Q(L) -* 0 as L -» oo. Finally, 1), 2), and 3) also give

£3^

I

^(U)ß(ll./-iH)7ß(L)2

I

N<||i||SJV + L !|j-<||>(||i||-N)

2

^ ß(L)- 2 X 8(JV +fe)sup k=l

ieS

^ 8(JV + l)ß(L)" 2 X fe sup k=l

-J(U)(i + log II; - 'ID2

X 2

\\j-i\\>k

AUJ) 1611; - 'II1/4

X

i e S ||j-i||>fc

2

186

Absence of symmetry breaking. Non-existence

S 21(N + l)Q(L)~2 t

k 2 SU

~

k=l

P

isS

Z

||j-ij|>k2 2

J(UMj - '117'4

^ 27(iV + l)K(7/4)ß(L)" 2 X k~ k>l

as L -> oo. Consequently, if L is sufficiently large then Ej + E 2 + E 3 ^ C. This proves the lemma. • It is now easy to complete the proof of Theorem (9.20). Proof of Theorem (9.20). We need to show that each x' preserves each ß £ ^(<ï>). This will be achieved by checking that T' and $ satisfy the assumptions of Proposition (9.3). To this end we can and will assume that t = 1. Let A £ y be given and N ^ 1 be so large that A a {|| • || ^ N}. We put C = 1. For this choice of iV and C we select an integer L according to Lemma (9.33), and we let A = {|| • || ^ N + L] and

x=

{x[^-N'L\ieS).

Clearly, f e TA°. From (9.32) we immediately see that f is a localization oft 1 in the sense of condition (i) of Proposition (9.3). We further deduce from Lemma (9.28) and Lemma (9.33) that the expression at (9.26) is not larger than 1. Thus condition (ii) of Proposition (9.3) also holds, with the constants c = C = 1/2. We therefore conclude that each ß e $(<&) is T1-invariant, D We conclude this chapter with two comments on the validity of Lemma (9.33) in dimensions d =£ 2. (9.34) Comments. (1) As we have mentioned before, a continuous symmetry breaking does occur among short-range systems in three or more dimensions. This will be proved in Chapter 20. Nevertheless, it is interesting to see how Lemma (9.33) fails in that case. So let S = Zd with d ^ 3, and suppose J is such that J(i,j)^. 1 for all nearest-neighbour pairs {i,j}, and J(i,j) = 0 whenever \i — j \ =£ 1. Such a J satisfies any (spatially homogeneous) decay condition. Let N ^ 1, rN = £ /T 2 , and t(-) = f*(||i|) be any real function k^N

on S that only depends on the maximum norm ||-|| on S, equals 1 on {II'II ^ N}, and vanishes outside some large cube A = {||-|| ^ N + L}. For such a f(-) and A the sum at (9.29) is at least 2d/rN. To see this we note first that | {Hi = k}\ ^ 2dkd~i ^ 2dk2 for all k ^ 0. Combining this and the Cauchy-Schwarz inequality we obtain

J&mm-tu))2

rN x {i,ijnA#0

£ 2drN £ k2(t*(k) - t*(k + l)) 2 k=N

Continuous symmetries in two dimensions

187

£ 2 d T | Lfe_1fc(t*(fc)- t*(fc + 1)) J = 2d(t*(N) - t*(N + L + l)) 2 = 2d. Since r^ -> 0 as iV ->• oo we conclude that in three or more dimensions it is impossible to choose a function t( • ) in such a way that (9.29) becomes bounded uniformly in N. (2) The proof of Theorem (9.20) does extend to one-dimensional systems. In fact, the main idea of the proof can be adapted to prove the absence of a continuous symmetry breaking for certain one-dimensional systems which are not covered by Theorem (9.11). To be precise let us assume that S = Z and that all other hypotheses of Theorem (9.20) hold, except that (9.21) is replaced by the condition that (9.35)

\i-j\2J(Uj)SKn

£ jeZ:0<\i-j\£n

for all i e Z and n ^ 1. (Note that this condition includes the quadratic falloff J(iJ) = \i — j \ ~ 2 which is just beyond the decay condition (9.12).) Then the conclusion of Theorem (9.20) is still valid. Since Lemma (9.28) neither depends on the dimension nor on the decay properties of J, the claim above will be proved once we have established a counterpart to Lemma (9.33). To this end we redefine q(k) by putting q(k) = 1 for all k, instead of (9.30). (9.31) and (9.32) remain unchanged. Let N ^ 1 be given, L = N, A = {|-| S 2N}, and t(-) = r(|-| - N,N). The sum at (9.29) can then be estimated as follows. As before, we split this sum into three parts Z 1 ; Z 2 , and S 3 . Di extends over all i, j eZ with |i| ^ N and \j\> 2N. Z 2 takes account of all pairs i, j eZ with N <\i\ <2N and |/| < N or | j \ > 2N, and Z 3 comprises all pairs i, j with N < |i\ < | j \ <, 2N. To estimate these partial sums we first conclude from (9.35) that (9.36)

n

X

J(UJ)£*K

jeZ:|i-j|än

for all i e Z and n ^ 1. Indeed, choosing an integer k ^ 0 with 2k ^ n ^ 2k+1 we see that the left side of (9.36) is at most 2t+1

I

L/-i|ä2 k

J{i,j)^2Y,2-m2-k-m mSO

^ 4K £

Y

2k +m g | j - i | < 2 k + m + 1

\i~j\2J(i,j)

2~m = 8JC.

(By the way, (9.36) also implies (9.35) with a different constant K.) Using (9.36), we obtain the estimate Zx S I

I

WSJV |>-i|>N

./(U) ^ (2JV + l)8K/N S 24K.

188

Absence of symmetry breaking. Non-existence

Similarly,

Z N<\i\S2N

Z, =N~

X J(UM\i\-N)2+

N

Z

k2 X J(i,j) + (N-k)2

2

s N~ x Z

\j-i\Zk

k = l i=±(JV + k)

J(i,j)(2N-\i\)2

X •'(U) \j-i\^N-k

< 16K. Finally, S 3 = iV-2

X

J(UJ)\\i\-\J\\2

N<\i\<\j\£2N

^N~2±

X *• N<]i\<2N

Z

J(i,j)\i-j\2

|j-i|g4JV

because of (9.35). Consequently, the expression in (9.29) is less than or equal to C = 44K. This result provides us with a counterpart of Lemma (9.33) and thus proves the claim that there is no continuous symmetry breaking in one-dimensional systems which satisfy the hypotheses of Theorem (9.20) with (9.35) instead of (9.21). o

Part II Markov chains and Gauss fields as Gibbs measures

As its title suggests, the main development of this part centres around the probabilistic key words Markov chains and Gauss fields. Of course, we shall consider these classical subjects from a Gibbsian point of view. That is, we shall be concerned with specifications y which are such that ^(y) contains a Markov chain or a Gauss field, respectively. The extreme boundary ex g(y) then consists only of Markov chains resp. Gauss fields, and this fact will enable us to analyse &(y) by Markovian resp. Gaussian techniques. Needless to say, the motivation for these investigations comes primarily from Probability Theory rather than from Statistical Mechanics. Chapters 10 and 11 deal with the case where y is a Markov specification with infinite state space and parameter set Z, the integers. The topics which are discussed there include the identification of the Markov chains in g(y), the existence and uniqueness of a shift-invariant Gibbs measure for y, and the non-existence of Gibbs measures for y. Some explicitly solvable examples of phase transition are also provided. Chapter 12 is devoted to Markov specifications and Markov chains on infinite trees. In particular, this chapter contains an analysis of the phase transition region of the Ising model on a Cayley tree. The subjects discussed in Chapter 13, the last chapter of this part, are Gaussian random fields and specifications. For a Gaussian specification y, the extreme boundary of &(y) consists of Gaussian random fields which can be described explicitly. The main problem is that of finding effective criteria on y for which ^(y) is non-empty. This problem admits a satisfactory solution when y has finite range or S = Zd and y is shift-invariant. Whilst Chapter 11 depends on Chapter 10, Chapter 12 can be read independently of Chapters 10 and 11. Likewise, Chapter 13 is independent of the Markov chapters 10 to 12. The only chapters of Part I which are indispensable in order to understand this part are Chapters 1,2, and 7. With a few exceptions which can be read when the need arises, the results of this part will not be used in Parts III and IV.

Chapter 10 Markov fields on the integers I

In this and the next chapter we shall be concerned with random fields which have parameter set S = Z, the integers. In this chapter, the state space will be any measurable space (E, S), which is endowed with an a priori measure X e Ji{E, S). In Chapter 11, £ will be countable. We shall look at ^-specifications y = pX_ which are Markovian in the following sense: For each interval A = {i, i + 1,..., k] in Z and each A e tFA, yA(A\a>) depends on co via c o ^ and a>k+1 only. This condition is clearly satisfied whenever y is Gibbsian for some nearest-neighbour potential. If y is Markovian then each \i e &(y) exhibits the property that KA\fA)

= ii(A\&{i-i,k+i))

|*-a.s.

whenever A and A are as above. This property is called the two-sided Markov property, and each /i that enjoys this property is called a Markov field. It is natural to ask how the two-sided Markov property is related to the familiar Markov property of Markov chains. The latter states that ^04|^{...,;-2,i-i}) = li(A\&{t-i})

|*-a.s.

for all i G Z and A e &{i
Two-sided and one-sided Markov property 10.1

191

Two-sided and one-sided Markov property

Throughout this chapter we set S = Z. We fix an arbitrary measurable space (£, S) and an a priori measure A on (£, S). As far as A-specifications are concerned, Remark (1.28)(3) asserts that there is no loss in assuming that A e Sf{E, S). We shall do so throughout the proofs of this chapter. Since S = Z, we shall often work with intervals in Z. It will therefore be convenient to also use the standard interval notations for lattice intervals in Z, in that (10.1)

V,kl={jeZ:i<j
{jeZ:i<j^k}

\_i,k-] =

{jeZ:iSjSk}.

Here — oo ^ i < k ^ oo. The following definition is basic to this chapter. (10.2) Definition. A specification y on Z is said to be a Markov specification if 7]<,k[(^l ') i s ^{j, ^-measurable for all A e ^ i i k [ and all i,keZ with i + 1 < k. A A-modification p is called Markovian if p]Uk[ is #[ ; k]-measurable whenever i,keZ are such that i + 1 < k. Clearly, a A-specification y = pA. is Markovian whenever p is Markovian. p in turn is Markovian whenever p = p® for some A-admissible nearestneighbour potential , and Corollary (2.32) provides a converse of this assertion: Each positive Markovian pre-modification is Gibbsian for some nearest-neighbour potential. What about Markovian A-modifications p which are not necessarily positive? Such p's are constructed in the example below. The details are rather lengthy and should be skipped at a first reading. (10.3) Example. Let A e Jl{E, S) be an a priori measure, and suppose that for each j e Z w e are given a measurable function py: E x E -> [0, oo [ such that Pj(x, • ) is a probability density with respect to A for all x e E. We write k

9iA°>)=

n Pj(<»j-i><»]) j=i+l

whenever i < k and coeQ, and we assume that Zik = X]iMgik < oo for all i < k. (Since A]i>ktöf£ik_1 = 1, the latter holds whenever all p/s are bounded.) We define a family p = (p A ) A e ^ as follows. For any finite interval A = ]i, /c[ we put = ] k[

''

\9i,Jzuk \di,k-i

ifZu>0, otherwise.

If A is the union of pairwise disjoint and non-adjacent intervals 7 l 5 . . . , IN, we put

192

Markov fields on the integers I N

PA

= EI Pi„n=l

(This construction generalizes that of Comment (3.8)(1). In fact, if pj > 0 for all j then p is the Gibbsian specification for the nearest-neighbour potential -i»o>) i f A = U \ 0 otherwise.)

hj},

We claim that p is a Markovian A-modification. Indeed, it is evident that p is Markovian and AApA = 1 for all A e ^ . We need only check the consistency of the pA's. We shall do this by verifying condition (c) of Proposition (1.30). So let A , A e ^ with A c A and a e Q b e given. We must show that J AA(dC|o>) J AA(d^|co)|pA(C)pA(^) -

PMPA(0\

=0

for AA\A(- |a)-almost all co. We only do this when A and À are intervals. The extension to general A and A is straightforward. So let m S i < k S n, A = ]i,/c[, and A = ]m,n[. We distinguish the following six cases for co: (i) ZmJco) > 0, Z u (eo) > 0, (ii) k < n, ZmJco) = 0 < ZtJco), (iii) fe = n, Z m ,„M = Ziik(œ) = 0, (iv) Zm>n(co) > 0 = ZLk{co), (v) k < n, ZmJco) = Z i,k(œ) = 0' (yi) ^ = n, Zmi„(o)) = 0 < Ziifc(co). In the first case we have PA(0PA(>?)

= ö'm,iMö't,nMö'i,k(0ö'!-,k(»7)/-Zm,nMZi,t(co) =

PMPÀQ

for all Ç,rçe Q with CS\A = 1s\\ = % \ A - The cases (ii) and (iii) are similar. In case (iv) we have PA(0 = 9mA(O)9i.k(0gkA(°yzm,«((O) = ° for AA(- |co)-almost all Ç because Zitk(co) = AA{gUk\co) = 0. Thus pA(£)pA(>?) = 0 = pA(rç)pA(Ç) for 1A(- |co)2-almost all (Ç,rj). Case (v) is similar. To treat case (vi) we first note that ^AV\(l{Z m „ = 0}^A9m,n)

= l{Z m „ = 0}Zm,n

= 0-

Thus AA(gm „ > 0|co) = 0 for AA\A(- |a)-almost all co e {Zmn = 0 < Zu„}. But for these co we have PA(0PA(>?)

= ö'm,;Mö'i,n-i(0ö'i,n('?)/Zi,nM = 0

for 1A(- |co)2-almost all (Ç,rç).This completes the proof that p is a Markovian 1-modification. o Having introduced the notion of a Markov specification we recall the notion of a (possibly inhomogeneous) Markov chain. In contrast to common usage,

Two-sided and one-sided Markov property

193

we shall use the term Markov chain for a probability measure on (E, S)z rather than a sequence of random variables on a suitable probability space. Of course, the former is to be thought of as the joint distribution (or "canonical version") of the latter. (10.4) Definition. Let (Pj);eZ be a family of probability kernels from S to S, A probability measure \i on (Q, J*) = (E,
fi(ai+J e Aj for all 0 ^ j ^ n) = 1 ^(A0(dxo) J ^ ( x o . d x j . . . J P i + ^ x ^ ^ d x J . A0

Ai

A„

(ii) For all i e Z and A e S, »fa

e

^l^-oo,i[) = Ptfa-^A)

M-a.s..

If all P;'s are equal to some P, p is called simply a Markov chain for P. Definitions (10.2) and (10.4) involve two different kinds of Markov property. Suppose y is a Markov specification and \x e ^(y). Then KA\^i,kl)

= y]Uk[(A\-)

/i-a.s.

for all i,keZ with i + 1 < k and A e i*j U [ . Taking conditional expectations with respect to !F{Uk} we obtain n(A\&rn,k}) = y]i,kl(A\-)

ju-a.s.

and therefore (10.6)

KAWun)

= ii(A\&{itk})

/i-a.s.

for all i, keZ with i + 1 < k and all A e J^i?k[. Property (10.6) is called the two-sided Markov property, and each \x which satisfies this property is called a Markov field. On the other hand, if n is a Markov chain then (10.4)(ii) implies the following left-sided Markov property: If i e Z and ^ e ^ i j 0 0 [ then (10.7)

KA^-^n)

= KAl&w)

/i-a.s..

It is well-known that the left-sided Markov property is equivalent to the right-sided Markov property which states that (10.8)

KAl&u,^)

= n(A\P{l])

H-SL.S.

for all ieZandAe ^_ 0 0 j i [ . (10.7) and (10.8) together are therefore called the one-sided Markov property. (To derive (10.8) from (10.7) let B e J 2 ^ and C e #j, t. Then

194

Markov fields on the integers I

n{Ar\Bnc) = p(iAnBp(c\&{i])) = =

fi(fi(A\^{i]nBfi(c\^{i])) li{li{A\&{{])\Bnc).

(10.7) entered into the first equality.) Recall that two further Markov properties were already discussed in (8.24): the global and the local Markov property. In fact, the two-sided and the one-sided Markov property are nothing other than the specialization to the simple graph Z of the local and global Markov property, respectively. This is shown in the remarks below. For each K c Z w e put dV = {i e Z\V: \i - j \ = 1

for some ; e V).

(10.9) Remarks. (1) The two-sided Markov property is equivalent to the local Markov property on Z which asserts that (10.10)

n(A\^rv) = n(A\^dv)

/i-a.s. for all Ae&Y

whenever V cz Z is finite. (2) The one-sided Markov property is equivalent to the global Markov property on Z which is defined by the requirement that (10.10) holds for all V c Z. (3) The one-sided Markov property implies the two-sided Markov property. Proof. 1) Suppose V, W <= Z are such that VD(WüdW) = 0 and (10.10) holds for both V and W. Then (10.10) holds for VUW.To prove this let B e &Y. C e J v , and A = BC\C. For any D e J KUff we have H(ADD) = =

n(n(B\^dvncr]D) H(n(B\Pgr)p(C\PdwnD)

and therefore p(A\#yUW) = n{B\^dv)n(C\^dw)

/x-a.s. .

Since d(V(J W) = dV(J dW, we conclude that ß{A\^vuw)

= ß{A\^d(VUW))

/i-a.s..

By a standard argument, the last identity extends to all A e JVuw2) According to Step 1) above, equation (10.10) holds for any finite union of non-adjacent finite intervals whenever it holds for finite intervals. This proves Remark (1). 3) Remark (3) follows directly from Remarks (1) and (2), but we shall prove (3) before (2). Suppose /i satisfies the one-sided Markov property, and let i + 1 < k, A e &v
Two-sided and one-sided Markov property

195

ß(c, n B, n A n B2 n c 2 ) = M i c ^ r w ^ M Q i J^ } )) = MMC1|^'{£})iBl/iMl^'{,.t})iB2/x(c2|^{t})) = /^(lcInji1^(^l^{i.t})1Ji2nc2)The first equality follows from (10.7), the second from (10.8), and the last from both (10.7) and (10.8). This proves (10.6) and thereby Remark (3). 4) To prove Remark (2) we first note that the one-sided Markov property simply means that (10.10) holds for any infinite interval V. By Steps 3) and 1) above, this implies (10.10) for every finite union of non-adjacent finite or infinite intervals. If V is an infinite union of arbitrary non-adjacent intervals, we let W be a finite subunion and A e !FW. Since dW cz dV and (10.10) holds for A and W, (10.10) also holds for A and V. A standard extension argument then gives (10.10) for V and all A e J'y. a For our purposes, the main point of the preceding remark is assertion (3): Every Markov chain is a Markov field. The converse does not hold in general. This is fairly clear from a comparison of assertions (1) and (2). In fact, we shall provide a counterexample in (11.33). We are thus led to ask if the converse holds, at least, under suitable conditions. The next section is devoted to this question. We conclude this section with a more explicit version of Remark (10.9)(3). (10.11) Example. Suppose n is a Markov chain for transition kernels (P;);6z, where P,(x, •) = p;(x, -)X for all x and i and the p/s satisfy the hypotheses of Example (10.3). Define p as in that example, and let y = pX.. Then ß e
J

d/x y,i,t[(ojj,t[ e B\ •)

whenever A,C eS,Be ê]i-K, and i + 1 < k. But using the notation of Example (10.3) we obtain from (10.5) ß{aieA,a]Uk[eB,akeC) =

J

n(dœ)^(dy)\{Zik(azx{k]y)>0]

{<7i
=

J

C

=

B

/i(dtö) J Â(dy)Zik(œzx{k]y)y]

{ateA}

$ *]i'k[(dQgao>z\v,k](y) e B\cozx{k}y)

C

J

n(dœ)\(Pi+i...Pk)(œi,dy)y]

{"it A]

1

C

v(dco)y]Uk[{<j]iMeB\œ).

{
This shows that ß is specified by y.

o

£B\uj_\{k]y)

196

Markov fields on the integers I

10.2

Markov fields which are Markov chains

Let p be a Markovian A-modification and y = pXm. We pose the following question: Are there any elements of <&(y) which have the one-sided Markov property? There is a natural approach to this problem. For given \i e @(y), we shall try to define transition kernels P; by the formula Pi(fi-i,A) = lim 7[iit[(ff, e A\ •)

/i-a.s..

Here i e Z and A e S. Of course, this is assuming that the limit on the right only depends on ai_1 /i-almost surely. Since this limit is measurable with respect to f] ^{i-i}u[k,œ[> this amounts to the requirement that there is no k>i

influence of the right tail f] # j t

œ[.

k>i

To be more specific we define (10.12)

p{Uk{ =

À]umj]p]itk[

whenever i < j < k. Since p is Markovian, pjik[ is measurable with respect to ^{ij,k]. Clearly, 7u.t[(o/e i4|eo) = j"

À(dx)p]iiM(xœ^{j])

A

for all A e S and œ eCi. Next, we fix any \i e <&(y) and put (10.13)

Vj = nX{]]

(jeZ).

(At this point it is convenient to assume without loss that A e 8?{E, ê).) Since

yw = P W % we have (10.14)

p{j}Vj = ßy{j] = ß.

On the other hand, we also have (10.15)

n = pjUk[Vj

on ^ . t [ \ W .

Indeed, let / be bounded and measurable with respect to &]i,k[\{j}- Then Vj(pjuk[f) = vA(h,k[\{j}Pn,k[)f) = vjh,k[\{j}(Pv,k[f) = Mi.*[(Pn.*[/)

=

^7n.t[(/) = M/)-

From (10.14) and (10.15) we conclude that (10.16)

pfuki = Vj(pu]\^]iM\U})

v,-a.s..

Taking conditional expectations with respect to ^{ij}u[k,œ[ w e a l s o obtain (10.17)

pik[ = Vj(p{J}\^{ijnKoo[)

v r a.s..

Consequently, for any two fixed integers i, j with i < j the sequence (p^k[)k>j

Markov fields which are Markov chains

197

is a backward martingale with respect to Vj. This martingale converges v,almost surely and in L^v^-norm towards a limit p{ij

(10.18) pioo[ = v M ; } i n ^ W k>j

= vMj}\f]

k>j

&r{uMk.aDi)

v r a.s..

Now take i = j — 1 and suppose that p/j_ 1]00[ does not depend on the right tail, in that (10.19)

v a s

pjj_Uao[ = Vj(P{J}|JVw))

r--

f o r a11

3e z-

Under this condition, n is a Markov chain. This is ensured by the lemma below. (10.20) Lemma. Let p be a Markovian X-modification, and suppose fi e ^(pX) satisfies (10.19). Then there exists a family (p,) jeZ of measurable functions pf. E x E -> [0, oo [ with the properties below. (0 Pj(
on E x S is a probability kernel. (Hi) ß is a Markov chain for (Pj)jeZ. Proof. Without loss we assume that X e 0>(E, S). For given j e Z we let q/. E x E -> [0, oo [ be any measurable function such that v

j(P{j]I^{j-uj}) = aMj-i, °))

v a

r -s- •

Define J

'

\l

otherwise.

Pj clearly satisfies (ii). Also, for each B e ^{j-X} we have j vj{dœ)À(qj(œJ_1, •)) = J /z(dco) J A(dx)q.(co,_t,x) B

B

= Jdv,.q>,._1,<7J.) = f dv,.p w B

B

= ß(B) = v,.(B) and therefore kiq^co^y, •)) = I for v,-almost all a>. Thus pj(aj_l,aJ) = q/oy-i.CT,-) v,-a.s., and this gives (i). Finally, (10.18) and (10.19) imply that

198

Markov fields on the integers I

k>j

and therefore Pj(°>-i» °j) = vj(P{j)l-^i-oo.j]) v r a.s.. Consequently, \{ A e S and B e 3F^_m^ then j d/z P/oy.!,^) =• v,(lBn{«,,e^}Pj(oy-i,oy)) B =

V/(lBn{ffj.e^}P{j})

= /z(Bn{o > e^}). This gives property (10.4) (ii) and completes the proof of the lemma,

o

By virtue of the lemma above, the problem of establishing the one-sided Markov property has been reduced to the problem of finding sufficient conditions for (10.19). We shall discuss two such conditions, namely (I) extremality of p in @(y), and (II) shift-invariantce of p, combined with an irreducibility condition on y. Case (I) is treated in the theorem below. (10.21) Theorem. Let p be a Markovian À-modification on Z and y = pX_. Then each peex <&(y) is a Markov chain for suitable transition kernels Pj of the form Pj(x, •) = pj(x, -)X. Here j e Z, x e E, and pj is a measurable nonnegative function on E x E. Proof. We verify (10.19). Let j e Z be fixed. We write A = {j — 1, j}, and we let / be any nonnegative function on Q which is measurable with respect to 0 "^AU[k, » [• We define a function g on EA by 9(0 = j p(dœ)f(ÇœZXA)

(C e EA).

We will show that / = g(aA) y,-almost surely. Applying this to / = Py-i>œ[ and using (10.18) we shall arrive at (10.19). We fix any Ç e EA. Since fis measurable with respect to pj ^AU[k m[, Fubini's k>j

theorem implies that /(£o-Z\A) is measurable with respect to the right tail f] 3Flk œ[ . By Theorem (7.7)(a), p. is trivial on ST and, a fortiori, on the right k>j

tail. Thus /(Ço-Z\A) is constant /^-almost surely, and therefore /(ÇOZ\A) = 9(0 p-a.s.. Since ( was arbitrary, we conclude from Fubini's theorem that XA x p((Ç, co)eEAx

Q: f(tcoZXA) ± g(Q) = 0.

This can be rewritten as X x Vj_t((x,co) E E x Q: f(xcozxlj}) # g(co^xx)) = 0. Because of (10.14), p. is absolutely continuous with respect to y,^. Thus y,--!

Markov fields which are Markov chains

199

may be replaced by p in the last equation, and this gives vj(f # # K ) ) = ^ x H((x, a>) 6 E x Q: /(xcoZNW) # ^(eo^x)) = 0. The proof is thus complete.

•

We note in passing that the preceding proof has shown that ^\ ~ fl ^Vj[k,oo[ k>j

whenever f] #" [t

œ[

/i-a.s.

is trivial and A = {j — 1, j}. The local absolute continuity

k>j

of p with respect to Xs is an essential prerequisite for this to hold. In general, the preceding statement fails. This problem has been discussed by von Weizsäcker (1983). One might suspect that Theorem (10.21) has a converse, that each Markov chain in &(y) was extreme in @(y). This is not the case, as we shall see in (11.33) and (11.39). Combining Theorems (7.26) and (10.21) we obtain the following corollary. (10.22) Corollary. Suppose (E, S) is a standard Borel space, and let p be a Markovian A-modification and y = pA_. If &(y) ^ 0 then &(y) contains a Markov chain. Turning to the above-mentioned case (II), we shall now assume that p is homogeneous (i.e., shift-invariant). We shall also need that p is irreducible in the sense below. (10.23) Definition. Let p be a homogeneous Markovian ^-modification. We say that p is irreducible if for each N ^ 1 there exists a set CN e S, an integer n(N) ^ 1, and a measurable function hN: £-> [0, oo[ such that CN1E as iV t oo, A(hN) > 0 for sufficiently large N, and P?-n(N)MN)[(«>) ^

K(C00)

whenever coeQ and N ^ 1 are such that co_n(N) e CN and œ„(N) e CN. The following examples should give some feeling for the condition of irreducibility. (10.24) Examples. (1) Let X e Jt(E, S) and 0> be a A-admissible shift-invariant nearest-neighbour potential. We may assume without loss that
Suppose that

j « ^ + (%} + 0>{i+1})/2 [0

if A = {/, i + 1}, otherwise.)

200

Markov fields on the integers I C = s u p Zf0}(co) < oo meiï

and cN^

sup

0 {0>1} (to)< co

co:coo,coi e CN

for some sequence CN ] E. Then p * is irreducible. For we can just take n{N) = 1 and hN = l c „e- 2c Vc (2) Let £ be a countable set and À be counting measure. Also, let P be a stochastic matrix on £. Define p in terms of P by putting pj(x, y) — P(x, y) in Example (10.3). Thus k

P];,k[M =

n

fK-i,fflj)

'^"'(to,-.^)

Lj=i+1

whenever P k '(GO,-, eoj > 0, co e Q, and i + 1 < k. In particular, we have for all n ^ 1 and to e Œ with P2n(co_n,co„) > 0. We claim that p is irreducible whenever P is aperiodic and irreducible in the usual sense. (See, for example, Chapter 7 of Breiman (1968).) Indeed, it is well-known that such a P enjoys the following property: For all x, y e E there exists an integer n(x, y) ^ 1 such that P"(x, y) > 0 for all n 5: n(x, y). Therefore we can take any sequence (Qv)jväi of finite subsets of £ with CN1E, together with n(N) = max n(x,y) x eC and -» " hN=lcN

min

Pnm(x,y)P"m(y,z)/P2nm(x,z).

o

x,y,zeC„

We now state the theorem which deals with case (II) above. Recall from (5.2)(1) and (5.13) that ^&(y) stands for the set of all shift-invariant elements of ^(y). (10.25) Theorem. Let (£, S) be a standard Borel space, X e Ji{E, S) an a priori measure, p an irreducible homogeneous Markovian À-modification on Z, and y = p/L Under these conditions, each ß e ^&(y) is a Markov chain for some transition kernel P satisfying P(x, •) = p(x, -)X for all x e £ and some measurable function p: E x £ -> [0, oo[. To prove this theorem we shall verify (10.19). A key to accomplishing this is provided by the proposition below. (10.26) Proposition. Suppose the hypotheses of Theorem (10.25) are satisfied, and define v = /J.À^0y Let {fk)k^i be a sequence of v-integrable functions which exhibit the following properties:

Markov fields which are Markov chains

201

(i) As k -> oo, fk tends in Ll{v)-norm towards some limit / œ . (ii) There is a set A e tf containing 0 such that fk is measurable with respect to ^&m (resp. J*Äu{-k}) far all k ^ 1. Then / œ is measurable with respect to J ^ v-almost surely. Proof. We assume without loss that X e &{E, ) of the conditional probability v(A\^A)(a>), where A e STA and a> e Q. We extend n to a proper probability kernel from J ^ to J^ in the natural way, and we define a probability measure v on (Q x Q, 3F x J^) by v = \ v(dco) n(-\œ) x 7i(- \CO). Clearly, v(A x Q) = v(Q x A) — v(A) for all A e #". We also have v((U) e Q 2 : CA = VA) è 1 v(dco)7r(cxA = cojco)2 = 1 because 7i is proper. We intend to show that (10.27)

lim \ v(d£ d»/)|/ t (0 - fM\

= 0-

In fact, combining (10.27) with assumption (i) and the inequality v(l/k - nfk\) = J v(dco) J 7T(dC|co)lJ n(dr,\œ)Uk(0 - /*(>?)] I

SlmCdnMiO-fMl we obtain that lim v(|/ œ - nfk\) = 0, t-»00

and this proves the proposition because each nfk is J^-measurable. The proof of (10.27) is broken up into three steps. Step 1. For given Ç, n e Q, x e E, k e Z and n ^ l w e define 0 =

P]k-n,k+n[(XCz\{k}%

(p*(x,C,i0 = (p„k(x,C) A ç>*(x,»/), and %k(C,>7) = « ( - , U ) ) . We claim that (10.28)

lim \ v(dC,dr,)cpnk-%,r,)\fk(0 - fM\

= 0

for all n ^ 1. Let /c be so large that [k - In, k] D A = 0. For all x e £ and Ç1, C2 e Q with Ci = CA w e then have

202

Markov fields on the integers I

\W)~W)\^t

i= l

\W)-fk-n{
Therefore the integral in (10.28) is at most Jv(dÇ\dÇ 2 )

ti*(Ax)ti)\fk(ti)-fk-n(xtiz\{k-n})\ i= l

= 2 J v(dC) J A(dx)^-(x,C)|(/ t - /t-,,)(xCz\{*-B})l = 2 vA{k_„}(p]([_"2nik[|_/^ — A-«l) = 2 V/l]t_2„?„[(P]t-2„,k[IA

—

A-nl)

= 2vy lik _ 2BiB[ (|/ k -/ t _J) = 2v(|/t-/t_J). In the above, the first equality comes from the fact that fk(co) does not depend on C0fc_„. The third equality depends on (10.12) and the fact that \fk — fk_„\ is measurable with respect to ^k-2n,ki\{k-n}- The final equality is justified as follows. Let g be any bounded measurable function and A = ]/c — 2n, /c[. Then vyA(g) = A**AU{O}(PA0) = A**A(PA*{O}0) = WA(^{O}ÔO =

v{g)

because pA is measurable with respect to J\UAA which is contained in &[0}The proof of(10.28) is completed by noting that lim v(\fk — / k _„|) = Obecause of assumption (i). *'^00 Step 2. For all e > 0 there is a number ô > 0 and an integer n ^ 1 such that sup v{q>* < ô) < e. |k|>«

This comes from the irreducibility of p. For let CN, hN and n(N) be as in Definition (10.23), and let a = 0 and 2<x{CN) — 1 > ^ / l — e. We put c5 = A(fejv) and n = n(N). For any fc e Z we then have { K , i f ) e ß 2 : i ; t . . , | ; t + „ i i n . „ ^ + , e Q } c {<äf ^ <5}. This is because $*(-,£,?/) ^ hN whenever (Ç,rj) belongs to the set on the left. Thus V(9n

^

Ô

) ^ J v ( d 0 > ) 7 r ( ( T t _ „ , (T t + „ 6 C w | o > ) 2

^vn(ak_n,ak+neCN)2 =

But if \k\ > nthen

v

(°Jt- n ,°Jt + n

e

Cjv)2-

Markov fields which are Markov chains v

203

K - n > ak+n e CN) = fj.(ok-„, ak+n e CN)

^ Mo*-« e Q ) + n(ak+n e CN) - 1 = 2«(CW) - 1 This completes Step 2. Step 3. It is fairly obvious how Step 2 can be used to derive (10.27) from (10.28). Let us introduce the abbreviation 9k(C,ri) = 1/(0 -fM\

(1 ^ fc ^ co,£»7 efl).

For fixed e > 0 we let c > 0 be so large that v(gœ 1^ àc j) < e. By virtue of Step 2, we can find an n ^ 1 and (5 > 0 such that v((p*~n < ô) < e/c for all k > 2n. Assumption (i) and (10.28) ensure that v(l/* - / J ) < e/2

and

?(%*-"&) < £<5

for all sufficiently large /c. For these k we obtain, using the triangle inequality

gk(C,rj) g âX»7) + |/k(C) - /„(Ol + UM - fJn)\, v(flk) ^ v(l { % -n < i } g t ) + <5-1 ?(
^ v{lm-r,
^ cv((p^

+ 2v(\fk-fao\)

<ô) + v(gœl{ë^c])

+E + 2e

This completes the proof of (10.27) and the proposition, D With the help of Proposition (10.26), the proof of Theorem (10.25) is straightforward. Proof of Theorem (10.25). Let p e %(y) be given. We apply Proposition (10.26) to the sequence fk = p?-lM, k ^ 1. Assumption (i) of (10.26) holds because ( / A ä i i s a backward martingale with respect to v = pX^y (Recall (10.17).) In particular, / œ = P]°-i>0O[ v-a.s.. Assumption (ii) of (10.26) holds with À = { — 1,0} because p is Markovian. Hence Proposition (10.26) implies that P]°-i,oo[ is equal to some ^-measurable function g v-almost surely. Consequently, (10.19) holds for j = 0. For arbitrary j e Z w e use the shift-invariance of p and p to deduce that v

MJj-i,oo[ = 9° 0-j) = v (pi-i.cot ° Oj = g)

= 1. The theorem thus follows from Lemma (10.20).

o

204

Markov fields on the integers I

10.3

Uniqueness of the shift-invariant Markov field

Throughout this section we shall assume that the hypotheses of Theorem (10.25) are satisfied. Thus (E, S) is assumed to be a standard Borel space, X is any a priori measure (without loss: probability measure) on (E, S), p an irreducible homogeneous Markovian A-modification, and y = pX_. We intend to show that there is at most one p e ^&(y). This will be achieved as follows. First of all, Theorem (10.25) asserts that any p e ^@(y) is a Markov chain for a suitable transition kernel P. Clearly, the one-dimensional marginal distribution a = at(p) is P-invariant, in that ccP = a. We shall show that P is even ergodic in the following sense: For a-almost all x, Pn(x, •) tends to a in total variation distance as n -> oo. Of course, this ergodic theorem is interesting in its own right. We then shall use the ergodic theorem to show that each p e %(y) has short-range correlations and is thus trivial on the tail u-field. By virtue of Theorem (7.7), this means that &@(y) c ex &(y). But this implies that \%(y)\ S l. To carry out this program we fix any p e &e(y). By Theorem (10.25), there is a measurable function p: E x E -> [0, oo[ such that P(x, A) 4 J X(dy)p(x, y)

(xeE,Ae

S)

A

is a probability kernel and p is a Markov chain with transition kernel P. Let us look at the iterates P" of P. For each n ^ l w e define a measurable function p": E x E -» [0, oo[ recursively by (10.29)

p\x,y)

= p(x,y),pn+1(x,y)

= X(p"(x, -)p(-,y)),

where x,y e E. Then P"(x, •) = p"(x, -)X for all x e E and n ^ 1. Looking at (10.5) we see immediately that P"(a-n,A) = n{p0 eA\&{-.„}) = p(a0 e A\^_x^n])

p-a.s.

for all ,4 e S and n ^ 1. Because of (10.14) this is equivalent to the statement that (10.30)

p"(o-n,<j0) = v(p {0} |#{_ ni0} ) = v(p {0} |^_ 00i _ n]U{0} )

v-a.s.

for all n ^ 1. Here, as before, v = pl{0}. Next we consider the one-dimensional marginal distribution a = ot(p) of p. We set (10.31)

r(x) = J cc(du)p(u,x)

(x e E).

Then (10.32)

a = rX, and

r(a0) = v(p{0}|Jr{0})

Indeed, for all A e S we have

v-a.s..

Uniqueness of the shift-invariant Markov

field

205

rX(A) = [a(du)P{u,A) = aP(A) = a{A), and (10.30) (for n = 1) gives j

r((T0) dv = j X(dx) j ^(da))^^..!, x) =

j

dvv(p{0}|Jjr{_1,0})

{a0EA}

=

I

dvp {0} .

We also remark that (10.33)

r>0

/l-a.s. on {hN > 0}

for all sufficiently large N, where hN is as in Definition (10.23). In particular, if {hN > 0} | £ (as is the case in the examples in (10.24)) then a and X are mutually absolutely continuous. To prove (10.33) it is sufficient to show that r 2: (2a(CN) — l)hN for all N^l.

A-a.s.

From (10.32) and (10.16) we obtain, writing n = n{N),

r(a0) = v(v(p{0}|^7_„;„[X{0})|^r{0})

^ /X(
v-a.s. .

The proof of (10.33) is completed by noting that _„,
n-»oo

and lim X(\p"(x, •) — r|) = 0

for oc-almost all x e E.

n-*oo

Proof First of all we note that j ct(dx)X(\pn(x, •) - r\) = j /x(dca) j A(dy)|p"(eo_B,y) - r(y)| =

v(|p"(«7_ B ,«7 0 )-r(«7o)|)

206

Markov fields on the integers I

for all n ^ 1. To prove the convergence to zero we put /„ = pn(ff_„, c 0 ). From (10.30) we conclude that (/„)„ai is a backward martingale with respect to v, and its limit in L1(v) is .7 00

V

(0}

»Oi JS5-œ'-"]ut°W-

Also, /„ is measurable with respect to 3F^_n^. That is, (/„)„ â i satisfies condition (ii) of Proposition (10.26) with A = {0}. We thus obtain, from that proposition, that / œ has an J^-measurable version. From this and (10.32) we conclude that /» = v(p{o}l^{o}) = r(a0)

v-a.s. .

This proves the first part of the theorem. To prove the second part we note that X(\p"(x, •) — r|) is decreasing in n for all x. Indeed, we have rX

= a = ap = (^ X(dy)r(y)p(y, •))}.

and therefore r = $ A(dy)r(y)p(y, •)

/l-a.s. .

Combining this with (10.29) we get A(|p"+1(x, •) - r\) = p ( d z ) | J A(dy)p(y,z)lp»(x,y)

- r(y)]\

^$A(dy)$A(dz)p(y,z)\p»(x,y)-r(y)\ =

A(\p"(x,-)-r\)

for all n ^ 1 and x e E. Using Fatou's lemma, we finally arrive at the conclusion Ja(dx)lim/l(|p n (x,-)-r|) n-*ao

^ lim J a(dx)A(|p"(x,-) - r|) = 0 n-»oo

which proves the second part of the theorem.

•

We observe that Orey's ergodic theorem for countable state Markov chains is a special case of Theorem (10.34). To see this let E be countable, X be counting measure, and P be an aperiodic irreducible stochastic matrix on E. Suppose P is positive recurrent. As is well-known, this means that there is a unique probability vector a on £ with aP = a. By virtue of (10.5), there is a unique Markov chain /x for P with
Uniqueness of the shift-invariant Markov field

lim X \Pn(x,y)-a(y)\ n-*oo

207

=0

yeE

for all x e E. This is Orey's ergodic theorem (cf. Blackwell and Friedman (1964)). Note that a special case of this theorem was already stated and proved in(3.A3)and(7.15). We are now ready to turn to the main result of this section. (10.35) Theorem. Let (E,S) be a standard Borel space, ke Jt{E,$), p an irreducible homogeneous Markovian À-modification, and y = pX. Then either y®(y) = 0 or \y&(y)\ = 1- In the latter case there exists a probability kernel P from S to S with the following properties: (i) There is a measurable function p: E x E -» [0, oo[ such that P{x, •) — p(x, -)h for all x e E. (ii) There is a unique a. e ^{E, S) with ctP = a such that lim P"(x, •) = a in n-*oo

total variation distance for a-almost all x. a has a k-density r which is positive on {hN > 0} for sufficiently large N. (Hi) y®(y) = {p.p} c ex^(y), where p.P is the Markov chain with transition kernel P and one-dimensional marginal distribution a. Proof. We shall prove that %{y) c ex3%). This will imply that \%(y)\ ^ 1. For if %{y) contained two distinct elements then any non-trivial mixture of these would belong to &B(y)\ex &(y). Let p. G %(y) be given. According to Theorem (10.25), p is a Markov chain for a suitable transition kernel P with A-density p. To show that p is extreme in
lim sup \p(A f)B)-

p(A)p(B)\ = 0

for all cylinder events A e &. Let us take any such A. We pick two integers i < k such that A e ^,-,k], and we choose arbitrary integers m, n with m k. We shall prove that (10.37)

sup \p{A^B)-

p{A)p{B)\

g j a(dx)A(|p i -"(x,-) -r\

+ |p"-*(x,-) - r\ + \p"~m{x,-) - r\),

where a, r, and pe are defined in terms of p and p as before. According to Theorem (10.34), the expression on the right of (10.37) tends to zero in the limit m -> -co, n^-oo. Thus (10.37) gives (10.36). To prove (10.37) we let A* e £{Uk] be such that A = {a[iM e A*}, and we choose any B e ^mM. For jeï and z e E we define a probability measure v/ on (EP- °°[, Su-œ[) by the requirement that

208

Markov fields on the integers I

vf((7j e dx0,...,oj+t

e àx() =

öz(x0)P(x0,dxi)...P{xe_x,dx()

for all £ 2: 0. Thus vf is a Markov chain for P which starts from z at time j . Then we can write H(A H B) = \ n(da>) \ l(dx)pi-m(œm, x J X(dz)P"-%k,z)

x) J v*(dOM£ [ijk] ) vzn(dri)\B(w^Mri)

\

and H(A)H(B) = \ X(dx)r(x) \ v?(dQU.(Cp.t]) x J /x(dca) j X(dz)p"~m(œm,z) I

v^drjUsiai^^r,).

Thus \H(A^B)-PL(A)H(B)\

^ \ Hideo) \ X(dx) \ v*(d£) J l(dz)|p- m ( füm ,x)p"- k (C k ,z) -r(x)p»-m(com,z)\ = J a(dw) J X(dx) j px-^dy)

\

X{dz)\pi-m{W,x)p"-k{y,z)

-r(x)p"-m(w,z)\. Estimating the last term by means of the triangle inequality \pi-m(w,x)pn-k(y,z)

-

r(x)pn'm(w,z)\

^\pt'm(w,x)-r(x)\p''-k(y,z) + r(x)\p"-k(y,z) - r(z)\ + r(x)\r(z) -

p"-m(w,z)\

and using the fact that J X(dx)r(x)Pk-i(x, •) = aPk~l = a we arrive at (10.37). This completes the proof that n is extreme and thus unique. The additional assertions (i) to (iii) of the Theorem follow from Theorem (10.25), (10.32), (10.33), and Theorem (10.34). o We conclude this chapter with a comment on Theorems (10.25), (10.34), and (10.35). These theorems were stated for a Markovian homogeneous Xspecification y. However, we made no use of the consistency property yAyA = yA, A <= A, of y. Moreover, it is possible to dispense with the assumption of absolute continuity with respect to X. (See Papangelou (1983).) Consequently, the conclusions of the above-mentioned theorems hold for any shift-invariant Markov field fionZ with a standard Borel state space (E, S) provided /j, admits suitable versions of the conditional probabilities n(- |^_„, B[ ) which satisfy an irreducibility condition similar to (10.23).

Chapter 11 Markov fields on the integers II

In this chapter we will continue the study of Markov fields on S = Z under more restrictive assumptions. The state space E will be countable, and we shall look only at Markov specifications y which are positive and homogeneous. Such specifications y are always Gibbsian for a suitable shift-invariant nearest-neighbour potential [- Whilst all Markov fields for Kalikow's Q are Markov chains, the Spitzer-Cox example provides a number of Markov fields which fail to be Markov chains. The third example of phase transition was also invented by Spitzer (1975b). It has the peculiarity that it exhibits a total breaking of shift-invariance: ^(Q) is nonempty, but although yQ is shift-invariant there is no Markov field for Q which is invariant under any shift. This is a consequence of the fact that Q is null recurrent.

210

Markov fields on the integers II

11.1

Boundary laws, uniqueness, and non-existence

Throughout this chapter we assume that S = Z and £ is a countable state space. The a priori measure X on E is counting measure. We look at Markov specifications y which are homogeneous (i.e., shift-invariant) and positive. Positivity means that pA(a>) = yA^A = a>A|a>) > 0 for all A e if and a> e Q. Recall from Remark (1.28)(5) that p = (p A ) A e ^ is the unique /l-modification with y = pi.. p is a Markovian pre-modification. In particular, p is quasilocal. Theorem (2.30), Corollary (2.32), and Corollary (5.9) thus show that p = p* for some A-admissible shift-invariant nearest-neighbour potential S> (which may be chosen to be a gas potential for some vacuum state a e E). It will be convenient to pass from S> to the positive matrix Q = (Q(x, y))x,yeE on E which is defined by ß((oi_1,o)£) = exp[-0 { i _ l i i } ((o) - iO{,-1}(o)) - i;.) Q is often called the transfer matrix associated with <S>. The ^-admissibility of S> implies that all powers Q" of Q are well-defined, in that (11.1)

Q"(x,y)
for all x, y e E and n ^ 1.

Indeed, for all co e fi and n > 1 we have Qn(œ0,œn) = ZjS,B[(a))exp [-•|) - i® w (û>)] < GO. The specification y can be re-expressed in terms of Q as follows. If co e Q and A = ]i,/c[ then k

(11.2)

yA(A\a>)

o*-,(û),-,û)t).

If A is the union of pairwise disjoint and non-adjacent intervals I1,...,IN

then

N

(11-3)

7AK

=

Û»AI«O)

= I l ^H(ffr„ = »rj« 0 )B= l

Q

We agree to write y = y if y is defined by means of Q via equations (11.2) and (11.3). Note that these equations are a special case of the construction in Example (10.3). Consequently, equations (11.2) and (11.3) establish a welldefined mapping Q -> yQ from the set of all positive matrices on E which satisfy (11.1) to the set of all positive homogeneous Markov specifications. The discussion above has shown that this mapping is onto. However, it is certainly not one-to-one. To see this more clearly we introduce an equivalence relation between positive matrices P and Q with finite powers by writing P ~ Q if and only if yp = yQ. Of course, this equivalence relation should be viewed as a special case of the equivalence relation between potentials which was defined in (2.33).

Boundary laws, uniqueness, and non-existence

211

(11.4) Remark. Let P and ß be two positive matrices on E which satisfy (11.1). Then P ~ ß if and only if there is a number q > 0 and a function r : E -> ] 0, oo [ such that (11.5)

P(x, y) = Q(x, y)r(y)/qr(x)

for all x, y e E.

Proof. Looking at (11.2) we immediately see that P ~ ß whenever (11.5) holds. Conversely, suppose P ~ ß. From (11.2) (as applied to a singleton A) we obtain that P(x,y)P(y,a)/P2(x,a)

=

Q(x,y)Q(y,a)/Q2(x,a)

for all x, y, a e E. We fix a reference element a e E and define q = ß( fl) a)/P(a, a), r(x) = ß 2 (x, a)/P2(x,a)

(x e E).

Then P(x,y) =

Q(x,y)Q(y,a)/P(y,a)r(x)

for all x, y e E, and for y = a we obtain P{x,a)/Q{x,a) = qlr{x). Replacing x by y in the last equation and inserting this into the next to last equation we arrive at (11.5). • If £ is finite, the Perron-Frobenius theorem (3.A1) implies that every positive matrix Q is equivalent to a stochastic matrix P; cf. Comment (3.8) (2). (This observation was a basic tool in the proof of Theorem (3.5).) In the case of an infinite E, an analoguous result was obtained by Vere-Jones (1962). To state it we need to introduce the quantity (11.6)

L(Q) = lim Q"(x, x)1/n e [0, oo]

(x e E).

(The existence of the limit follows readily from the supermodularity relation ß n + m ( x , x ) ^ ß"(x,x)ß m (x,x), m, n^l. The limit does not depend on x because ß n + 2 (x, x) ^ ß(x, y)Q"(y,y)Q{y, x) for all n ^ 1 and x, y e E) We note that L(ß)" 1 is the radius of convergence of the power series £ ß"(x, x)z". nâl

The result of Vere-Jones now asserts that every positive matrix ß with L(ß) < oo is equivalent to a substochastic matrix P with L(P) = 1. As we shall not need this result, we refer the interested reader to Chapter 6 of Seneta (1973). What we shall need is the simple remark below which states that equivalent positive stochastic matrices P and ß with L(P) = L(ß) = 1 exhibit the same recurrence properties. Let us recall that a positive stochastic matrix P either satisfies £ P"(x, y) = n£l

oo for all x, y e E, or £ P"(x, y) < oo for all x, y e E. In the former case, P is »gl

212

Markov fields on the integers II

called recurrent, whilst in the latter case P is said to be transient. P is recurrent if and only if Hp(ty < GO) = 1 for all (or some) x, y e E. Here /i£ e é?(Ez+, é>z*) is the Markov chain with transition matrix P and initial point x, and T = min {n ^ 1: an = y} is the time of the first visit at y. A recurrent P is called positive recurrent if Hp(xy) < GO for some (and thereby for all) x, y e E or, equivalently, if there is a probability vector a on E such that a.P = a. Otherwise, P is said to be null recurrent. (A reader who is not familiar with all this is referred to Breiman (1968), for example.) We note that a recurrent P always satisfies L(P) = 1. (11.7) Remark. Suppose P and Q are positive stochastic matrices with L(P) = L(Q) = 1 and P ~ Q. If P is recurrent then P = Q, and if P is transient then so is Q. In particular, a positive stochastic matrix Q which is null recurrent or transient with L(Q) = 1 can never be equivalent to a positive recurrent positive stochastic matrix. Proof. Suppose P and Q are related to each other by (11.5). Then Pn(x, x) = Qn(x, x)/qn for all x e E and n ^ 1. Hence q = L{Q)/L{P) = 1. Thus £ P"(x, x) = Y, Q"(X> X ) f° r a ll *•> a n d this shows that P is recurrent if and nil

nil

only if Q is. To show that in this case P — Q we need to prove that r is constant. But for all x, y e E and n ^ l w e have from (11.5) r(x) = Q"r(x) = ^ ( r ° *„)

^ J ^(T, = k)Q-kr(y) k= l

= r(y)/ig(^ ^ n). Since Q is recurrent, we conclude that r(x) ^ r(y) for all x, y e £. This completes the proof, D From now on we shall assume that we are given a positive matrix Q for which (11.1) holds, and we shall investigate the set ^(Q) = ^(yQ) of all Markov fields for Q. Our first result provides a characterization of all Markov chains in ^(Q) and, in particular, a description of all extreme elements of <&(Q). In (11.31) and (11.39) we shall see how this description can be used to determine ex <&(Q) in particular cases. The characterization of the Markov chains in <&(Q) involves the notion of a boundary law for Q. (11.8) Definition. Let Q be a positive matrix. A boundary law for Q is any family {
Boundary laws, uniqueness, and non-existence

213

Suppose Q is stochastic and {/;,r;: i e Z} is a boundary law for Q. If rt = 1 for all i e Z, then {/;: i G Z} is a family of probability vectors on E such that — oo and/,, -> oo. This is a consequence of the consistency equations /,Q = / i + 1 and gr,. = r,_1. (11.9) Theorem. Lef Q be a positive matrix on E which satisfies (11.1). (a) Each boundary law {/;,r;: i e Z} for Q defines a unique Markov chain fie
Ufa = x„,..., ffi+„ = x„) = ^(x 0 )6(x 0 , x j . . . Q(xB-1,xB)ri+H(xn).

Here i G Z, M ^ 0, and x 0 , . . . , x„ G £. (b) Every Markov chain \i G ^(Q) admits a representation of the form (11.10) in terms of a boundary law for Q. (c) Every \i G ex^(Q) has a representing boundary law {^i,rt: i G Z}. Moreover, for any fixed ae E there exist two sequences (x„)nfel and (j n ) n è i "i E such that ^(x)/^ 0 (a) = lim ô n+, (x„,x)/ô n (x n ,a) n-*oo

/or all x e E and i < 0, and ri(y)/r0(a) = lim ß » - ' ^ , j n )/ô"(a, y„) n-*cc

for all y G E and i > 0. Proof, (a) By Kolmogorov's extension theorem, there exists a unique \i G ^(Q,J^) which satisfies (11.10). Comparing (10.5) and (11.10) we see that ß is a Markov chain with transition matrices (11.11)

Pi =

(Q(x,y)ri(y)/ri_l(x))x,yeE,

i e Z. From Example (10.11) we know that /J. G <S{y), where y is defined in terms of the functions (x,y) -> Pfay) = 6(*.3')',/(j')/rI--i(x) as in Example (10.3). But a glance at (11.2) shows that y = yQ. Hence ^ G &(Q). (b) Let ^ G ^(Q) be a Markov chain with transition matrices (P ; ) i6Z . We look for vectors rt G ]0, CO[ £ which are such that (11.11) holds for all i G Z. This will give us the required representation of \i. Indeed, since Pt is stochastic, (11.11) implies that Qrt = ri_1 for all i G Z. Also, putting ^(x) = /x(<7, = x)/r,(x) we have «f£rÉ = 1, SiQ(y)=

I M ^ = x)fJ+i(*.3')/ri+i(3')

214

Markov fields on the integers II

for all y and i, and Ufa = x0,...,ai+n

= x„) = ß(at =

x0)Pi+1(x0,x1)...Pi+n(xn_1,xn)

= fi(x0)Q(x0, xt)...

Q(xn_t, x„)ri+n(xn)

for all i e Z, n ^ 0, and x 0 x„e£. To find the vectors rt we first note that each Pt is positive. This is because 0 < W{?-i.i}(ff|-i = x,at = y) = n(ot„i =x,ol = y) = A*(o"«-i = x)Pj(x,3>)

for allieZ (11.12)

and x, y e £. Next we observe that Q(x,y)Q(y,a)/Q2(x,a)

= n(at = j ' k i - i = x,ai+1 = a) =

P,(x,y)Pi+1(y,a)/PtPi+1(xta)

for all x, y, ae E and i e Z. We exploit this identity in a similar fashion as in Remark (11.4). We fix an a e E and define numbers qi > 0 in such way that li/Qi-i = Pi+1(a,a)/Q(a,a) for all i e Z. We put r,(x) = qi+1Q2(x,a)/Pi+iPi+2(x,a)

(ieZ,xe

E).

Equation (11.12) can then be rewritten as Pi(x,y)Pi+1(y,a)/Q(x,y)Q(y,a)

=

qjr^x).

Setting y = «we get Pi(x,a)/Q(x,a) =

q^Jr^x).

Replacing x by y and i by i + 1 in the above and inserting this into the next to last identity we obtain (11.11). (c) Fix any ß e ex ^(Q). According to Theorem (10.21), ß is a Markov chain. Assertion (b) thus shows that ß satisfies (11.10) for a suitable boundary law {th rt: i e Z). Let a e E, i < 0, and x e E be given. For /^-almost all co e {a0 = a} we have ^ W f i ' ^ ^ o W = MCT; = x\a0 = a)

=

l i m ji(ffj = n-»oo

X|^_00>_I1]U{0})(ÛJ)

= lim y{L„,o[(?i = x\co) = lim Q n + i (œ_ n ,x)Q- i (x,a)/Q>_„,a).

Boundary laws, uniqueness, and non-existence

215

The third equality follows from the backward martingale theorem and the ^-triviality of the left tail (cf. the proof of Theorem (10.21)). Since /J(CT0 = a) — W{o}(°b = a) > 0, there is an a> e {a0 = a} such that the preceding identities hold for all x e E and i < 0. Putting xn = a>_n we arrive at the statement for the / / s . The statement for the r;'s is obtained similarly by looking at fi(Gt = x\a0 = a) for i > 0. a We now turn to the questions of uniqueness and non-existence of Markov fields for a positive matrix Q. As far as shift-invariant Markov fields are concerned, a complete answer to these questions is provided by Theorem (10.35). Let us restate that theorem in the present setting. Recall that a positive recurrent positive stochastic matrix P admits a unique probability vector a e ]0, oo IE with a.P = a. Putting /,• = a, ri = 1, and Q = P in (11.10) we obtain a Markov chain fi e ^(P) which is the unique shift-invariant Markov chain for P. This Markov chain will be denoted by \iP. (11.13) Theorem. Let Q be a positive matrix on E which satisfies (11.1). Then either ^©(Q) = 0 or |^©(Q)| = 1. The latter case occurs if and only if Q is equivalent to a positive recurrent stochastic matrix P with positive entries. In this case P is unique and ^©(Q) = {nP} c: ex^(Q). Proof. If Q ~ P for some positive recurrent positive stochastic matrix P then iip e ^©(P) = ^®(Q), as was shown above. Conversely, suppose there is some n e ^©(Q). By Example (10.24)(2) and Theorem (10.35), /i is unique, extreme in ^(Q), and a Markov chain for some transition matrix P. P is positive (cf. the proof of Theorem (11.9)(b)) and admits an invariant probability vector. Hence P is positive recurrent and \i = \iP. Also, since \i e ^(Q) fl ^(P) we have Q ~ P. Indeed, we can either refer to Theorem (2.34), or simply note that y\(°\ = (t>K\w) = fj,(aA = oiA\adA = œdA) = y^(aA = wA\w) for all A e y and œ e Q. (dA was defined above (10.9).) The uniqueness of P follows from the uniqueness of fi and also from (11.7). a As a consequence of Theorem (11.13) we obtain the result that ^(Q) is either empty, or a singleton, or infinite dimensional. Later on we shall, in fact, see that all these possibilities occur. (11.14) Corollary. Let Qbe a positive matrix which satisfies (11.1). (a) For each /J. e $(Q) we have the following alternative: Either \i is shiftinvariant, or its translates 9^) (i e Z) are pairwise distinct. (b) If Q ~ P for some positive recurrent stochastic matrix P with positive entries then either ^(Q) = {/iF} or |ex^(Q)| = oo.

216

Markov fields on the integers II

(c) / / Q is not equivalent to any positive recurrent stochastic matrix with positive entries then either ^(Q) = 0 or |ex^(Q)| = oo. Proof. To prove (a) we let p e &(Q) be given. We assume that 6>;(/i) = 6}(n) for distinct integers i, j . We will show that p is shift-invariant. First of all we note that p is periodic with period p = \i — j \ ^ 1, in that 6p(p) = p. Next we put p' = dxifi). From Remark (5.10) we know that p' e ^(Q). We need to show that p' = p. This will follow once we have shown that p' = p on J*p2. For this gives that ß = W]?m,P„[ = ß'y$m,Pni = I*'

on JV,, p n [

for all integers m < n, and therefore p = p'. We let v and v' denote the distribution of (erp;); 6 z under /i and p', respectively. The assertion p' = p on J*pZ is equivalent to the statement v = v'. This identity will follow from Theorem (11.13) provided we can show that v, V e ^@(QP). To see this we first note that v and v' are shift-invariant because p and thus p! are periodic. To show that v, v' e f£(Qp) we let j e Z, x e E, A = [ j - nj + ri]\{j}, and £ e EA be given and define Çp e £ p A by C£ = C; (i e A). Then V?U}VA = C, <Jj = x)

=

J

v(dœ)Qp(œj.1,x)Qp(x,(uj+1)/Q2p(œj_1,œj+l)

= K(yPA = Cp,(Tpj = x) = V(<7A = C, <7j = X).

Hence vy?„ = v for ally, and Theorem (1.33) implies that v e &(QP). Similarly, v' e ^(Qp). This completes the proof of (a). To prove (b) and (c) we suppose that either Q is equivalent to some positive recurrent stochastic matrix P and ^(Q) ^ {UP}, or Q is not equivalent to such a P and
Boundary laws, uniqueness, and non-existence

217

(11.15) Theorem. Let Qbe a positive matrix on E which satisfies (11.1). If inf X

Qn(x,x)>0

xeEn=l

for some integer N ^ 1 then ^(Q) = %(Q). Proof. We put p = N\. Then inf Qp(x, x) ^ inf max Q"(x, x)pl" xeE

XEE

lgngN N

^ inf AT1 X xeE

Q"(x,xyln

n—1 N

^ inf AT1 X (1 A ß"(x,x)) p xeE

n=l

^ inf A T ' ( £ 1 Aß-(x,x)Y xeE

\n =l

/

>0. We shall use this inequality to show that 9p(p) = p for all p e ex ^(Q). Corollary (11.14) then will imply that ex^(Q) <= %(Q), and this will prove the theorem because of the extreme decomposition theorem (7.26). So we take any p e ex^(Q). According to Theorem (11.9)(c), p admits a representation of the form (11.10) in terms of a boundary law {V,-,^: i e Z}. This representation shows that 9p(p) = p provided we can find a constant c > 0 such that £t_p = c£t and ri+p = cri for all i e Z. To show the latter we choose any number 8 with 0 < 3 < inf ß p (x, x). For all x e E and i e Z we have *e£ '*(*) = ' i - p o ' t o ^ ^-pWß p (x,x) >

ô^.p(x)

and r ; W = Qpri+P(x) 2: Qp(x,x)ri+p(x) > öri+p(x). Thus (11.16)

ti>Mi_p

and

r, > Sri+p

for all i e Z.

We put c = ^iri+p. This is independent of i because ^ri+P = Vt-iQ)ri+p = ^-i(ßr i + p ) = / H ^ - H , for all i e Z. (11.16) shows that 0 < cô = ^(<5ri+p) < ^

= 1.

Now we define t\ = ft_p/c and *f," = (/, - <5/,_p)(l - c<5)"\ i e Z. The systems {S'i,r,: i e Z} and {«f",^: i e Z} are boundary laws for Q. Indeed, (11.16) en-

218

Markov fields on the integers II

sures that each (•' is positive, and we have

and similarly /f"Q = (f"+1 and
for all i e Z,

we have /^ = c<5// + (1 — c<5)//'. As /^ is extreme and 0 < cô < 1, we conclude that n = n' = p.". This shows that (f • = (f; and therefore (f^p = C(f; for all i e Z. A similar argument works for the r;'s: The boundary laws {/„ ri+p/c: i e Z} and {/;,(?•,• — <5ri+p)(l — cä)""1: i e Z} define two measures v', v" e ^(Q) which satisfy /^ = c^v' + (1 — cô)v". The extremality of \i gives /^ = v', whence ri+p = crt for all i. The proof is thus complete, D Combining Theorems (11.13) and (11.15) we arrive at the following result. (11.17) Corollary. Let Qbe a positive matrix on E which satisfies (11.1) and is such that N

inf X Qn(x,x)>0 xeEn=l

for some N ^ 1. Then ^(Q) = {fiP} when Q is equivalent to a positive recurrent stochastic matrix P with positive entries, and ^(Q) = 0 otherwise. This corollary provides both: a sufficient condition for the existence of a unique Markov field for Q, as well as sufficient condition for the non-existence of such a Markov field. Some comments on these conditions are in order. (11.18) Comments. (1) The uniqueness condition of Corollary (11.17) certainly holds whenever Q can be written as Q = tP + (1 — t)I with 0 < t < 1, P being any positive recurrent stochastic matrix with positive entries, and / being the identity matrix. Indeed, such a Q satisfies inf Q(x, x) ^ 1 — t > 0 and is xeE

positive recurrent because the probability vector a with txP = a. also satisfies a.Q = a. If £ is finite, each positive stochastic matrix Q can be written as above. We thus obtain Theorem (3.5) again. (2) It is interesting to compare the uniqueness condition of Corollary (11.17) with the uniqueness condition which is obtainable from Theorem (8.39) (resp. Comment (8.41)(3)): The identity |^(Q)| = 1 holds whenever there are functions u, v from E to ]0, oo[ and a number C > 1 such that

C-^Q{x,y)lu{x)v{y)^C for all x, y e E. (To check this it is sufficient to note that yQ coincides with

Boundary laws, uniqueness, and non-existence

219

the Gibbs specification for the bounded nearest-neighbour potential $

f-log[ß(ffi,oi+i)MoiMoi +1 )] { 0

=

^

if A

= {'»' + !}» otherwise

and the (necessarily finite) a priori measure (uu)A.) The condition above implies that N

Y

£ Qn(x, x) < oo

for all iV ^ 1

xeE n=l

because

Z 6"(x,x)^

C YJ U(X)V(X) xeE

xeE

for all n ^ 1, and Clu(a)v(a)

YJ U(X)V(X)

^ Q2(a, a) < oo

for any a e E. Consequently, if E is infinite then the uniqueness conditions of Theorem (8.39) and Corollary (11.17) exclude each other. (3) The non-existence condition of Corollary (11.17) is satisfied whenever Q = tP + (1 — t)I for some 0 < t < 1 and any positive stochastic matrix P which is either null recurrent or transient with L(P) = 1. In fact, the equation Qn(x,x)=

t

{nk)tkPk(x,x)(\-tr-k

k=0

is readily seen to imply that L(Q) = tL(P) + 1 — t = 1, and Q is not positive recurrent because otherwise there is a probability vector a with <xQ = a and thereby ccP = a. Remark (11.7) thus implies that Q is not equivalent to a positive recurrent positive stochastic matrix, o The non-existence condition of Corollary (11.17) also holds when Q is the transition matrix of a random walk on ZN, N ^ 1. This will be shown in the final corollary below. Under a growth condition on Q, the same result was already obtained in Example (9.17). Recall that a matrix Q on ZN is said to be homogeneous if Q(x, y) = g(0, y — x) for all x, y € ZN. (11.19) Corollary. Let £ = ZN for some N ^ 1 and Q be a homogeneous positive matrix on E such that

C= Z ß(0,x)
Then <${Q) = 0. Proof. 1) Since inf Q(x, x) = *g(0,0) > 0, the result will follow from Corollary xeE

220

Markov fields on the integers II

(11.17) provided we can show that Q is not equivalent to a positive recurrent stochastic matrix. According to Remark (11.7), the latter certainly holds when Q is equivalent to a positive stochastic matrix P with L(P) = 1 which is not positive recurrent. It is well-known that a homogeneous P can never be positive recurrent. (For suppose there was a probability vector a with aP = a. The ergodic theorem (10.34) then would imply that <x(y) = lim Pn(0,y) = lim P"(-y,0) n —•co

= a(0)

n~*co

for all y e E. This is impossible.) To prove the corollary it is therefore sufficient to find a homogeneous positive stochastic matrix P with L(P) = 1 and P ~ Q. 2) For each s e UN we define
Q(0,x)exp(s-x)

xeE

and, if q>(s) < oo, Qs(x,y) = Q(x,y)exp [s • (y - x)]/(?(s)

(x, y e E).

(The dot denotes the inner product.) Qs is stochastic and homogeneous, and Remark (11.4) shows that Qs ~ Q. Therefore the corollary will be proved once we have found an s 0 such that (s0) < oo because Q"(0,0) S C for all n, and therefore L(Q) ^ C. Since ßS0(0,0) = ß"(0,0)M So )" for all n ^ 1, this will also show that L(QSo) = L(Q)/(p(s0) ^ 1. Since L(P) ^ 1 for each stochastic matrix P, we shall arrive at the desired conclusion L(QSo) — 1. (By the way, the argument above shows that s 0 will be characterized by the equation L(Q) = (p(s0) = inf (p(s).) seW

3) In order to find s0 we shall use an argument which is well-known in large deviation theory. We fix an integer k ^ 1 and introduce the truncated matrix Q(xy) = \Q{X'y) (0

i f

l*-^/c, otherwise,

as well as the associated generating function q>. We note that cp(0) < C. Putting 6 = min ß(0, x), we have

#(s)^^ X eSX =23 Z c o s h s -->^^l 2 1*1 = 1

i= l

for all s = (Si,..., sN) e IRN. As ^ is continuous, ^ attains its minimum at some S with |s\ 2 S C/ö. Also, 0 = grad^(s)= ^ x (5(0, x)exp(s-x).

The Spitzer-Cox example of phase transition

221

Consequently, the vector Qs(0, • ) belongs to the set K of all probability vectors a on £ which are such that £ x cc(x) = 0 and a(x) = 0 when |x| > k. The set xeE

of all such a with rational coordinates is dense in K. For each n ^ l w e can therefore find an a„ e K such that na„(x) e Z and lim a„(x) = Qs-(0,x) for all n-*oo

x e E. Looking at the paths of length n in E from 0 to 0 which contain nct„(x) jumps of size x we obtain the inequality

&5(o,o) ^

n

n &(o,xr«<*>. 11 (najx))\ V X

Using Stirling's formula, we arrive at the inequality L(QS) ^ 1 which is equivalent to the inequality L(Q) ^ cp(s). Since ß"(0,0) ^ ß"(0,0) for all n, we end up with the inequality L(Q) ^ cp(s). Finally, we let k tend to infinity. The associated minimal points s have a cluster point s 0 . Letting k run through an appropriate subsequence and using Fatou's lemma we obtain the desired result
k

^liminf
This proves the corollary, D In the next three sections we shall present three examples of positive stochastic matrices Q which exhibit a phase transition. In each of these examples, the state space will be E = Z+, the set of all nonnegative integers.

11.2

The Spitzer-Cox example of phase transition

We begin by introducing the notations (11.20)

â{n,p,k) = {l)pk{\ -p)"'k

(n,k^0,0SpS

1)

and (11.21)

/t(q,k) = e-qqk/k\

(k^0,q>0)

for the binomial and Poisson distributions, respectively. Later on, we shall use the elementary formulas (11.22) (11.23)

6(m,p,-)*â{n,p,-)

= 6{m + n,p,-)

X t(n,pi,k)6{k,p2,-)

= ê{n,p1p2,-)

(m,n ^ 0,0 ^ p ^ 1), (n ^ 0,0 ^ Pl,p2

JcâO

(11.24)

Mli,-)*M92>-)

= M9i+Qi,-)

(Si,q2 >0),

^ 1),

222

Markov fields on the integers II

and (11.25)

£ Av,W(k,P,

•) = Mw, •)

(« > 0,0 ^ p s i).

Here a*a' denotes the convolution of two probability vectors a, a' on Z+. That is, a * a'(fc) = £ a(Oa'(/c - f)

(k ^ 0).

The preceding formulas are easily checked using moment generating functions. For example, (11.25) is verified by noting that

I

£

fiiq,k)6(k,P,t) z' = £ /t(«,fc)(l-p + pz)*

k>0

tf>0

= e x p [ - ^ + ?(l - p + pz)]

for all 0 ^ z ^ 1. After these preliminaries we can define a matrix Q which will be shown to exhibit uncountably many phases. This example of phase transition was invented by Spitzer (1975b) and solved completely by Cox (1977a). We put E = Z+, fix a parameter 0 < p < 1, let q = 1 — p, and define a stochastic matrix Q on E by (11.26)

Q(x,-) = t(x,p,-)*t(q,-)

(xeE).

More explicitly, (11.27)

Q{x,y) = Xf!t{x,p,k)t{q,y-k)

(x,yeE).

k=0

(The choice q = 1 — p is just for convenience. For arbitrary q > 0, the ratio q(l — p) _1 would appear in many of the expressions below.) We note that Q may be viewed as the transition matrix of a Markov chain which describes the number of individuals in a population that evolves as follows. At each time unit, the individuals die independently of each other with probability q, and survive with probability p. After that, but at the same time unit, an independent Poisson number of immigrants is added to the population. Q is evidently positive. Moreover, Q is positive recurrent. Indeed, the probability vector a = ^ ( 1 , •) satisfies a.Q = a. In fact, even more is true, a is a reversible probability measure for Q in the sense that (11.28)

a(x)Q(x,y) = a(y)Q(y,x)

for all x, y e E.

This follows from (11.27) because for each 0 ^ / c ^ x A y w e have

The Spitzer-Cox example of phase transition

Ml,x)t(x,p,k)Mq,y

223

~k) = ckqx+>/(x - k)\(y - k)\,

where ck = e~1~qpkq~2k/k\ is independent of x and y. Next we introduce a two-parameter family {n"-v: u, v ^ 0} of probability measures which will turn out to coincide with ex @(Q). Let u, v ^ 0 be given. For each i e Z and x e E we put (11.29)

/f(x) = /i(l + up\x), r»{x) = >(1 +

vp-\x)/a(x).

Then t"Q=

I

>*(i + u p ' , x)*(x,p, •)*/*(«,•)

= / ( p + up i + 1 , -)*/t(q,

•)

i+1

= /,(p + q + up ,-) —

c

i + l-

The second equality comes from (11.25), and the third from (11.24). Combining this and (11.28) we also obtain Qri°(x)= I Q(x,yWLt(y)/<x(y) yeE

= I

Q(y,x)r^(y)/oi(x)

yeE

= ^ ; Q(x)/a(x) = r_i+1(x)/a(x) = tf-iM for all i e Z and xe E. Finally, a short computation shows that W

= I

Mx)r?(x) = e"°

for all i. Consequently, the system {£",çruvr?: i e Z) is a boundary law for Q which defines a unique Markov chain ß"-v e ^(Q) via (11.10). We note that Bi(tiu-V) = pup "'•'"'' for all i e Z a n d u , ^ 0. In particular, 0,0 (i = /xß, the unique shift-invariant Markov chain for Q. Moreover, the measures nu'v are pairwise distinct. This can be seen from the equation (11.30)

nu-"((ri = x) =

fï(x)rtv(x)e-uv

= /fc((l + up')(l + t;p-'),x) which holds for all i e Z, x e E, and u,v^0. We are now ready to state the theorem of Cox which gives a complete description of &(Q). In particular, this theorem shows that ex ^(Q) has the cardinality of the continuum. But pi0,0 is the only element \i of ^(Q) which is tempered in the sense that supieZn(oi)\i\~r < oo for some r ^ 1.

224

Markov fields on the integers II

(11.31) Theorem. Consider the matrix Q at (11.26). Q satisfies ex#(ß)

=

{fiu'v:u,v^0}.

Moreover, for each \i e ^(Q) the limits U = lim <7;P~'

and

V= lim otpl

£-* —oo

i->oo

exist ^-almost surely, and \i has the representation \i=

n"'vm(du,dv),

\ [0,oo[

2

where m is the distribution of (U, V) under /i. Before entering into the proof we observe that (11.32)

ß"(x,-) = <*(*,PV)*/fe(l-pV)

for all n ^ 1 and x e E. For n = 1, (11.32) is identical with (11.26). Thus suppose that (11.32) holds for some n ^ 1. Then (11.32) holds for n + 1 because

ß"+1(x,-) = £ Q"(x,y)Q(y,-) yeE

= £

6{x,pn,k)fi{\-pn,t)6{t

= £

^ ( x , ^ f c ) X l - p V W , p , - ) * ^ , p , •)*>(«.•)

+ k,p, •)*/*(«,-)

= X ^(x,p",/c)/(p-p" + 1 ,-)*^(/c,P,-)*/te-) = «*(x,p"+1, • ) * / * ( ! - P B + 1 , - ) for all x e E. The third equality follows from (11.22), the fourth from (11.25), and the last from (11.23) and (11.24). Proof of Theorem (11.31). The proof is broken up into four steps. In the first and second step, we shall use (11.32) and Theorem (11.9)(c) to show that ex^(ß) <= {[i"'v: u, v ^ 0}. In Step 3 and 4 we shall establish the reversed inclusion and the integral representation in terms of U and V. 1) Let (x„)„ âl be any sequence in E, and suppose that c = lim Q"~l(xn,0)/ n->oo

ß"(x„,0) exists and is positive. Then u = lim x„p" exists, and for all i e Z and x e E we have "~>°° limß"+,'(x11>x) = (7(x). To prove this claim we first note that the sequence {xnp")n^l is bounded. This follows from the hypothesis c > 0 and the estimate

The Spitzer-Cox example of phase transition

225

ß"" 1 (x B ,0)/ß"(x ll ,0) = 6(xn,p"-\0)fi(\

- p- 1 ,0)/^(x ll ,p-,0)/i(l - p",0)

n~œt{xn,p"-\m{xn,P\Q)

= (1 - p"_1)*"(l - p") - *" = [1 - p " _ 1 g ( l - p " ) ^ 1 ] " " gexp[-x„p" _ 1 qf]. In the above, we have used (11.32) for the first equality. We conclude that the sequence (x„p")„ âl has cluster points. Let u be any cluster point. Letting n run through a subsequence with x„p" -» u, we conclude from the Poisson convergence theorem that t(x„,pn+i, -)-»/i("P V ) and therefore Q" +i (x n ,-) = ^ x „ , p " + \ • ) * / ( ! - p " + ' , - )

=c Using this for i = — 1 and i = 0 we obtain c = r i ( 0 ) / a O ) = exp(-uq/p). Thus u is uniquely determined by c. This shows that the sequence (x„p")„ äl has a unique cluster point and is therefore convergent. 2) Let n e ex^(Q) be given. We show that pi = /J,"'" for suitable u, i; ^ 0. According to Theorem (11.9)(c), n is represented by a boundary law {^,r;: i e Z ) which satisfies 4 = 4(0) lim Qn+i(x„, • )/Q"(x„, 0)

(i < 0)

n-*oo

and r, = ro(0) lim ß - ' ( •, yH)/Qn{0, yn)

(i > 0)

for suitable sequences (x„)„ ôl and Cyn)„fel in £. Step 1) shows that u = lim x„p" exists and ^ = c x C for all i < 0, where cL = oo

other hand, we conclude from (11.28) that Q"-'{-,yn)/Q''{0,yn) = «(0)Q"^(y„,

-)K-)Qn(yn,0)

for all n and i > 0. Thus Step 1) again shows that v = lim ynp" exists and «-»•oo

r, = c ^ f o r a l h ' > 0, where c 2 = r 0 (0)a(0)/^(0) = ro(0)/r£(0). Since StQ = Si+l, C o = C n Q n = r«-i, and gr" = r ^ for all i, we arrive at the conclusion that

226

Markov fields on the integers II

4 = cxQ and r, = c2r? for all i e Z. But 1 = £QrQ = c^c^rl = c1c2eul'. Equation (11.10) thus shows that the boundary laws {/;,r ; :ieZ} and {V",e-l"Vj': i e Z} represent the same measure. Hence /i = /i"'". 3) For fixed u, v ^ 0, the limits [7 = lim otp~l and F = lim atpl exist /i u '"i->—oo

i->oo

almost surely, and fi"'v(U = u,V = v) = l.To see this we introduce the shorthand notation mi = (I + up')(l + vp~'), i e Z. Clearly, lim m;p~' = u and i->—oo

lim m ; p' = v. From (11.10) we know that Oj(/i"'") = /è(m;, •) for all i e Z. In i->oo

particular, we have /iu,"(<7;) = m; and /iu'"(|<7; — m,|2) = m, for all i. Hence X n'-WOi - mjp-' ^ e) ^ e" 2 X P~2im< < oo i<0

i<0

for all e > 0, and the Borel-Cantelli lemma gives Hu'v( lim |<x,p_i - m £ p _i | = 0 ) = 1, whence U exists and equals u /iu'"-almost surely. The result for V is obtained similarly. 4) We define a map q> from ex ^(Q) to [0, oo [2 by —oo

i->oo

(v e ex S(ß)).

/

cp is well-defined. This is because Step 2) has shown that each v e ex ^(Q) is equal to some n",v, and in Step 3) we have seen that
j" v w(dv) ex»(Q)

which exists because of Theorem (7.26). We put m = (p(w). Then ^=

J

/i*
ex»(Q)

/i">um(du,di?).

| [0,oo[

2

Together with Step 3) we obtain fi{U and F exist) =

j" fiu'v(U and F exist)m(du, dv) = 1.

[0,oo[ 2

Moreover, for each Borel set B cz [0, oo[2 we have H((U, V)EB)=

\ [0,oo[

=

2

{

[0,oo[ 2

HU'V{{U, V) e B)m(du, dv) lB{u, v)m(du, dv)

= miß). This shows that m is the distribution of (U, V) under /i.

The Spitzer-Cox example of phase transition

227

It remains to show that each nu,v is extreme in ^(Q). From Step 3) we know that ô(uv) is the distribution of (U, V) under n"'v. By the above, ô{u
wu-v{{nu'v}nexg(Q)).

This shows that fj."'" e ex ^(Q), and the proof is complete.

•

Theorem (11.31) enables us to characterize the class of all Markov chains in y(Q). This characterization shows that neither is any Markov field for Q a Markov chain, nor is any Markov chain in ^(Q) extreme in @(Q). Thus both the converse of Remark (10.9) (3) and the converse of Theorem (10.21) fail in general. (11.33) Corollary. Let Q be given by (11.26), and let n e &(Q). The following statements are equivalent. (a) \i is a Markov chain. (b) The random variables U and V are independent with respect to p. = ÜK e- /x//x(e-ÜK). (c) \i has a representation of the form H=

lml(6.u)\m2(6.v)euvnu'v

with suitable measures m1, m2 on [0, oo[. Proof. The equivalence of (b) and (c) is immediate from Theorem (11.31). We thus need only prove the equivalence of (a) and (c). (a) implies (c). Suppose [i is a Markov chain, and let m be its representing probability measure on [0, oo[2. We putm(du,di;) = e~a+u)(1+v)m(du,dv). Then H{UeA,o0

= 0,VeB)

= l m(du, dv)n"-v(U e A,a0 = 0,V e B) =

J

m(du,dv)e-(1+u)il+v)

AxB

= m(A x B) for all Borel sets A, B c [0, oo[. Since U is .^j-«,>0[-measurable and V is J^ 0oo[ -measurable, the one-sided Markov property (10.7) of ft implies that m(A x B)/m(A x [0, oo[) = n(V e B\a0 = 0,U e A) = n(V e B\a0 = 0) = m([0,oo[ x B)/m([0, oo[2) for all A, B with m{A x [0, oo[) > 0. This shows that m = m1 x m2, where m1 = m(- x [0, oo[) and m2 = w([0, oo[ x -)/m([0, oo[2). Putting mt(du) = e 1+ "w 1 (du) and m2(dv) = e"w2(dy) we arrive at statement (c). (c) implies (a). Suppose (c) holds. For i e Z we define

228

Markov fields on the integers II

/, = Im^duWr^

im2(dv)ry.

Then li(oi = x0,...,oi+n

= x„)

= J mi(du) J m 2 (du)e u Vr(x 0 )Ô(x 0 ,x 1 ).. • Q{xn_uxn)e-"vr°+n{xn) whenever j ' e Z , n ^ 0 , and x0,...,xne cf. Theorem (11.9) (a), a

11.3

E. This shows that /lis a Markov chain;

Kalikow's example of phase transition

As before, we take E = Z+. We fix two numbers p, q with 0 < q 0 by the two requirements (11.34)

a{\ - p)"1 - b{\ - q)~l = 1.

a/b = p/q,

Thus a = p{\- p)(l - g)(p - q)~l and 6 = g(l - p)(l - q)(p - q)~\ We also put c = a — b = (I — p)(\ — q). Next we introduce a (row) vector a e ]0, oo[ £ by (11.35)

a(x) = apx - bqx = c(px+1 - qx+1)(p - q)~l X

= c Y, pkqx~k

(x e E).

k=0

The equality of the first and second expression on the right comes from the first requirement in (11.34), whilst the second requirement in (11.34) ensures that a is a probability vector on E. Looking at the second expression for a we realize that a satisfies the recursion relation (11.36)

«(0) = c, a(x) = poc(x - 1) + cqx

(x ^ 1).

We define a positive matrix Q on E by (11.37)

Q(x,y) = poi(x - l)ot(xylô^iy)

+

cqxz(xy1a(y)

= [pàx-i(y) + cqx~]u(y)/a{x). Here x, y e E, and ôx_1 is Kronecker's delta. The matrix Q was invented and studied by Kalikow (1977) for a specific choice of p and q. We note that Q(0, •) = a because cL^y) = 0 for all y e E. Moreover, Q is stochastic. Indeed, (11.36) shows that Q(x, •) is a convex combination of the probability vectors öx^ and a for all x ^ 1. To get a feeling for the characteristic properties of Q it is useful to look at the case q = 0, although this case

Kalikow's example of phase transition

229

was excluded at the beginning because Q then fails to be positive. In the case q = 0, Q is given by Q{x,y) = \S^{y) i f ^ 1 ' ^ , y y y * ' \(l-p)p iîx = 0,yeE and can be thought of as describing the evolution of the number of inhabitants of a fixed territory: The population loses one individual per time unit until the time of extinction, at which time a geometrically distributed number of immigrants enters the territory. In the case q > 0, this process is shortened, in that the inhabitants of the territory may be dislodged by an invading population of size distribution a even before extinction (with a positive probability which depends on the number of inhabitants). It is clear from the preceding interpretation that Q is positive recurrent. In fact, a is the equilibrium distribution of Q, in that ocQ = a. To check this we fix some y e E. Then aQ(y) = a(0)a{y) + pa(y) + £ = [c + p + cq(\ 1

= [c(l - qy

cq*a(y)

qy'My)

+ pMy)

= *(y). According to Theorem (11.13), the positive recurrence of Q implies that %(Q) = {J"Q} <= ex^(g). Kalikow's interesting discovery was that Q admits a non-trivial entrance law {a;: i e Z} which reaches equilibrium, in that a; = a for all i ^ 1. Let us introduce this entrance law. We put s = q/p. For i e Z and x e E w e define (11.38)

a(x) a,(x)= „ , _,_.,, ,__, , ^ (1 - s 1 _ i ) M x ) + s 1 - 'a(x)

if i ^ 1, if i ^ 0.

Clearly, each a, is a probability vector on E, and the a,'s with i ^ 0 are pairwise distinct. Let us check that a,Q = ai+1 for all i e Z. We already know this when i ^ 1. So let i ^ 0. For each y e E we can write *iQ(y) = (1 - s 1_, ')ß(-».30 +

s^'aQiy)

= (1 - s ' - ^ p a f - i - l)a( —O - 1 ^-,-^^) + [(1 - s^cq-'ai-i)-1

+ s 1_ ']a(y)

because ocQ = a. A glance at the second expression on the right of (11.35) shows that cq-'ai-i)-1

= q-\p - qVp'^l = s _ i (l - s ) / ( l

- s 1 "') -s1-1).

230

Markov fields on the integers II

Consequently, the expression in square brackets equals s '(1 — s) + s 1 ' = s~'. Thus a .Q

= (1 - s 1 _ i )pa(-J - l M - i T 1 ^ , - ! + s~'a.

Since a,Q and a are probability vectors, we conclude that a*ô = (1 - s~0^-i-i + s~*a = «,-+iWe have thus proved that {a;: i e Z} is an entrance law for Q. We let p0 denote the Markov chain for Q which is defined by this entrance law. That is, p0 is given by (11.10) with ^ = a; and ri = 1. Theorem (11.9) (a) asserts that p0 e ^(Q). The properties of {a;: i e Z] imply that 9j(p0) ^ 9k(p0) for all j , k e Z with j =£ k. In particular, p is not shift-invariant. But p0 = pQ on ^o.oot- Thus /i 0 is one-sided shift-invariant, in that p0(9iA) = p0(A) whenever i e Z and A is such that A e ^j0,oo[ a n d fy^ e ^ 0 ,œ[- Moreover, lim 9j(p0) = pQ. Since /x0 e #(ß)\# e (6)> Corollary (11.14)(b) shows that j->-CO

|ex^(Q)| = oo. However, much more can be said. (11.39) Theorem. Consider the matrix Q at (11.37), and let p0 be as above. Define pj = 9j(p0),j e Z. Then ex3(ß) = {/xQ}U{/x,:./eZ}. For each p e ^(Q), the limit Z = lim (<j; + i) exists in {— oo} U Z p-almost surely, and the unique extreme decomposition of p is given by p = p(Z=

- cc)pQ + X KZ =j)fij-

In particular, each p e ^(Q) is a Markov chain. Proof This proof is similar to that of Theorem (11.31). 1) For all n 2: 1 and x, y e E we have Q"(x,y) = lp"öx.n(y) + q<^" +1)v0 a(x A (n - l))]a(y)/a(x). By the definition (11.37) of Q, this equation holds for n = 1. The induction step runs as follows. For any x, z e E we can write Qn+1(x,z) = a(z)a(x)"1 X lP"àx-n(y) + 9 ( *-" +1)v0 a(* A (n - 1))] yeE

x C P ^ - I ( Z ) + c
A (n - l))(p + c(l - q)" 1 )]

= a(z)a(x)-1[p"+1<5;c_„_1(z) + 9<*-">v°a(* A n)].

Kalikow's example of phase transition

231

The last equality comes from the identities p + c(l — q)~l = 1 and cp V~"l[o,:c](") + qlx~H+l)v0a(x

A (n - 1))

k= 0 x An

= c £ PV * = q(x'n)v0a(x

A n).

The induction proof is thus complete. 2) Let fi e ex ^(ß) be given. Theorem (11.9)(c) asserts that /i is represented by a boundary law {/j,^: £e Z} via (11.10). Moreover, there are sequences (x„)ni ! and (y„)„z t in £ such that (11.40)

4/4,(0) = lim en+,'(x„, -)/e"(^,0) n-*oo

for i < 0, and (11.41)

rt/ro(0) = lim Q-\ •, yn)/Q"(0, yn) n->oo

for i > 0. Finally, there is no loss in assuming that r0(0) = 1. Combining Step 1) and (11.41), we obtain r,(x) = lim Lp'-'ôx-^iy,,)

+ g ( *-" + m > v 0 a(x A (n - i - l))]/a(x)

n->oo

= a(x)/a(x) = 1 for all x e E and i > 0. Thus r; = 1 for all £ e Z. Next, we identify the / ; 's. From (11.40) and Step 1) we conclude that (11.42)

ctt(y)/t0(0)a(y)

= lim t„,,(y)/t„,0(0). n-*oo

for all y e E and i < 0. Here t-.i(j') = Pn+%K-n-i(y) + ^ — i + 1 ) v 0 a ( x „ A (n + i - 1)). Suppose now that lim sup (x„ — n) = oo. Then for each £ < 0 and y e E n->oo

there are infinitely many n (namely all n with x„ — n — i > y) such that t nAy)ltn,oiS>) = q~'cc(n + i — l)/a(n — 1). Since a(/c) ~ apk as k -» oo, it follows from (11.42) that c/;(}>)//o(0)a(}>) = as _ i for all £ < 0 and y e E. As r, = 1 and /;?-,• = 1, a summation over y shows that c//0(0) = as~' for all £ < 0. Since this is impossible we conclude that lim sup (x„ — n) < oo. n->oo

Suppose next that liminf(x„ — n) = — oo. Then for each i < 0 there are n->oo

infinitely many n such that t„ t = a(x„). Inserting this into (11.42) we obtain c{i = 4)(0)a for all £ < 0. Since / ; and a are probability vectors, we arrive at the conclusion that 4 = a for all £ < 0. This implies that ß = ßQ.

232

Markov fields on the integers II

Finally, suppose that the sequence (xn — n)„ êl has a cluster point j e Z. Then xn — n =j for infinitely many n. For these n and all i < j A 0 and y e E we have '-.«(30 = Pn+iSj-t(y) + qj-i+l°L(n + i - 1). Using (11.42), we conclude that ctt(y)/*(y)m

= lim tn
= lim [g'--'- 1 p" + 'a(n + i - l ) - 1 ^ - , ^ ) + 1] n-»oo

=

9'--'-

1

pfl-1Vi(3')+l

for all i < j A 0 and y e E. Multiplying this equation by a(y) and summing over y we obtain c/4(0) = q'-'b^aU

- j) + 1 = qi-'b-^ap'-1

= s'-'-1.

Hence 4 = a - 1 p'--'a(j - 0^,-i + s 1 _ i + J a for all i <j A 0. Since *f;, <5y_;, and a are probability vectors, we finally arrive at the result that

for all i < j A 0 and therefore all i e Z. This shows that /x = /^. To summarize the results at this stage, we have proved that each fie ex ^(Q) belongs to the set {nQ} U {[if j e Z}. 3) We have /iQl lim (at + i) = — oo 1 = 1. This follows from the BorelCantelli lemma because

X HQ(at + i ^ k) = X

Z a(x)

< a(l — p) * £ p* ; < oo

for all k eZ. Similarly, ii}1 lim (<X; + i) = y 1 = 1 for ally e Z because

Together with Step 2), this shows that the limit Z = lim ((j, + i) exists in i-> — oo

{— oo} U Z /i-almost surely for all fie ex ^(Q). In view of the extreme decomposition theorem (7.26), we further conclude that Z exists /i-almost surely for

Spitzer's example of totally broken shift-invariance

233

all 11 e &(Q), and the extreme decomposition of any ß e ^(Q) is directed by the distribution of Z under \i. This in turn implies that each /Xj is extreme in &(Q); cf. Step 4) in the proof of Theorem (11.31). Alternatively, we can argue as follows. Since ^{Q)\{ixQ} # 0, there is some /x e ex^(Q)\{fiQ}. By Step 2), H = Hj for some j e Z. Thus /Xj e ex &(Q) for some j e Z. But the /x/s are translates of each other. Remark (7.2) therefore shows that fXj e ex ^(Q) for all j e Z. To prove the final sentence of the theorem, we note that each [x e ^{Q) admits a representation of the form (11.10) with 4 = fi{Z=-

oo)a + £ I*{Z

=j)a,-j

and rt = 1 (i e Z). Hence [i is a Markov chain. This completes the proof, D

11.4

Spitzer's example of totally broken shift-invariance

In the preceding examples of phase transition, Q was a positive recurrent matrix. Therefore it is natural to ask if there is any matrix Q which exhibits a phase transition but is not equivalent to a positive recurrent stochastic matrix. In view of Theorem (11.13), such a Q can never admit a shift-invariant Markov field. Since yQ is shift-invariant, one might wonder if such a Q can admit any Markov field at all. In fact, Corollary (11.14) asserts that such a Markov field, if it exists, has pairwise distinct translates. Is this case possible? F. Spitzer (1975b) has answered this question in the positive by providing an example which we are now going to discuss. It is clear from the above that in contrast to the preceding examples the main point of this example is the existence problem. As in the examples above, we take E = Z+. We consider a stochastic matrix P of the form P ( „ ) = f;*;"' rt * x i l ; y e f la(y) it x — 0, y e E. Here 0 0 for all x,yeE. In particular, P is irreducible. As in the preceding examples, it is convenient to think of P in terms of population dynamics. P describes the evolution of the number of inhabitants of a territory. At each time unit, the inhabitants survive independently of each other with probability p and die with probability 1 — p. At the time of extinction, a new population of size distribution a immigrates into the territory.

(11.43)

234

Markovfieldson the integers II

In the following we shall also need to look at the process with the same survival mechanism but without immigration. This process is described by the stochastic matrix (11.44)

P(x,y) = 4(x,p,y)

(x,yeE).

According to equation (11.23), the powers of P are given by (11.45)

Pn{x,y) = 6{x,p\y)

(n^l,x,yeE).

It is evident from the intuitive description of P that P is recurrent. A formal proof runs as follows. For each x e E, we let fip e ^(Ez*,éz*) denote the Markov chain with transition matrix P and starting point x. We look at the extinction time T = min {n ä: 1: er„ = 0). We obtain HP(t < oo) = a(0) + Y, oc(x)fip(t < oo) = a(0) + Y

a x

( ) hm

x^t 1

^P(T

= n)

n->oo

= a(0) + £ a(x) lim fâa, = 0) x^.1

n-^oo

= a(0) + X a(x) lim (1 - pnf = 1. The next to last equality is a consequence of (11.45). Since P is irreducible, the equation HP(T < oo) = 1 implies that P is recurrent. Next we will show that a can be chosen in such a way that P is null recurrent. Indeed, for each x ^ l w e have ^(T)= £ * > « )

= I

(1-(1-PT)

and therefore, by Fatou's lemma, liminf/ip(x) = oo. Consequently, we can find an increasing sequence (x(k))kèl in E such that /ip(fc)(T) ^ 2k for all /c. Thus, if a is any positive probability vector with ac(x(k)) ^ c2~k for some c > 0 and all k then A#(T) = 1 + E « W # ) ^ I

a(x(/c))/i^")(T) = oo,

and this means that P is null recurrent. Finally, we observe that Q is null recurrent whenever P is null recurrent. For on the one hand we have 2 n (0,0) = P 2n (0,0) ^ a(0)P 2 " _1 (0,0) for all n ^ 1 and therefore 2 £ ß"(0,0)^a(0) £ P"(0,0) = oo.

Spitzer's example of totally broken shift-invariance

235

Thus Q is recurrent. On the other hand, Q is null recurrent because

oo = /I°(T) = X tib > » ) ^ 2 I Ufa > 2k) ^ 2 X

/i°(T >k)

= 2/I°(T).

We are now ready to state Spitzer's result. (11.46) Theorem. Let P be given by (11.43), and suppose a is chosen in such a way that P is null recurrent. Define Q = P2. Then |ex ^(Q)\ = oo, but ö,(/i) ^ /i for allj e Zand ne ^{Q). Proof. Since Q is null recurrent, Remark (11.7) ensures that Q is not equivalent to a positive stochastic matrix. Thus ^®(Q) = 0. Consequently, the theorem will follow from Corollary (11.14) provided we can show that ^(Q) ^0. The proof of this is broken up into four steps. 1) The powers of P satisfy the equation (11.47)

P»(x,y) = â{x,p",y) + " f ^ ( T = n - k) [Pk(0, y) -

ô0(y)l

k= l

Here n 5; 1, x 5; 1, y 2Ï 0, and ô0 is Kronecker's delta. For we can write Pn(x,y) = n*(on = y,x>n)

+ n$(on = y,x ^ n).

The first term on the right is equal to Hxp{an = y,T>n)

= (l-

ô0(y))^(an

= y)

= P"(x,y)-ô0(y)fif(<7n =

= 0)

Hx,p",y)-ö0(y)fi*P(^n).

Because of the Markov property of /i£, the second term can be written as t

^ ( T = m)ß°P{an.m = y) = " f fifc = n-

m=l

k)Pk(0,y).

k=0

Combining all this and noting that P°(0,y) = ö0(y) we arrive at (11.47). 2) Let (xn)nèl be any sequence in E such that x„p" ^ 1 and xnp" -> 1 as n -> oo. (For example, x„ might be the smallest integer exceeding p~") Then the limit ai(y)=

limPn(xn_hy) n-*oo

exists for all y e £ and i e Z, and cct is a probability vector on £. To prove this we put x = x„_; in (11.47). Since lim x„_,p" = p1', the Poisson convergence theorem implies that "^°° lim t(xn-l,pK,y)

= fc{p\y\

236

Markov fields on the integers II

where /(p 1 , • ) is given by (11.21). As for the second term on the right of (11.47), we note that fipni(x = n — k) = HP"'(T 5= n — k) — HP"'(T ^ n — k — 1) Pn-k(x^i,0)-Pn-k-1(xn.i,0)

=

— H — nn~kYni

— (1 —

nn~k~1)xn-i

Hence lim fipni(T = n — k) = exp( — p'~k) — exp( —p'"*"1) for all i e Z and k ^ 1. Moreover, /X*"-'(T

= n - k) ^ (1 -

n k in p

-y

^ exp(-p'"*)

and

Z

ex

P(-?'-*)< °°-

The dominated convergence theorem thus gives = » - k)[_Pk(0,y) - ô0(y)]

lim " f nï-ii

[exp(-p;-k)-exp(-pi-'t-1)][P't(0,>')-^0(>')]

= Z

for all i e Z and y e E. This proves the existence of a,-. The sum over _y of the last expression equals zero, and fe(p\ •) is a probability vector. Hence a,- is a probability vector. 3) {<x;: i e Z} is an entrance law for P. Indeed, from Fatou's lemma we conclude that alP{y)=

Z \im xeE

P"(x„_i,x)P(x,y)

n->oo

^ lim infP n + 1 (*„_;, y) n-*oo

= <*t+i(y)

for all i e Z and y e E. The strict inequality is impossible because cctP and ai+1 are probability vectors. Hence <xtP = <xi+1 for all i. In particular, at-2Q = «; for all i, and this implies that the a;'s are strictly positive. 4) To complete the proof we put ^ = <x2i and rt — 1, i e Z. According to Step 3), \ßi,r{. i e Z} is a boundary law for Q. Theorem (11.9)(a) thus shows that <S(Q) ^ 0. a In the proof above, we have constructed an element fi1 e ^(Q) by means of a sequence (x„)„ äl in E with x„p" -> 1. The same reasoning applies to any sequence (*„)„>! with x„p" -> c for some c > 0. This gives us a Markov chain

Spitzer's example of totally broken shift-invariance

237

lic e ^(Q). The construction shows that 0,(/ic) = // c p 2 j for all j e Z and c > 0. Moreover, using the fact that Z i<0

Z l^'fa = *) - >(cp 2 ',x)| < °o J E £

and applying a Borel-Cantelli argument similar to that in Step 3) of the proof of Theorem (11.31), we see that W

limCT;/?"2'= c I = 1

for all c > 0. In view of the extreme decomposition theorem, this implies that ex @(Q) is uncountable. It would be interesting to know if the Markov chains fic are extreme in ^(Q), and if in this case there are any further extreme elements of ^(Q). We suggest this problem to the reader. Finally, we note that the construction of nc does not rely on the hypothesis that Q is null recurrent. Consequently, choosing a in such a way that Q is positive recurrent we obtain a further example of phase transition which exhibits uncountably many phases but also admits a shift-invariant phase.

Chapter 12 Markov fields on trees

In the preceding two chapters we have seen that Markov chains are very useful in the study of Markov fields on the simple graph Z. We are thus led to ask if the set Z of integers can be replaced by a parameter set S with a richer graph structure which is still simple enough to admit the notion of a Markov chain. Since it is impossible to define Markov chains on graphs containing loops, we must require that S be a tree. A particular tree of special interest is (ßSTid), the Cayley tree of degree d ^ 1. ^STid) is the unique connected tree with d + 1 edges emanating from each vertex. Note that ^^(1) = Z. It will turn out that the Cayley trees of degree d > 1 are large enough to admit a phase transition for Markov fields even when the state space E is finite. For this reason we shall be content with looking at finite state spaces only. In Section 12.1 we shall first introduce a suitable notion of a Markov chain on a tree. Then we shall extend two basic results of Chapters 10 and 11 from the case S = Z to the case when S is a tree. In analogy with Theorem (10.21) we shall show that a Gibbs measure /i for a Markov specification y is a Markov chain whenever fi is extreme in @(y), and as a counterpart to Theorem (11.9) we shall obtain a characterization of the Markov chains in &(y) in terms of a suitable concept of a boundary law. In Section 12.2 we shall apply these results to the Ising model on a Cayley tree ^^~(d). We shall again obtain the result of Section 3.2 that no phase transition occurs when d = 1. In the case d > 1, however, the Ising model exhibits a phase transition for suitable values of the coupling constant and the external field. These critical values can be specified explicitly. In addition, we shall identify some special elements of the associated simplices of Gibbs measures.

12.1

Markov chains and boundary laws

Let £ be a finite state space and S the vertex set of a locally finite connected tree. By definition, this means that there is a distinguished set B cz {b cz S: \b\ = 2} of "bonds" or "edges" b = {i,j} between "adjacent" sites i, j e S which exhibits the three properties below. (i) Local finiteness. For each i e S, the set di =

{jeS:{i,j}eB}

of all neighbours of i is finite. Of course, this implies that

Markov chains and boundary laws

239

5A 4 (J di\A ie A

is finite for all A e ^ . (ii) Connectedness. For any two sites i,j e S there is a sequence i = i0, i x ,..., i„ = 7" in S such that {ik_l5 ik} e ß for all 1 ^ /c ^ n. Such a sequence is called a path from i toj. (iii) Tree property. For all i, j e S, there is only one path from i to 7. Consequently, we can introduce a metric d on S by letting d(ij) be the length n of the unique path from i to 7. In this chapter we shall be concerned with specifications on S which are Markovian in the following sense. (12.1) Definition. Let y be a specification for E and S. y is said to be a Markov specification if yA(cA = CI ') is J^ A -measurable for all £ e £ A and A e y . Clearly, each Gibbs specification for a nearest-neighbour potential is Markovian. Also, if y is Markovian then each fj. e $(y) is a Markov field, in that ^ satisfies the local Markov property A*(o"A = C r a = M*A = C I ^ A )

A*-a.s.

(Ce£A,Aen

Besides this Markov property, we can also introduce a stronger Markov property which corresponds to the one-sided Markov property of a Markov chain. In order to do so we need some further notation. For each bond {1,7} e B we let 17 denote the associated oriented bond which points from i to 7. The symbol B will stand for the set of all oriented bonds. Each site ke S induces a splitting of B into the sets Bk = {y e B: d{k,i) = d(k,j) + 1} and k

B = {ijeB:d(k,j)

=

d(k,i)+l}

of oriented bonds that point towards k and away from k, respectively. Similarly, each oriented bond ij e B defines a splitting of S into the "future interval" ]ij, col =

{keS:ijeÏÏk}

and the "past interval" ] - o o , y [ = {fceS:ye*#}. (12.2) Definition. A probability measure JX on (Q, &) will be called a Markov chain if

for all y e ß and j^ e E. Any stochastic matrix Pf • on E with

240

Markov fields on trees

H(a} = y\&[l]) = Py(o-„ y)

M-a.s.

for all y e E will then be called a transition matrix from i toj for ß. A Markov chain \x will be said to be completely homogeneous with transition matrix P if ß(aj = y\^{i])

= P(ai,y)

/i-a.s.

for all y e E and all ij e B. (12.3) Comments. (1) Standard arguments show that every Markov chain ß satisfies ß(A\^i_XtiJ[) = ß(A\^[i}) ß-a.s. for all A e ^y >00 [ and all ij e B. (2) Let ß be a Markov chain with transition matrices (Py-)y6g, and let a k = ak(fl) be the marginal distribution of ß at fc e S. Then (12.4)

A*(*A

= 0 = «*(£*)

El

ijekH:i,jsA

Pttfi'tj)

for all connected sets A e ^ and all Ç e -EA and fc e A. This is easily seen by induction on |A|. (3) Let ß be a Markov chain, and suppose that V is a copy of Z which is imbedded (as a graph) into S. Then the marginal distribution av(ß) of /i on V is a Markov chain in the sense of Definition (10.4). This follows readily from equation (12.4). (4) Let (Pij)iJSB be a family of stochastic matrices on E. (P y ) ije B is a family of transition matrices for a Markov chain ß if and only if there exists a family ( a A e s of probability vectors on E such that (12.5)

a,.(x)Py(x, 30 = a / y ) / ^ x)

(y e S, x,y e £).

This is because (12.5) is equivalent to the statement that the expression on the right of (12.4) is independent of the choice of k e A for all connected sets A e ^ and all Ç e EA. (5) Let P be a positive stochastic matrix on E. P is the transition matrix of a completely homogeneous Markov chain ß if and only if P is reversible, in that there exists a probability vector a on £ such that <x(x)P(x, >>) = a(y)P(y, x)

(x, y e E),

and in this case we have a = ak(ß) for all fceS. This follows from (12.5). Moreover, equation (12.4) shows that ß is invariant under the group 1(B) of all graph automorphisms of S. 1(B) is defined as the set of all transformations T e T of the form TO» = (a>r;i,)ieS(a> e Œ) with a bijection T*: S -> S which is such that {i^i, T^ j} e B if and only if {i,j} e B. In the special case S = Z (with the usual graph structure), 1(B) consists of both translations and reflections.

Markov chains and boundary laws

241

This is the reason why we speak of a completely homogeneous (rather than a homogeneous) Markov chain. (6) Every Markov chain /i is a Markov field. For let A e y be given, and let A e £f be a connected set with A U dA cz A. Equation (12.4) shows that /i((jA = (corç)/iK = C W ) = M K = C'?)MCTA = C ^ ' )

for all Ç, C e £ A , to e £ÔA, andrç,rç'e £A\(AUÔA) Summing over C and r/' we obtain /i(crA = (co7/)/i(<7aA = to) = /i(<7MA = (or])ß(aAUdA = Ceo). Therefore, if /i(cA\A = corç) > 0 then M(CTA

= CkA\A = oiri) = /i(ffA = C|ffaA = to),

and this means that

Since ^ is generated by the union of all these ^A\ A 'S, we conclude finally that ^ K = CI ^A) = /X(OA = CI ^3A)

M-a.s..

Hence ß is a Markov field, o In Section 12.2 we shall see that the converse of the last comment fails: There exist examples of finite state Markov fields on a tree which are not Markov chains. Nevertheless, the following analogue of Theorem (10.21) holds. (12.6) Theorem. Let y be a Markov specification. Then each ß e ex^(y) is a Markov chain. Proof. Let ij e B and y e E be, given. For each n ^ l we write A(n) = {k e S: d(k,j) ^ n} and A(n) = A(n) fl ]i/', oo[. In view of Theorem (7.7), ß is trivial on the tail c-field ST. A fortiori, /i is trivial on the smaller er-algebra ( | &)ij,

co[\A(n)-

»ai

This implies that Indeed, the u-algebra on the left is clearly contained in that on the right. Conversely, if/ is bounded and measurable with respect to the latter u-algebra then f(x
/x-a.s.

for all x e E because /(xersMi}) is measurable with respect to (~) &\i}< œ[\A(ll). Using the backward martingale convergence theorem, we conclude that

242

Markov fields on trees H((7j = y\&{(i)

= Hm n(oj = y\&{i]UVj,OO[\AW)

M"a-S-•

n-»oo

Since {1} U ~\ij, oo[\A(n) => dA(n) and n e &(y), the expression on the right is almost surely equal to lim 7M„Mj = y\-) = Hm n((7j = n-*oo

y\rMn))

n-*oo

= filaj = y n>l

By virtue of the inclusion

this implies that Mo-, = y|.F w ) = ii(Oj = yl^-«,,^)

/x-a.s..

Hence n is a Markov chain, D We now work towards obtaining a characterization of the Markov chains in <&(y) which is similar to Theorem (11.9). For simplicity we shall only consider positive Markov specifications. In view of Corollary (2.32), a positive specification y is Markovian if and only if y = y* for some nearest-neighbour potential . Setting (12.7)

Qb(C) = exp[-{;}(£;)]

when b = {i,j} e B and ( e Eb, we see that each positive Markov specification y can be written in the form (12.8)

yA(aA = œA\œ) = ZA(œr1

ft

Qb(œb),

ftHA#0

where A e ^ , a> e Q, and ZA{w) is a normalizing constant. It will often be convenient to think of Qb as a transfer matrix along the bond b. To stress this aspect we introduce a family {Qtj: ij e B} of positive matrices by writing (12.9)

Qij(x,y) = Q}i{y,x) = Qb{t)

whenever b = {i,j} e B, Ç e Eb, and x = Çt, y = £,-. (12.10) Definition. A family {tftj: ij e B) of (row) vectors *fy e ]0, oo[ £ will be called a boundary law for {Qij: ij e B} (or y) if for each ij e B there is a number Cij > 0 such that W

= c«,- EI

'*iß«M

forallxeE.

kcdi\{j}

This definition is consistent with Definition (11.8), as is explained in the example below.

Markov chains and boundary laws

243

(12.11) Example. Let S = Z (with the usual tree structure) and Q be a positive matrix on E. Define a family {Qi}: ij e B} by Q (M+1) = Ö and Q ^ - D = QT, where i e Z and QT is the transpose of Q. Let {/,-, r;: i e Z} be a boundary law for Q in the sense of Definition (11.8). Then {fi}: ij e B} defined by 4u+i) = A and 4i,i-i) = r7 is a boundary law for {Q^: ij e B} in the sense of (12.10). This is a straightforward computation, o The next theorem extends Theorem (11.9) to Markov specifications on trees. To state it we observe that if A e ^ is connected and k e dA then A fl dk consists of a unique element which will be denoted by kA. (12.12) Theorem. Consider a Markov specification y of the form (12.8), and let {Qtj: ij e B} be the associated family of transfer matrices. (a) Each boundary law {ß^: ij e B} for {(?y: ij e B } defines a unique Markov chain \i e ^(y) via the equation

(12.13) n(aAUdA = C) = zA J ] 4kA(Ct) f l Qbttb)kedA

fcHA#0

Here A e y is any connected set, £ e EAUdA, and zA > 0 a suitable normalizing constant. (b) Each Markov chain \i e @(y) admits a representation of the form (12.13) in terms of a boundary law {^ij: ij e B} which is unique in the sense that each £tj is unique up to a positive factor. Proof, (a) The first step in the proof of assertion (a) consists in showing that the expressions on the right of (12.13) are consistent. This means that

(12.14)

E ^ n « Ç K e£^

J n QAM = ZA n **&*) n e«.&)

teflA

fcnA#0

kedA

fcHA#0

whenever A, A e
?A n ^jQkjiCj) n keV

kedA\{i}

4, A (cj n QM>)bHA#0

Since {/l7: y e B} is a boundary law, the last expression coincides with the right side of (12.14) up to a factor zJcuzA. Summing over CAUAA w e s e e that this factor is 1. This establishes (12.14). As a consequence of (12.14), equation (12.13) defines a unique finitely additive measure on the algebra of all cylinder events, and thereby a unique probability measure fj. on (Q, #"). By definition, \x is positive on cylinder events. To show that \i is a Markov chain we fix any ij e B, x, y e E and œ e Q, and we let A e y be a connected set with i e A c ] — oo, y[. We set A = A U BA\{j}. Equation (12.13) shows that

244

Markov fields on trees H(a} = x\aA = oiA)/fi((Tj = y\aA = oiA)

= Sjt(x)Qji(x, oJty/jiWQjAy, <",-)• Summing over x e E we obtain

The expression on the right depends on at via co; only. We thus conclude that n(o} = yli^-oo,^) = /x(o) = y\^{i})

/x-a.s..

It remains to prove that /z e ^(y). Let A e if be given. Take any two configurations £, co e Q with £S\A = cos\A. Let A e y be an arbitrary connected set with A c A . Then we can write, using (12.13) and (12.8), M ° A = CAICT(AUflA)\A= — ^(^AUflA

= n

=

0i

(AödA)\A)llJ-$TA

^\Ud$ln(a\Ud\

=

ffl

A I CT(AUflA)\A =

^(AUflAjXA)

— ^AUAA)

QtiCVQM

bnAjt0 bnAjtQ

= ttvK = C A M / ^ A K = OJA\(O). Summing over £A e £ A we see that n e &(y). (b) To prove assertion (b) we fix any Markov chain n e &(y). [i is positive on cylinder events because y is positive. For ij e B and x, y e E we define Pij(x,y) — n(pj — y\a{ = x). Let A e if be connected, £ e Q, and a e £ be any fixed reference state. We write MffAuav =

CAU*A)

= /x(^)/x(5|A)/x(C|B)//x(^|B),

where A = {aA = a},

B = {a8A = ÇdA},

C = {aA = £A}.

Equation (12.4) shows that H(B\A)=

fi

PkJa,Ck).

kedA

On the other hand, we conclude from (12.8) that H(C\B)/n(A\B) =

yA(C\0/yA(A\0 fcnA^0

/

Consequently, equation (12.13) holds with ZA = /*(öA = a ) / I l ôfc( aa ) / kA

fcczA

kedA

Markov chains and boundary laws

245

and 4 ; (x) = Pjt(a, x)/Qß(a, x)

(ij eB,xe

E).

In particular, replacing A by A = A U {/} in (12.13) and comparing the resulting equation with (12.13) we obtain the consistency equation (12.14). But (12.14) implies that {£{j: ij e B} is a boundary law; cf. the first part of this proof. To prove the uniqueness of 0. Applying (12.13) to a singleton A = {i} and a configuration £ with Cj = x for some j e di and Çk = a for all k e di\{j} we obtain the equation

4(x)/4,(x) = (z{l]/z'{t]) n 4(Û)/<;,-(Û)kedi\{j]

It follows that ^ is a positive multiple of fy. This completes the proof of Theorem (12.12). • We have just seen that a boundary law is only determined up to a positive factor. It is sometimes useful, therefore, to introduce a normalization. The simplest normalization is that at a given reference state a e E. We will say that a boundary law {
4,(x)=

n

4ß«W/4ß«(a)

i}jeB,xeE).

kedi\{j}

We conclude this section with two corollaries of Theorem (12.12). The first deals with completely homogeneous Markov chains in a homogeneous setting. Suppose there is an integer d ^ 1 such that \di\ = d + 1 for all ie S. S is then called the Cayley tree or Bethe lattice of degree d and will be denoted by ^^(d). Clearly, ^^"(l) equals Z with its usual graph structure. Two embeddings of a part of <^"(2) into the plane are shown in Figure 1 below.

vy yy yy YY

Y YY

(o) (b) Figure 12.1 Two embeddings of the Cayley tree
246

Markov fields on trees

A Markov spécification y on ^STiß) will be said to be completely homogeneous with transfer matrix Qify satisfies (12.8) with functions Qb which are given by (12.9) with Qij = Q for all ijeB. (Note that the matrix Q is necessarily symmetric.) Let such a y be given, and suppose n e &(y) is a Markov chain. An inspection of the proof of Theorem (12.12) shows that ft is completely homogeneous if and only if the unique boundary law {/y: i / ' e ß } which is normalized at some a e E and represents ft, via (12.13), is completely homogeneous, in that / y = ( for all ij e 0 and some t e ]0, oo[£. In view of (12.15), { is then a solution of the equation (12.16)

£{x) = ^Q(x)//Q(a))d

(x e E).

We thus arrive at the following conclusion. (12.17) Corollary. Let d ^ 1 and S = <^~(d), and consider a completely homogeneous Markov specification y with positive transfer matrix Q. For any fixed reference state ae E there is a one-to-one correspondence between the completely homogeneous Markov chains /J. e @(y) and the solutions tf e ]0, oo[ £ of equation (12.16). This correspondence is established by equation (12.13) with £.. = t far all ij e B. The second corollary will provide a condition under which a convex mixture of Markov chains fails to be a Markov chain. This condition will be used in the next section to obtain examples of Markov fields which are not Markov chains. (12.18) Corollary. Let S = ^^{d) for some d^\, and let y be a completely homogeneous positive Markov specification. Also, let /i1, . . . , fiNe^(y) be pairwise distinct Markov chains and {///': ije B}, . . . , {^
Proof Let 0 ^ t l s . . . , tN < 1 be such that ]T t„ = 1, and suppose that n = ]T tn\in is a Markov chain. Let {*fy: y e ß} be the boundary law which n= l

_

describes \i and is normalized at a. Take any ij e B and let k e di\{ j} be as in the hypothesis. Applying (12.13) to the measures n, / ^ , . . . , fiN, the singleton A = {i}, and all configurations Ç with Çm = a unless m =j or k, we see that there are numbers c1,...,cN > 0 (depending on i) such that

WM)

= t tncJ}?(xV}?(y) n= l

The Ising model on Cayley trees

247

for all x, y e E. Since all boundary laws are normalized at a, it follows that

t ^^vn[4m»(z)-4"»(z)]2 = o m,n = l

for all z e E. Let m # n be such that tmtn > 0. Then ^ m ) = $•> for all ;ï e 5 , and therefore //m = //„, in contradiction to our assumption, a

12.2

The Ising model on Cayley trees

In this section we assume that S = ^^(d) for some d ^ 1. Thus S is the unique connected tree with \di\ = d + 1 for all i e S. For the state space we choose E = { —1,1}. We shall sometimes write + and — instead of 1 and — 1, respectively. The a priori measure on E is counting measure. For given parameters J e R (the "coupling constant") and heU (the "external field") we shall consider the Ising potential (12.19)

' -Ja^j <&£•*= J -for, 0

iîA = {i,j}eB, if,4 = {i}, otherwise

as well as the associated set ^(J,/i) = (§(<&J'h) of Gibbs measures. We shall use Theorem (12.12) to identify some particular Markov chains in <&(J, h). By virtue of Theorem (12.6) and some specific properties of the model we shall then be able to determine the phase transition region {(J,h)eR2:\${J,h)\

> 1}.

To begin with, we note that the Gibbs specification yJ-h for Q>Jh is a completely homogeneous Markov specification with a transfer matrix Q = QJth which, in view of (12.7) and (12.9), is given by <2(-,-) = exp(J-2/z(d+l)-1), (12.20)

ß ( - , + ) = ß( + , - ) = e x p ( - J ) , Ô( + , + ) = exp(J + 2h{d+ I)" 1 ).

According to Corollary (12.17), there is a one-to-one correspondence between the completely homogeneous Markov chains in g(J, h) and the numbers s > 0 which satisfy the equation (12.21)

s =

' ß ( - , + ) + sß( + ,+) N d

ß ( - , - ) + sß( + , - )

(Just put a = — 1 in the corollary and identify s with the vector / = (1, s).) To exploit this equation we pass from s to the new variable

248

Markov fields on trees

t = h(d + l)'1 + ^logs. Inserting (12.20) into (12.21), expressing s in terms oft, and taking logarithms, we arrive at the equation (12.22)

t = h + d q>j(t).

Here (12.23)

with w = tanh J. This gives us the following result. (12.24) Proposition. For given parameters d ^ 1 and J, h e U. there is a one-toone correspondence t -> ßt between the real solutions of equation (12.22) and the completely homogeneous Markov chains in ^(J,h). /x, is determined by equation (12.13) with *f y (-) = 1, *fy( + ) = exp(2t - 2h(d + l)" 1 ), j / ' e l The transition matrix Pt of jxt is given by ( eJ~'/2 cosh (J - t) e'-J/2 cosh (J - t)\ ' ~ Ve_J,~'/2 cosh (J + t) e J+ '/2 cosh (J + t)/ ' and the one-dimensional marginal distribution a, = <7;(/xr) (i e S) o/ jjr is a( = (2e" 2J + 2cosh2t) _1 (e^ 2J + e~2',e~2J + e2t). Proof. We only need to check the formulas for Pt and <xr. The first follows from the equations Pr( + ,+)/P r ( + , - ) = exp(2r + 2J), P t ( - , + ) / P t ( - , - ) = exp(2t-2J) which are easily derived by looking at the proof of Theorem (12.12)(a). The second formula then follows from Comment (12.3) (5). D The formula for Pt has the surprising consequence that »M = °j = +)Ht(<*i = °j = -)lnA°i = ~
+)

= exp 4J for all {i,j} e B. In other words, the coupling constant J is easily recovered from the bond marginal distribution of an arbitrary completely homogeneous Markov chain in &(J,h). A similar formula holds for exp4f, and this determines h via (12.22). For later purposes we also note that the magnetization of

The Ising model on Cayley trees

249

H, is given by (12.25)

n,(ot) = (e~ 2J + cosh 2() _1 sinh It

(i e S).

In particular, JU,( 0. Therefore we will now assume that J > 0. (The antiferromagnetic case J < 0 will be discussed at the end of this section.) The function q>j is evidently odd. Since J > 0, q>j is increasing and concave on [0, oo [. This can be seen from its derivative (12.26)

The maximal slope of q>j is j(t) -> J as t -> oo. As an immediate consequence of these qualitative features of q>j we obtain the following lemma. (12.27) Lemma. For given d ±t \ and J > 0 we define h{J,d) = max \_dcpj(t) — t] = min [t — dq>j{t)~\. The fixed point equation (12.22) has (i) a unique solution t^ = t^(d, J, h) when \h\> h(J,d) or h = h(J,d) == 0, (») two distinct solutions £_ < t + (depending on d, J, h) when \h\ = h(J,d) > 0; and (iii) three distinct solutions t_ < t # < t+ (depending on d, J, h) when \h\ < h(J, d). We will now compute the critical external field h(J, d). We put J(\) = oo and, for d > 1, (12.28)

J(d) = arcothd = ~ l o g ^ ^ - . 2 d— 1 If J ^ J(d) then dw rg 1. Since dw — 1 is the maximal slope oî dcpj(t) — t, this means that d(pj(t) — t is decreasing in t. Hence h(J,d) = 0 when J :g J(d). In the other case J > J(d), the function dcpj(t) — t has a positive slope at t = 0. Thus h(J,d) > 0. To find the explicit value of h(J,d) we note that d(pj(t) — t has a local maximum at 1/2

w w 1 where w = coth J = w" . Indeed, using (12.26) we see that dcp'j(t) = 1 if and only if sinh 2 1 = (dw — 1)/(1 — w2) and therefore (12.29)

t J d = artanh

250

Markov fields on trees

tanh 2 t = (1 + sinh 2 t) - 1 sinh 2 1 = (d - w)/(d - w). Evaluating q>j at tJtd we finally get '0

(12.30) h(J,d) = i

ifJ^J(d), 2

Jdw-

2

\Y

(d - wV'

d ar tanh -— — ar tanh if J > J id). \dw — 1/ \d — w J In particular, h(J, 1) = 0 for all J > 0. In the case d > 1, a straightforward calculation shows that h(J,d) = (d- 1)J + ~^—\og(d

- 1) - ^logd + o(l) as J -^ oo

and h(J, d) = \(d2 - l) 1 / 2 (i - J{d)f'2{\ + o(l)) as JI J(d). So far we have discussed equation (12.22). Now the point is that \@(J, h)\ > 1 if and only if equation (12.22) has more than one solution. This will be shown in the following theorem. In other words, the thorn-shaped region {(J,h)eU2: J> J(d), \h\

^h(J,d)}

is the phase transition region of the ferromagnetic Ising model on ^^(d). (This region is marked with an F in Figure 2 below.) In addition, this theorem will clarify the rôle of the completely homogeneous Markov chains in 'S{J, h). In order to state it we need some notation. We let ë?IiB)(Q, &) denote the set of all \i e ^(Q, &) which are invariant with respect to all x e 1(B); cf. Comment (12.3)(5). Similarly, we put %B)(J, h) = %(J, h) n ^I(B)(^i, F). Moreover, combining the notations of Proposition (12.24) and Lemma (12.27) we shall write H*, ß-, fi+, and \i# instead of nu, \it , /i r+ , and /i r# , respectively. That is, \i^ is the unique, /i_ the minimal, fi+ the maximal, and n# the intermediate Markov chain in ^I(B)(J, h). (12.31) Theorem. Consider the ferromagnetic Ising potential (12.19) with parameters J > 0 and he U on the Cayley tree S = ^^(d) of degree d^.1. Let J(d) and h(J, d) be defined by (12.28) and (12.30). (a) J ^ J(d) or\h\> h(J,d) then 9(J,h) = 9I(B)(J,h) = {^}. (b) In the opposite case we have ß-, p+ e ex^(J,h) fi ex^I(B)(J,h) and \ex9(J,h)\ex&I(B)(J,h)\

= oo

Also, if \h\ < h(J,d) then \i# is a third extreme element of 'Si^J, h).

The Ising model on Cayley trees

251

Proof. 1) For given J and h we let £_ be the smallest and t+ be the largest solution of (12.22). (The case t_ = t+ = t^ is not excluded.) Also, we let {yy: ije B} be any boundary law which is normalized at — 1, and we put fy = A(d + i r 1 + i l o g ^ ( + )

(ijeÏÏ).

We claim that £_ ^ ttj ^ f+ for all ij e If. This follows from the equation (12.32)

ty = A+

I

%(tj

(ye ß )

fce3i\{j}

which is obtained from (12.15) in the same way as equation (12.22). Indeed, since q>j ^ J we conclude from (12.32) that ttj ^ h + dJ = i^(oo) for all ij e B. As (p7 is increasing, a second application of (12.32) shows that ttj ^ iA2(oo) for all y, where \j/ = h + dcpj. Repeating this argument, we see that ttj £j i^"(oo) for all n ^ 1 and ij e B. The sequence (i/'"(oo))näx is decreasing and bounded from below by t+. Its limit is a fixed point of ^ and thus equal to t+. This proves that ttj 5£ t+ for all y e B. The lower estimate of t; • is similar. 2) To prove assertion (a) we assume that J ^ J(d) or \h\ > h(J,d). Lemma (12.27) then shows that f_ = t+ = t^. Step 1) above and Theorem (12.12) thus imply that there is only one Markov chain in <&(J,h). In view of Theorem (12.6), this is only possible when \ex&(J,h)\ = 1. Using Theorem (7.26), we conclude that \&(J,h)\ = 1. 3) Next we assume that J > J(d) and \h\ S h(J,d). Then t_
= EI 4,A(CJ kedA

EI

-1)

LQ(Ci,Cj)/Q{-,-)l

{i,j'}nA#0

According to Step 1) above, {fu: ijeB} is dominated by the normalized boundary law defining p+. Hence rAC(p) ^ rAC(ß+ ). In view of Theorem (7.26), this inequality even holds for all n e &{J, h). Suppose now that n+ was not extreme. Then ß+ is a non-trivial mixture of two distinct measures \i, p' e ^(J, h). This implies that rAJ-(n+ ) is a non-trivial convex combination of the numbers rA^{ß) and rA^(p') which are at most >A,Ç(M+)- From this we conclude that rAI-(p+) = rA
252

Markov fields on trees

tn = h+dY is strictly monotone with limit in one of the (two or three) fixed points. We fix an arbitrary "root" k e S and put t(j = t„ if d(k, j) + 1 = d(k, i) = n. This defines ttj for all ij e Bk. For ij e kB, ti} can be defined uniquely by means of the consistency equation (12.32). The family {fu: ij e B} defined by ^ ( - ) = 1 and ^.( + ) = exp(2t y 2fc(d + l)" 1 ) is then a boundary law. The associated Markov chain vr belongs to <§(J, h) but fails to be completely homogeneous. To complete this step we note that the family {vt:t_ < t < t+} is linearly independent. This can be seen by a straightforward modification of the proof of Corollary (12.18). The result thus follows from Theorem (7.26). 5) To prove the final statement we let \i e <&(J, h) be any completely homogeneous Markov chain. Let P be its transition matrix and a its one-dimensional marginal distribution. We show that n is extreme in &I(B)(J,h). From Comment (12.3)(5) we know that /JE
zeI(B,n)

In view of standard extension arguments, it suffices to check this when A and C are cylinder events. In this case we can find connected sets A, À e y with Ae ^A and C e J ^ . For given k ^ 1 we let n ^ 1 be so large that ^(T" 1 A, A) ^ k for all T e I(B, n). Formula (12.4) then shows that sup luiAHT^B)

- n(A)n(B)\ ^max

veI(B,n)

xeE

£ \Pk(x,y) - oc(y)\. yeE

By Theorem (3.A3), the last expression tends to zero as k -> oo. This establishes the mixing property of \i, Now let A e SF be such that ^(-T 1 AAA) = 0 for all T e 1(B), and put C = A. The mixing property then implies that ß(A) = ß(A)2 and thus ß(A) = 0 or 1. Applying Corollary (7.4) to the set 11 = {Qx ^3(œ,A)-+

lA(xa>): x e 1(B)}

of all deterministic probability kernels which are induced by the transformations in 1(B), we conclude that ijeex ^ /(B) (Q, J5"). This completes the proof of the theorem, D As a by-product, the preceding theorem provides us with examples of Markov fields which belong to ^ /(B) (Q, #") but are not Markov chains. Indeed, if

The Ising model on Cayley trees

253

J > J(d) and \h\ < h(J,d) then Corollary (12.18) ensures that a non-trivial mixture of /i_, \i#, and /i + can never be a Markov chain. But each such mixture belongs to ^HB)(J,h). This observation shows that Theorem (10.25) fails when Z is replaced by a Cayley tree of degree d > 1. It is worthwhile to compute the magnetization [i+io^ in the critical case J > J(d), h = —h(J,d). In this case we have t+ = tj d; cf. (12.29). The derivation of (12.29) shows that sinh 2tj d = 2 sinh2 ts d coth tj d = 2w(l - w2)~l(d - w)l/2(d - w)1/2 and e" 2J + cosh 2tJd = (1 - w)/(l + w) + 1 + 2 sinh2 tJ>d = 2w(l - w 2 ) - 1 ^ - 1). Equation (12.25) thus implies that (12.33)

/x+(oi) = (d - w)1/2(d - w)l,2(d - I)"1

(i e S)

when J > J(d) and h = — h(J, d). Note that the expression on the right of (12.33) is positive, tends to 1 as J-> oo, and behaves like (d + l)d~U2(J — J(d))m as J | J(d). Let us summarize our results in the ferromagnetic case J > 0. If d > 1 and J > J(d), the Ising ferromagnet on ^^(d) exhibits the phenomenon of spontaneous magnetization against an external field. More precisely, if h < — h(J,d) then <&(J,h) = {/!„,} for some completely homogeneous Markov chain p.^ with magnetization /^(o,-) < 0; cf. (12.25). At the negative external field h = —h(J,d), there exists a completely homogeneous phase \i_ of negative magnetization. This \i_ is the limit of the unique phases for h < —h(J,d). However, in addition to \i_ a new phase /i + e ^I{B)(J, —h(J,d)) emerges. /i + has a positive magnetization which is given by (12.33). If h increases, /i + splits into two completely homogeneous Markov chains / i # and /i + .The magnetization /i+(<7;) increases with h, whilst ß#{at) is a decreasing function of h and reaches 0 at h = 0. The negative phase /i_ continues to exist, and besides /i_, / i # , and /i + there is an infinite number of inhomogenous phases. At h = h(J, d), the Markov chains /i_ and \i# coalesce. All phases except /i + disappear when h > h(J, d). Finally, let us turn to the antiferromagnetic case J < 0. In this case q>j is decreasing. As a consequence, equation (12.22) has only one solution t*. In view of Proposition (12.24), this means that there exists only one completely homogeneous Markov chain /i^ in ^(J, h) for all J < 0 and he U. However, in contrast to the ferromagnetic case this does not imply that the Gibbs measure is unique.

254

Markov fields on trees

In order to determine the antiferromagnetic phase transition region we introduce the notion of an alternating Markov chain. This notion is suggested by the fact that S — ^^(d) is a bipartite graph. By definition, this means that S can be decomposed into two disjoint subsets S0 and S t such that \b (1 S0\ = \b Pi S t | = 1 for all b e B. A boundary law {
if J e S 0 ,

°' Vi

tf/e^.

=

Similarly, a Markov chain in <£(J, h) will be called alternating whenever it can be represented by an alternating boundary law. An obvious modification of the derivation of equation (12.22) shows that an alternating boundary law for Q>Jh is described by a pair (t0, tx) of real numbers satisfying (12.34)

t0 = h + dcpjit,), t, = h + dq>j(t0).

Of course, t0 then solves the equation (12.35)

t = h,hJt)

4 h + d
Conversely, if t solves (12.35) then the pair (t, h + dq>j{t)) defines a unique alternating boundary law. Consequently, there is a one-to-one correspondence between the solutions of (12.35) and the alternating Markov chains in ^(J,h). Since \I/J 1 if and only if h e I(J,h). To see this we note that (12.35) has more than one solution if and only if there exist two real numbers t0 < t± which satisfy (12.34). In geometrical terms, the latter means that the line {(xl,x2) e U2: x t — x 2 = h] meets both the left lower and the right upper vertex of a square in U2 the other two vertices of which belong to the graph of the decreasing function dcpj. When h varies, the centers of all such squares form a connected curve M(J, d) in U2 which links (but excludes) the points ( - t-Jtd, dq>j( - t-.Jid)) and (f_Jid, dq>j(t-Jid)). (Note that j and recall (12.29).) We thus conclude that (12.35) has more than one solution if and only if h belongs to the interval I(J,d) = {*! - x 2 : (x1,x2) e M(J,d)}. l{J,d) is non-degenerate because sup/(J,d) ^ h(J,d) 4 t_Jj(t„J
+

2t„Jtd>0.

The Ising model on Cayley trees

255

In fact, numerical calculations suggest that I(J,d) =

]-h{J,d),h(J,d)l.

An explicit formula for h{J,d) is easily derived from (12.29). The resulting picture of the antiferromagnetic phase transition region is shown in Figure 2 below.

-

2

-

1

0

1

2

Figure 12.2 The phase transition regions of the Ising model on <^~(2). The antiferromagnetic region AF is open, whilst the ferromagnetic region F includes its boundary line except for the singular point. In analogy to Theorem (12.31) we can also obtain some information about the structure of phases when (J, h) belongs to the antiferromagnetic region. Suppose that J < —J(d) and h e I (J, d), and let t0 and t1 be the smallest resp. largest solution of (12.35). Then (12.34) holds, and t0 < t* < t1. Whilst ^ corresponds to the unique completely homogeneous Markov chain / ^ in & (J, h), the pairs (t0, fx) and (f1; f0) define two alternating boundary laws and thereby two distinct Markov chains /*_+, /*+_ e ^(J,h)\^IiB)(J,h). With the help of ideas similar to those in Step 3) of the proof of Theorem (12.31), one can show that /i_ + and /x+_ are extreme in 1.

Chapter 13 Gaussian fields

The theory of Gaussian fields is itself a well-known branch of Probability Theory. In this chapter we will show how this classical subject fits into our Gibbsian framework. Section 13.1 will be devoted to the question of whether a given Gauss field / i o n a countable parameter set S is a Gibbs measure for a suitable specification y. This will turn out to be the case whenever ß is non-degenerate and satisfies a weak kind of Markov property. The specification y with ß e @(y) will be Gaussian, in that each yA(-\m) is Gaussian field. In addition, y will be Gibbsian for a pair potential O. 0 . (ii) \i'S{yJ,h) # 0 then ex ^(yJ'h) is isomorphic to Mjo in that e x ^ ( y ^ ) = {T m ( M ):meM,, 0 } for each fi e ex^(yJ'h). In particular, if MJ<0 =£ {0} then yJ'h exhibits a continuous symmetry and ex^(y J ' Ä ) is either empty or uncountable. We shall also see that each \i e ex^(yJ'h) is a Gaussian field. Its covariance function depends only on J, whilst its mean belongs to an affine space Mj h c IRS which is characterized by an infinite system of linear equations. In view of these results, the existence problem for ^(yJ,h) is equivalent to the question of whether it is possible to construct an inverse C of J which can serve as a covariance function for the Gauss fields in ex^(yJ,h). This question is easy when J has finite range. In the general case we shall, at least, provide a sufficient condition. Combining all these results, it will be simple to construct Gaussian examples of continuous symmetry breaking. In Section 13.3 we shall turn to the special case when S = Zd and yJ'h is

Gauss fields as Gibbs measures

257

shift-invariant. In this case, the existence problem admits a satisfactory solution in terms of Fourier analysis. We shall prove that &(yJ'h) # 0 if and only if Mj h # 0 and the reciprocal J" 1 of the Fourier transform J of J is integrable with respect to Haar measure. One corollary of this theorem will assert that there is a unique shift-invariant Gibbs measure ß e ^{yJ,°) with /i(CTo) < °o if and only if J has no root. At the end of this chapter there is an appendix which contains the proofs of some auxiliary results about Gaussian random vectors and positive definite functions on Zd.

13.1

Gauss fields as Gibbs measures

Throughout this chapter, the state space E is chosen to be U, the real line. E is equipped with the Borel a-algebra S, and the a priori measure A is Lebesgue measure. In this and the next section, the parameter set S is an arbitrary countably infinite set. (13.1) Definition. A probability measure \i on (Q, #") = (£, S)s is called a Gaussian field or a Gauss measure if all finite dimensional marginal distributions <JA{ß) (A e y ) are Gaussian. The vector m = {mt)ieS = (^(o-;));eS is then called the mean of \i, and the symmetric function C(i,j) = HÜOi - mjicj - m,-)) = H^j)

~ mi™j (Uj e S)

is called the covariance function of \i. ß is said to be centered if m = 0. As is well-known, the preceding definition can be restated in Fourier analytic terms as follows. A probability measure \i on (Q, 2F) is Gaussian with mean m and covariance function C if and only if (13.2)

^(exp

t.-CT, ieS

exp - ~ Z hC(i,j)tj + i £ t,mt •£ iJeS

ieS

for all real sequences (£,.),.6S for which {ie S: t{ # 0} is finite. Here i stands for the imaginary unit. Recall that a symmetric function C: S x S -» M is said to be nonnegative definite resp. positive definite if (13.3)

£

UiC{i,j)Uj^0

resp. > 0

iJeS

for all complex sequences (u;), eS with {i e S: ui # 0} e Sf. It is well-known, and easily seen, that a covariance function is always nonnegative definite. If a Gaussian field p has a positive definite covariance function then each marginal distribution oA(n) of ß has the well-known Gaussian density with respect to 1 A (A e Sf).

258

Gaussian fields

In this section we shall address ourselves to the following question. Is every Gauss field a Gibbs measure? More precisely, is it possible to associate to a given Gaussian field fi a Gibbsian specification y in such a way that \i e <&{y)1 To answer this question we shall proceed in three steps. First, we shall look at the conditional expectations (13.4)

tf

= ufal^)

(ieS),

and we shall assume that fi(at ^ Çf) > 0 and that each £f only depends on finitely many spins. By virtue of general properties of Gaussian distributions, this will imply that (13.5)

ai-Z?

= J(i,i)-1 K + X J{i,j)oj

/i-a.s.

for a suitable vector h = (/j;)ieS e Q and a positive definite symmetric function J: S x. S ->M. (Equation (13.5) asserts in particular that the conditional expectation £f is affine.) We then shall conclude from equation (13.5) that the conditional distribution of ci given $~^ is Gibbsian for a quadratic pair potential $ = 0J,h. Finally, we shall conclude from the positive définiteness of J that $ is 1-admissible. This will allow us to define the Gibbsian specification y®, and Theorem (1.33) will imply that fi e ^(y®). Turning to the first step of this program, we introduce the covariance function (13.6)

T(i,j) = n((ai - tf)(d, -

tf))

(ij e S)

of fi. (F(i, i) is often called the mean square interpolation error.) (13.7) Proposition. Let \i e ^(Q, IF) be a Gaussian field with mean m. Suppose (i) T(i, i) > 0 for all i e S, and (ii) n is Markovian, in that for each i e S there is some di e £f with i $ di such that Çf has an ^di-measurable version. Define

j(i,j) = T(i,j)/r(i,i)ru,j)

(ijes)

and /!,= - £

J(i,j)mj

(i e S).

jeS

Then J(i,j) = 0 unless j e {i} U di, J is positive definite, and equation (13.5) holds for all i e S. Proof. First of all, hypothesis (ii) implies that £f = /^il^ai) A*-a.s. for all i e S. Therefore, if j <£ {i} U di then ai — Ç? is ST^ -measurable, and this gives Thus J(i,j) = 0 unless; e {i} U di, and this shows that the sums in the definition oîhj and on the right of (13.5) are finite.

Gaussfieldsas Gibbs measures

259

Next we take advantage of the fact that a^di{ß) is a Gaussian distribution. As is well-known, this ensures the existence of real numbers ut and vtJ such that P{ai\&rdi)= - « £ - X vuaj

A*-a.s..

See Proposition (13.A4) in the appendix. Letting vu = 1 and vtj = 0 when j $ {i} U <9i, this can be rewritten as (13.8)

/*-a.s..

In order to identify the vy's we observe that H(ok(o} - $)) = n(ak nipj - ^\3r{j))) = 0

(k ± j)

and Hiafa - £/)) = r ( ; , ; ) + täftaj - £/)) = r(j,j)

(j e S).

Equation (13.8) thus implies that

raj) = JLt + Zs w V - - tf)\ keS

for all i,j e S. To identify the M/S it is sufficient to integrate equation (13.8) with respect to \i. Equation (13.5) then follows from (13.8). It remains to show that J is positive definite. It is evident from the definition of J that J is nonnegative definite. To prove the strict inequality in (13.3) we look at the symmetric matrices fA = (J(ij')),^-6A> A e y . The nonnegative definiteness of J just means that all eigenvalues of all ,/ A 's are nonnegative, and J is positive definite if and only if all eigenvalues of all ,/ A 's are positive. Consequently, we need only show that each ßK is invertible. To this end we fix any A e ^ and define a matrix TA = (rA(z'J)); jeA by TA(fj ) = ptfa - nt)(oj - >7A)) = Hifiiioj -

tf))

(U e A )-

Here rçtA = \i{a^^). We claim that TA is an inverse of ßK. Indeed, for each i, k e A we can write X J{iJ)TAU,k)= j'eA

Z

J(i,j)n(aj(ak-^))

jEA

= X J(Uj)nH
= j(uM(°i-zn(°k-ri£)). The last equality follows from (13.5). If i # k then Ufa - #)(** - ,,*)) = KKOi - m^{i}H°k

- rf)) = 0.

260

Gaussian fields

In the case i = fc we have

This shows that ,/ A r A is the identity matrix and proves that J is positive definite. • Next we will use equation (13.5) in order to show that the conditional distribution of ai given 3~^ is Gibbsian for a pair potential O = 0 J ' h . For later purposes, we shall forget about the particular definition of J in Proposition (13.7). In particular, we shall dispense with the finite range property of J which was a consequence of the Markov hypothesis (ii) at (13.7). However, we still need to make sure that the sum on the right side of (13.5) is well-defined almost-surely. We shall do this by simply assuming that the set (13.9)

Qj =
iesl

has probability 1. Note that Qj e ST. (13.10) Lemma. Let J: S x S ->U be symmetric and positive definite, and let heQ. Also, let fi e £?(Q, J^) be a Gaussian field with the properties

(0 M " J ) = i. (ii) n({pi - £f) 2 ) = J(U 0" 1 f°r

al1

» e s>

and

(Hi) fi satisfies (13.5) for all i e S. Then, for each i e S, the conditional distribution of oi given 3~^ with respect to fi is Gaussian with expectation £f and variance Jfciy1. More explicitly, for each A e J^ we have H{A\Zr{i]) = A{i](p{i]lA) H-O.S.. Here p^y. Qj -> [0, oo[ is given by P{i} = Z ^ e x p

\J(i, i)af - Ufa - £ J(i,;>iOj ./Vi

with a ST^ -measurable normalizing function Z^ Proof Fix any i e S. By definition (13.4), the random variables ai — £f and Oj are uncorrelated for all j ^ i. Also, for each A e ^ the random vector (a,. — £f, t7A) has a Gaussian distribution. This follows from (13.5) because the image of a Gaussian distribution under a linear mapping is again Gaussian, and the almost sure limit of Gaussian random vectors is also Gaussian. (Look at (13.A1) for the first statement, and Proposition (13.A5) for the latter.) Remark (13.A3) thus implies that at — Ç? and
Gauss fields as Gibbs measures

nW{t})

261

^K^ai~iv\^{i))

=

= e"«ï/i(e' t( ^-^)

= exp[^r-it 2 M(^-^) 2 )] = exp [it^f - {t2J{i, i)" 1 ]

p-a.s..

But this means that the conditional distribution of 0 because J is positive definite.) We now can use (13.5) in order to replace £/* on Qj by the random variable

Zi=-W)

-1

hi + X •*(U)o)

With the help of the Gaussian density function p{i] = (J(M)/27r)1/2exp[-(<7£ - £,)2./(M)/2] on Qj, the last equation can then be rewritten as

P{j} clearly coincides with the function p ^ in the statement of the lemma. The proof is thus complete, G The conclusion of the preceding lemma can be restated as follows. For each i e S, the conditional distribution of at given 3~^ is Gibbsian with respect to the potential
\J{i,i)af+ hiai iîA = {i}, (13.11)
\{A = {i,j}, i =£j, otherwise.

There is, however, one technical point to which we must pay attention. If J has infinite range then the Hamiltonians HJ^h{co) for <&Jh do not exist for all œ e Q. Thus <J>Jh is not a potential in the strict sense of Definition (2.2). Nevertheless, H{'h(co) clearly exists for all co G fij5 and this will be sufficient for our purposes. The main question concerning (bJh is this: Does (5)J,h define a Gibbsian specification? For this we need to check if (J-7-'1 is A-admissible in the (slightly modified) sense that the partition functions Z^'h for
A = (J(iJ)) U eA

(AeSn

are invertible. (Cf. the proof of (13.7).) (13.13) Proposition. Let J: S x S ->Mbe a symmetric function, heQ, and
262

Gaussian fields

only if J is positive definite. In this case, the Gibbs distribution yi'k(-\o)) for O J ' / I and A e £f is the unique Gaussian field with mean

- £ /Ä1(Uk)(hk+

m;(A, CÜ)

keA

\

Y J(k,j)œ) i$A

ifieA,

J

ait

ifi£A

and covariance function {/Ä'iUj) [0

ifUjeA, otherwise.

Proof. Let A e 5^ be fixed. We will show that Z3^h(œ) < oo if and only if the matrix ^K is positive definite. Since m e Q J ; the Hamiltonian

^ n c ^ v O = E httt + \ E JdJKiCj + E d E J(Uj)o>j ieA

i,je \

ieA

j $A

is well-defined for all ( e £ A . Let us think of each ( e EA as a column vector, and define a row vector a(A, œ) — (Ö;(A, œ))ieA by a;(A,CÜ) =fc,-+ E

J

{hj)o3y

Then we can write tfA'h(CcosXA) = a(A,eo)C + K r A C , where £ r is the transpose of (. Let 0 A be an orthogonal matrix such that @>A = GTKßK(9K is a diagonal matrix. The diagonal entries qt (i e A) of @>A are the eigenvalues of ßK. We put b(A, œ) = (b;(A,co))ieA = a(A, œ)(9A. As AA is invariant under orthogonal transformations, we obtain Z

A

' » = j 2A(dC)exp [-a(A,co)C - K V A C ] = jA A (dC)exp[-fc(A,eo)C-K r S>AC] = ]^[ j A(dx)exp[ — bi(A, œ)x — ^ ; x 2 /2]. i'e A

The last product is finite if and only if qt > 0 for all i e A. This proves the first part of the proposition. We now assume that ßK is positive definite. Then we can write, putting m(A, o>)= - / A 1 a(A, œ)T, H{'"(CcoSSA) = HC - m(A,œ))TMC T

- jm{A, œ) fAm(A,

- m(A,œ)) œ).

The second term on the right depends only on œs\A, whilst the negative exponential of the first term is equal, up to a normalizing constant, to the Gaussian density with mean m(A, co) and covariance i/A"1. Combining this with the fact that yi'h(-\o}) is supported on { the second part of the proposition follows, D

Gauss fields as Gibbs measures

263

We note that yi'h(-|co) depends on ca via its mean only. The formula for this mean admits a nice interpretation in potential theoretic terms, as we shall explain now. (13.14) Comment. Consider the setting of the preceding proposition. The formula for the mean m(A, ca) immediately shows that m(A, eo) is the unique solution of the equations Z J{UJ)mj(A, eu) = - h{

(i e A)

jeS

m,(A, co) = eo;

(i £ A).

These equations can be considered to be a discrete Poisson boundary value problem. (If ht = 0 for all i, we have a discrete Dirichlet problem.) It is a standard fact of potential theory that such a boundary value problem can often be solved by means of a related Markov process. To be precise, let us assume that J is symmetric and such that (i) J(i,j)S0 for all i^j, (ii) £ J(i,j) ^ 0 for all i, and JeS

(iii) J is irreducible, i.e., S is connected if any two distinct sites i, j with J(i,j) # 0 are joined by an edge. These conditions imply that J is positive definite. Indeed, if {ut)ieS is any complex sequence with at most finitely many non-zero terms then condition (ii) gives

Because of (i), the expression on the right is nonnegative and vanishes if and only if U; = u,- whenever J{i,j) =£ 0. In view of (iii), the latter means that u{ = 0 for all i e S. By virtue of our assumptions (i) and (ii), we can define a stochastic matrix P o n S = SU{oo}by ' - J(Uj)/J(i, 0

if U j e S , i / j ,

X J(i,j)/J(i, i) if i e S, j = co, (13.15)

P(i,j) = 1 1 .0

if i = j = oo, otherwise.

Our Poisson equation for m(A, co) then takes the form (13.16)

X (P(i,j) - SJmjiKco) = -9i m;(A, co) = a)t

(i e A), {i e S\A).

Here gt = —hJJ(i,i), coœ = 0, and <5y is Kronecker's delta. To solve this equation we fix an arbitrary i e S and let (I„)„ Ê 0 be a Markov chain on S

264

Gaussian fields

with transition matrix P and starting point i. The underlying probability measure will be denoted by ß'P. Let TA

= min {n ^ 0: Xn e S\A}

be the first exit time from A. Since P is irreducible, each i e A is a transient state for the Markov chain with transition matrix P and absorption in S\A. As A is finite, we conclude that A4( T A) < °° f° r all i e S. Suppose now that m(A, co) is a solution of (13.16). Then for each i e A and N ^ 0 we have /(TA-1)AJV

ml(\,oi) = ßÜ

Y^

\

3xn + mx^A(N+l)(A,(o)J.

Indeed, for N = 0 this is identical to the first equation in (13.16), and the induction step is an immediate consequence of (13.16) and the Markov property of (X„)„^0. The Markov property also shows that ßU\(0XzJ)=

Z /4(l{rA>„,*n+1*A}Nxn+1l)

= Z 4(i{ t A >.} Z ^^(xA)max Z

fftjii^-i)

\J(k,j)cOj\/J(k,k)

< 00

and thus yuj,

sup |mXt

AJ ,(A,Û))|

< GO

N>0

for all i e A. Letting N tend to infinity, we can therefore apply the dominated convergence theorem to get the formula (13.17)

m;(A, co) = / 4 ( V

9x

^ + ^(c« XtA )

(i e S)

for the mean of yi'h{-\(o). This formula is completely analoguous to the solution of the Poisson boundary value problem in terms of the Wiener process. (See Section 4.6 of Chung (1980), for example.) o After digressing to potential theory we now return to our main point, the definition of a Gibbsian specification for the potential <^J,h in (13.11). In view of Proposition (13.13), we need to assume that J is positive definite. We can then take the natural Gibbsian definition whenever the boundary condition œ belongs to Q 7 . What do we do when co $ Q,? The answer is simple: Since Qj is tail measurable, the definitions of yJ'h on Q, and Q\Q 7 do not affect each other. yJ,h can thus be defined on Q\Q 7 in any consistent way we like.

Gauss fields as Gibbs measures

265

(13.18) Definition. Suppose J: S x S -> U is symmetric and positive definite, and let he Q. The Gaussian specification yJ'h for J and h is then defined by

the equations .7.*,.,^ y A , 0 4 M

,Z^»-U A (l y l exp(-H^*)|û,)

ifweflj,

1<W^)

otherwise

In the above, A e y , a» e Q and ^ e !F are arbitrary, H^' 1 and Z^,h are respectively the Hamiltonian and the partition function with respect to the potential Q>J,h at (13.11), <50ACOS\A i s t n e Dirac measure at the configuration which is zero on A and equal to œ on S\A, and Q 7 is defined by (13.9). As Qj e ST, the same arguments as in Proposition (2.5) and Remark (1.32) show that yJ,h is indeed a specification. Proposition (13.13) asserts that each yi'h('\co) is a Gaussian field. Moreover, the definition of yJ,h on Q\Qj has been tailored in such a way that (13.19)

fi(Qj)=l

for all n e $(yJ-h).

In other words, the Gibbs measures for yJ,h "feel" only the Gibbsian part of their specification yJ'h. To prove (13.19) we assume there was some fi e $(yJ-h) with n(Qj) < 1. Then v 4 n(-|Q\Qj) G 9(yJ-h) because Slj e ST. Thus v(<jA = 0) = vyl'h((TA = 0) = 1 for all A, and therefore v = <50, the Dirac measure at the zero configuration 0. This is impossible because 0 e Q 7 and v(Qj) = 0.

We are now ready to state two theorems on the representation of Gaussian fields as Gibbs measures. The first deals with the Markovian case in which Q , = Q.

(13.20) Theorem. Let \i e ^(Q, SF) be a Gaussian field. Suppose that (0 n(at + ^(CTJI^;})) > 0 for all i e S, and

(ii) for each i e S there exists some di e if with i $ di such that ^(c;!^;}) = H(oi\&rai)H-a.s.. Define J: S x S ->U and h e Ü. in terms of the conditional covariances and the mean of fi as in Proposition (13.7). Then yJ'h is well-defined and \i e ^(yJ,h). Proof. Combine Proposition (13.7), Lemma (13.10), Proposition (13.13), and Theorem (1.33). D In contrast to the preceding theorem, the next one will become important in the next section. It characterizes the Gibbsian Gauss fields in terms of their means and covariances. It implies that a centered Gauss field \i with covariance function C is a Gibbs measure if and only if C, considered as an infinite matrix, has a positive definite inverse J such that /*(fij) = 1. We need to introduce some notation. For given J: S x 5 -> U. and h e Q we put

266

(13.21)

Gaussian fields

MJih = \ m e Qy. ht + £ J(Uj)m} = 0 for all i e S1.

Note that M, 0 is a linear subspace of Q = Us, and M, h = m + Mj 0 for each m G M,,„. (13.22) Theorem. Let n be a Gaussian field with mean m and covariance function C. Also, let h e Q and J: S x S ->M be a positive definite symmetric function. Then the following conditions are equivalent. (a) n e g(yJ-h) (b) H(£lj) = 1, m e MJth, and X J(i,j)C(j,k)

= ôik

(i,keS),

JeS

where aik is Kronecker's delta. (In either case, the sum above is absolutely convergent.) Proof, (a) implies (b). At (13.19) we have already shown that /i(Qj) = 1. Consequently, the limits Xt = lim X J(Uj)oj Aey

(i e S)

jeA

exist /i-almost surely. Since /i is Gaussian, these limits also exist in the L2(/i)-sense and, a fortiori, in the L 1 (/i)-sense; see Corollary (13.A6). Hence VL(Xt) = lim X J(iJ)mj

(i e S).

By virtue of a well-known theorem of Riemann, the existence of the limit on the right side implies that m e Qj. Similarly, we have fi((Xt - « ) ) f o - mk)) = lim X J(Uj)C(j, k)

(i, k e S).

4ey jeA

Since \L e (£(yJ'h), we conclude from Proposition (13.13) that Z* = m^i}, •) = -J(i,i)'1

(ht + X. J{i,j)oj\

/z-a.s.

and therefore a{ — £f — (ht + Xi)/J(i, i) /i-a.s.. Taking expectations with respect to fi, we obtain that ht + ^(Xt) — 0. When combined with our previous equation for /i(X;), this shows that m e Mj h. We also obtain, using Proposition (13.13) in the last step, that H((Xi - fi(Xd)K

- mk)) = J(U)K(Oi - Z?)(ok - mk)) ôikJ(i,m°î-(St)2)

= =

ôikJ{i,iMy{t)\<7?)-yfo\<7i)2)

= öik.

Gibbs measures for Gaussian specifications

267

Together with our previous equation for the covariance of Xt and ak, this proves that J is an inverse of C. (b) implies (a). Since fi(£lj) = 1, the random variable tl=-J{i,i)-1(hi+Z

J(i,j)o)

is well-defined /i-almost surely, and we can write °i - & = JiiJ)'1

X J{Uj){Oj - rrij) jeS

because m e MJh. By Corollary (13.A6), the sum on the right also exists in L2(fi). Thus /i((7,- — £;) = 0 and J(i,i)n((Oi - tufa - mk)) = Z J(UJ)C(j,k) = ôik jeS

for all i, k e S. In particular, if i =£ k then at — ^ and ak are uncorrelated, and the proof of Lemma (13.10) has shown that in this case ai — £,- and 2T^ must be independent. Hence £f - it = fi(at - Q^{1})

= ii(at - &) = 0

H-SL.S.,

and this establishes equation (13.5). We also have WMiot

- tf)2) = J(Ui)n((at - Q(at - mt)) = 1

for all i. Thus all hypotheses of Lemma (13.10) are satisfied. From this lemma and Theorem (1.33) we then conclude that \i e <S(yJ'h). (Theorem (1.33) does not apply as it stands because yJ,h is a positive /l-specification on Qj only. However, since fi(£lj) = 1 we are allowed to modify the definition of yJ'H on £l\Çlj at (13.18) in such a way that yJ-h becomes a positive ^-spécification. For example, this can be achieved by letting a be the standard normal distribution and putting yJ'h = a. on Q\Qj.) a

13.2

Gibbs measures for Gaussian specifications

In the preceding section we started from a given Gaussian field n and sought a Gaussian specification yJ'h with /i e &(yJ'h). Here we shall take the opposite point of view: For a given Gaussian specification yJ'h we shall consider the associated set (S(yJ,h) of Gibbs measures. We begin with a remark which shows the significance of the sets Mj h in (13.21) for the specifications yJ'h (provided these sets are non-empty). For each m e f l we let xm e T denote the spin translation by m, i.e., TmCO = CO + m = (CO; + W ; ) ; e S

(<X> G Q ) .

268

Gaussian fields

(13.23) Remark. Let J: S x S -» R be symmetric and positive definite, and let h,heQ. Then the following conclusions hold. (a) If m e Mj h then ^•h+fi(-|Tmt0) = T m ( ^ ' Ä ( » )

for all A e y and co e Qj. (b) For each m e M, h, £(/•*) =

{rm(n):ne%(yJ'0)}.

(c) <&(yJ'h) is preserved by the transformations xm with m e Mj 0 . (d) If Mj 0 contains an element m ^ 0 then either ^{yJ,h) = 0, or ex &(yJ,h) is uncountable. Proof, (a) Let A e y , co e Qj and m e Mj h be given. From Proposition (13.13) we conclude that yJK'h+f,(-\xmœ) and Tm()>;(,Ä(-|co)) are Gaussian fields with the same covariance function. It is therefore sufficient to prove that the mean values of these fields coincide. This is immediately checked, using the explicit expression for these mean values. (b) (c) In view of (13.19), these statements are obtained by an obvious modification of Remark (5.10). (d) Mj 0 is a linear space. Therefore, if Mj 0 ^ {0} then yJ,H exhibits a continuous symmetry, because of (c). The assertion thus follows from Theorem (7.26) and Remark (7.2). D Our first theorem on <&(yJ'h) will show that a probability measure JJL belongs to y(yJ'h) if and only if /z admits a representation as a random translation of a particular Gaussian field JJLC. This means that H = | v(dm)Tm(//c) for a suitable v e ^(Cl, J27). Obviously, the expression on the right is just the convolution /J.C * v of /J,C and v. (Recall that for any two probability measures Hi, n2 on (Q, J*) the convolution JJL^ * JJL2 is defined by the formula

Mi * M2(/) = 1 M d O J H2(dri)M + V). Here / is any bounded measurable function, and Ç + r\ = (£,- + /?;),-eS-) We remind the reader that the matrix fA was defined in (13.12). (13.24) Theorem. Let J: S x S -> [Rfeesymmetric and positive definite, and let heQ. Suppose that <${yJ'h) ^ 0. Then the limits (13.25)

C(i,j) = lim X\i,j)

(i,j e S)

AeSf

exists, and a random field ß belongs to ex (S{yJ'h) if and only if ß is a Gaussian field with covariance function C and a mean m e M} h. Consequently,

Gibbs measures for Gaussian specifications

269

9(yJ-h) = W * v : v e 0>(Q,F), v(MJih) = 1}, where \xc is the unique centered Gaussian field with covariance function C. Proof. Theorem (7.26) ensures that ex #(yJ-*) # 0. So let p e ex #(/•*). If ( A J n è l is any increasing cofinal sequence in y, then Theorem (7.12) implies the existence of an co e Qj such that \i = lim yi'*{ • |a>) in the topology of n-*oo

local convergence. In view of Propositions (13.13) and (13.A5), this implies that the limiting covariance function C at (13.25) exists and \x is a Gaussian field with this covariance function. Theorem (13.22) then shows that the mean m of n belongs to MJh. Now let \ic be the centered Gauss field with covariance function C. The random field Tm(jxc) = \ic * dm is then again Gaussian and has the same mean and covariance function as \i. Hence \x = \xc * ôm. In the above, we have shown that In particular, this implies that \ic = i~m{jx) for some m e M, h and \i e ex^(y 7 '"). Remarks (13.23)(b) and (7.2) thus imply that ficeex^(yJ'°). The same remarks then allow to conclude further that nc*ôm = x"'(lxc) e ex &(yJ,h) for all m e Mj h. Hence exV(yJ'h) =

{Hc*àm--rneMJth}.

The final statement of the theorem now follows from Theorem (7.26). D Theorem (13.24) provides a complete description of (S{yJ'h): <&(yJ,h) is isomorphic to the set gP{MJh,!F\Mjh) of all probability measures on Mj h, provided (S(yJ,h) =£ 0. We are thus left with the existence problem. Combining Theorems (13.22) and (13.24) with Remark (13.23)(b) we see that &{yJ'h) # 0 if and only if M, h =£ 0 and J has an inverse C such that nc(Q,j) = 1. (As before, Hc stands for the unique centered Gauss field with covariance C. The existence of JXC is guaranteed by Proposition (13.A7).) Moreover, this inverse C should be given by (13.25). If J has finite range, the condition ^ c (üj) = 1 is trivial. Thus in this case we have ^(yJ'h) ^ 0 if and only if MJh # 0 and the limit (13.25) exists. The next theorem is a slight improvement of this statement. (13.26) Theorem. Let heÇï and J: S x S -> 1R be a positive definite symmetric function such that {j e S: J(i,j) j= 0} e 9> for all i e S. Then <${yJ'h) =£0if and only if Mj h # 0 and (13.27)

sup fA\i,

i) < oo

for all i e S.

Proof In view of Theorem (13.24), we need only prove the sufficiency. We will show that condition (13.27) implies the existence of the limits in (13.25). To

270

Gaussian fields

this end we shall establish the following monotonicity property: If 0 ^ A c A e y and t = (tt)ieA e UA then

To see this we recall from Proposition (13.13) that ßAx is the covariance of y^,h(- |CÜ) when œ e Qj. Therefore we have, putting / = £ liai> I

fÄ1(UJ)titj

= yA(f-yAf)2

onQj.

i, j e A

(We omit the superscripts J and h of y, for simplicity.) As the function on the right is constant, it coincides on Qj with yAyA(/ - y A /) 2 = TATAC/2) - VACTA/)2By Jensen's inequality, this is at most TATAC/2) - (TATA/) 2 = yÀP)

- (TA/) 2

i.jeA

This completes the proof of the monotonicity property. Our first conclusion is that fAl{i, i) is an increasing function of A. Assumption (13.27) thus implies that C(i, i) = lim fAl{i, i) exists for all i e S. Secondly, if i ^j then

fÄHU) + fÄHjJ) + 2fAl(Uj) is an increasing function of A. But (13.27) implies that this function is bounded. For, the Cauchy-Schwarz inequality gives

fAl (i,j)2 S fAl(U OA" * ( J J)

(hi e A)

v

because ßA is a covariance. We thus conclude that the limiting function C at (13.25) exists. Since J has finite range, C satisfies

X J(i,j)C(j,k) = lim X J(i,j)fAHj,k)

= ôik

for all i, ke S. We also have Qj = Q. Moreover, C is evidently nonnegative definite. Thus there exists a centered Gauss field \ic with covariance C; cf. (13.A7). fic satisfies condition (b) of Theorem (13.22) with h = 0. Thus pc e <$(yJ-°), and Remark (13.23)(b) implies that xm{nc) e %{yJ-H) for each m e MJth. This completes the proof, D (13.28) Comment. Suppose J: S x S -> IR has finite range and satisfies the hypotheses (i) to (iii) of Comment (13.14). Let P be the stochastic matrix of (13.15). We claim that condition (13.27) is equivalent to the statement that each i e S is transient for P. Of course, the latter means (in the notation of Comment (13.14)) that

Gibbs measures for Gaussian specifications

271

00

for all i e S. Indeed, it is easily checked that

f^(i,j) = n\, f £ hxn=^jjj(hi)

(Uj e A).

(For example, you might set co = 0 and gk = ökj in (13.17) and insert the result in Proposition (13.13).) Thus (13.27) is equivalent to the condition sup til £

1

{x„=i}) < °°

(*'ÊS)-

Since sup TA is the hitting time of the additional site oo, the claim follows. We thus obtain the result that ^(yJ'h) ^ 0 if and only if MJyh ^ 0 and all i e S are transient for P. In this case, the covariance function C of the Gibbs measures for yJ,h is given by C(i,j)=

£ P»(i,j)/J{j,j)

(i,jeS).

We note that the transience condition holds trivially whenever £ J(i,j) > 0 jeS for at least one i e S. o (13.29) Example. The harmonic oscillator. Let d ^ 1, S = Zd, ß > 0, and r-ßßd J(i,J) = -\ ß [_ 0

i f | i - ; i = i, ifi=j, otherwise.

We are thus in the setting of Comment (13.28). The stochastic matrix P at (13.15) is just the transition matrix of the simple random walk on Zd. By Pôlya's well-known theorem, P is transient if and only if d ^ 3. Thus (&(yJ'h) = 0 for all h when d ^ 2, whilst ex <3{yJ,h) is isomorphic to M, h when d ^ 3. The set Mj 0 contains all constant configurations. By Remark (13.23), this implies that the specifications yJ,h exhibit the continuous symmetry co -> (co; -(- Oie s (t e R). It is also easy to see that MJh ^ 0 for all h. Thus, if d ^ 3 then yJ'h exhibits a continuous symmetry breaking. We shall return to this example in (13.43). o (13.30) Example. Let S = Z + be the set of nonnegative integers and (c„)„â0 be any sequence of positive real numbers. We define -c.-Aj

J(iJ) = i

if \i-j\

= 1,

Ci + c^

if

0

otherwise.

i=j,

272

Gaussian fields

(Weputc.j = 0.)Then Y.JiUj) = 0 for alii. The stochastic matrix P a t (13.15) is given by P(i j) = J c ' A j/( c i }0

+ C;

-^

if | !

~-J'l = *' otherwise.

This is the transition matrix of a so-called birth and death process on Z + . A theorem of Harris (1952) states that P is transient if and only if

" P(fc,fc-1) ntitÀP{k,k+l)

C0

"

(See also Breiman (1968), Proposition 7.37, for example.) The product term in the sum above equals c0/cn. Thus P is transient if and only if £ l/c„ < oo. Suppose now we are given any h e Q. It is easily seen that each m0 e R can be continued to a unique m e Mv>Ä. Thus MJh is isomorphic to R. In particular, Mv>0 consists of all constant configurations and yJ,h exhibits the continuous symmetry &)-•(&); + t)ieS(te R). We thus arrive at the following conclusion: (&(yJ,H) is non-empty if and only if £ l/c„ < oo, and in this case ex (&(yJ'H) is isomorphic to R. o "êl If J has infinite range, the existence problem is more difficult. A necessary and sufficient condition for existence in the infinite range case is only known when S = Zd and J is homogeneous; see Theorem (13.36) below. In the nonhomogeneous case, the next theorem provides a sufficient condition, at least. (13.31) Theorem. Let J: S x S -> R be a symmetric positive definite function, and let he Q be such that Ms h j= 0. Suppose J satisfies the conditions 1

c^supjao" £|j(i,j)i
j^i

and Z \J(iJ)\J(Jjr112

< oo

(icS).

3*i J h

Then <${y - ) j= 0. Proof. We introduce the matrices v

' - J(i,j)/J(i, i) if i, j e V, i j= j , ' 0 otherwise

with V <=. S, and Q(i,j) = \Ps(i,j)\, i,j e S. By hypothesis, the powers Q" of Q satisfy £ g"(U) Û cn for all i e S and n ^ 0. Thus JtS

£

X Q"(i,j) £ (1 - c)' 1 < oo

(i e S).

The homogeneous case

273

Since \Pv(i,j)\ ^ Q"(i,j) for all n, i,j, the matrices Cr(i,j) = I

Pv(i,j)/J(j,j)

(hj

eS,V^S)

are well-defined. It is easily checked that CA(i,j) = ^^{i,}) when A e £f and i,j e A. The dominated convergence theorem therefore implies that the limiting covariance C of (13.25) exists and is equal to Cs. (13.25) implies that C is nonnegative definite. Thus there exists a centered Gauss field ßc with covariance C. \xc satisfies condition (b) of Theorem (13.22) with h = 0. Indeed, it is obvious that C = Cs is an inverse of J, and the property ßc(^j) = 1 follows from the estimate

0 c ( x \J(iJ)°j\)=

Z l^(U)IMkjl)

^ Z |J(U)|C(;J)1/2 jeS

^ Z l ^ " J M ; j r 1 / 2 ( z ß"(7'J))1/2 jeS

\«ä0

12

lß

/

g (i - c)- / z \J(uj)\J(jjr JsS

< 00.

Theorem (13.22) and Remark (13.23)(b) thus show that &(yJ-h) # 0. a Using Theorem (13.36) below, one can easily see that the hypothesis of Theorem (13.31) is far from necessary. For example, if d — 1 then the function

J(Uj)

1 1/2 1/4 0

when i = j , when \i — j \ = 1, when | i - ; | = 2, otherwise

violates the condition of Theorem (13.31) but satisfies the existence condition of Theorem (13.36). (The latter holds because J ^ 1/4.)

13.3

The homogeneous case

Here we assume that S = Zd, the d-dimensional integer lattice. We shall look at homogeneous Gaussian specifications. In view of Proposition (13.13), a Gaussian specification yJ,h is homogeneous if and only if the underlying quantities J and h are homogeneous, in that there exists an even function J': S ->• M and a real number h' such that J{i,j) = J'(i — j) and h} = h' for all i, j e S. In the following, we shall assume that J and h are homogeneous.

274

Gaussian fields

We shall replace J and h by J' and h', and we shall drop the primes. In other words, we shall use the letters J and h in a new meaning which can be related to their earlier meaning in a unique way. It is therefore clear what the positive definiteness of an even function J: S -> R means, and we can still use the notations Qj, MJh, and yJ'H when J and h have their new meaning. The sets Qj and Mj h are then invariant under the shifts of S, and the Gaussian specifications yJ,h are homogeneous. As a consequence of our homogeneity assumptions, we can now use the efficient tools of Fourier analysis. We let K = { z e C : | z | = 1} denote the complex unit circle, and we put G = Kd. Note that G is the dual group of S = Zd. We shall write (13.32)

zi =

zii...z^

whenever z = (z1,...,zd)e G and i = (i1,...,id) e S. The normalized Haar measure on G will be denoted by dz. By definition, dz is the image of the normalized Lebesgue measure on ] — 1, l ] d under the group isomorphism (13.33)

] - U ] d 3 p = ( p 1 , . . . , P a ) ^ z p = (e , "^...,e" t *').

Now let J: S -> R be any function which is absolutely summable, in that

(13.34)

X WOI < » . ieS

We then can define its Fourier transform (13.35)

J(z) = X z'-/(0

(z e G).

J is a continuous function on G. If J is even then J is real and satisfies J(zp)=

£./(i)cos7rp-i

(p6]-l,in

i eS

(The dot denotes scalar product.) In this case, J is positive definite if and only if J is nonnegative and not identically zero. A proof of this fact is given in (13.A8) of the appendix. If J ^ 0, we can define its reciprocal J'1 by using the convention 1/0 = oo. The main theorem on homogeneous Gauss specifications connects the existence of a Gibbs measure for yJ,h with the integrability of J - 1 . Recall from (5.2)(1), (5.12) and (5.13) that ^©(O,^) resp. %(yJ'h) stand for all the sets of all shift-invariant random fields resp. Gibbs measures for yJ,h. (13.36) Theorem. Let S = 1d, h e R, and J : S -> R be a positive definite even function which satisfies (13.34). Then (&{y]'h) ^ 0 if and only if MJh ^ 0 and j J(z)" 1 dz < oo. G

In this case we have

The homogeneous case

» ( / ' * ) = (Mc* V: v e P(Q,&),v(Mj,k)

275

= 1}

and %(yJ'") = U c * v: v e ^ 0 (Q, #"), v(M,,fc) = 1}. Here [ic stands for the centered Gauss field with covariance function (13.37)

C(iJ) = J z ' - ' ^ z ) " 1 dz

(i,jeS).

G

Proof 1) Suppose J1 is integrable, and let C be given by (13.37). C is symmetric and positive definite. So we can look at the centered Gauss field Hc with covariance C (cf. (13.A7)). For each i e S w e have

HC(T \JU - *m) ^ z \Ju - i)\cuj)112 \jeS

/

jeS

= c(o,oy>2 x w)i
Thus Hc(Qj) = 1- Moreover, we can write X J(7 - i)C(j, k) = X •/(./ - 0 1 ^ ^ z " ^ J ( z ) - 1 dz = |j(z)zk-iJ(z)^1dz

= Jzk-jdz = <5.-t

for all i, fceS. We have used that 0 < J < oo almost surely. Therefore, if m e Mj Ä then Theorem (13.22) ensures that [ic*ôme <${yJ'h). 2) Conversely, suppose that <$(yJ,h) # 0. Theorem (13.24) then shows that Mjh # 0 and the limit C in (13.25) exists. C is evidently homogeneous and nonnegative definite. A well-known lemma of Herglotz (cf. Proposition (13.A9)) thus ensures the existence of a finite measure a on G such that (13.38)

C(i,j) = J z^a(dz)

(ij e S).

To identify a we use the fact that C is an inverse of J in the sense of Theorem (13.22). (This theorem applies because Theorem (13.24) shows that '&(yJ'h) contains a Gaussian field.) For each k e S we have | zkJ(z)a(dz) = J X J(j)z k ^a(dz) = XJ(j)C(J,/c) = <50k = j V d z . Using the fact that the trigonometric polynomials on G are dense in the set of all continuous functions on G, we conclude that J(z)a(dz) = dz. We can thus write

276

Gaussian fields

a(dz) = l { j >0} (z)a(dz) + l{J-=0}(z)a(dz) = J(z)" 1 dz+ l {J=0} (z)a(dz). Since a(G) = C(0,0) < oo, the integrability of J - 1 follows. 3) To prove the final statement we will show that the right sides of equations (13.25) and (13.37) coincide. In view of (13.38), this will follow provided we can show that a(dz) = Jiz)"1 dz. In Step 1) above we have seen that <&(yJ'h) contains a Gauss field p with the covariance in (13.37). Let p=

J

v w(dv)

be the extreme decomposition of p. Combining Theorem (13.24) and Step 2) above, we obtain that V(T02)

- v(a0)2 = J J{zYl dz + a(J = 0)

for all v e ex (S(yJ'h). We can thus write, using Jensen's inequality, j J ( z r 1 d z = / l( ( T 0 2 )-M ( T 0 ) 2 = jv( ( T 0 2 )w(dv)-[Jv( ( T 0 )w(dv)] 2 ^j[v((To 2 )-v( ( T 0 ) 2 ]w(dv)

= J J{z)~l dz + a(J = 0). Hence a(J = 0) = 0, and this implies that the right sides of (13.25) and (13.37) are identical. The representation oieS(yJ'h) thus follows from Theorem (13.24). To obtain the characterization of <&&(yJ'h) we need to show that a Gibbs measure pc* v e (S(yJ'h) is shift-invariant if and only if v e ^©(Q,^). This follows from Corollary (7.28) because v can be identified with the probability measure w on ex (§{yJ'h) which represents jic * v. The proof is thus complete, D Theorem (13.36) has several implications which will be discussed now. We start with a remark on the sets Mj h. (13.39) Remark. Let he M and J: S -* M be even and positive definite. Suppose J has finite range, in that J(i) = 0 when |i| is large enough. Then MJh =£ 0. Moreover, if J(i) =£ 0 for at least one i j= 0 then Mj h is uncountable. Proof. 1) For each p e ] — 1, l ] d we let zp be defined by (13.33), and we look at the function J(p) = J(zp) on ] — 1, l] d . J is even, analytic, and not identical zero. Hence there exists an n = (n1,..., nd) e Z+ such that

The number ||n|| = n1 + ••• + nd is necessarily even. Without loss we can assume that ||n|| is minimal. Let m e Q be defined by

The homogeneous case

mt= -c(nr 1 hre»" l l (-l) w i / 2 i" 1 ...ij -

277

(ieS).

Then m e MJh because X J(j - i)mj JeS

--«">~ 1 *£ J( '-<£^••(£)"' ( •• ), d V

1

p=0

d

( d \"

p=0

for all i e S. 2) We next assume that J(i) ¥= 0 for some i # 0. Without loss we can assume that i1 # 0. Let N = m a x ^ : J(i) # 0}, and let z 2 , zd be any complex numbers # 0. The function /(w)

ii + N „ i 2

(weC)

on C is a non-constant polynomial. z 2 , ..., zd can be chosen in such a way that /(0) # 0. By virtue of the fundamental theorem of algebra, / has a root z1 ^ 0. Consequently, there exists some z e (C\{0}) d such that J(z) = 0. Here J, as defined at (13.35), is considered to be a function on (C\{0})d. We let É%S{W) denote the real part of a complex number w, and we put m = (ffie(zl))ieS. Then m e Mj 0 because

I '(J

i)ntj = ®e{zlJ(z)) = 0

(i e S).

jeS

Thus Mj 0 contains a nonzero element. Since MJ0 is a linear space and Mj,h = m' + MJ0 for every m' e MJth, we conclude that M, h is uncountable. n

Combining the preceding remark with Theorem (13.36) we see that only three cases are possible when J has finite range. Either
278

Gaussian fields

m = ( — h/J{l))ieS Remark (13.23)(b) thus shows that there is a one-to-one correspondence between ^&{yJ'h) and ^&(yJ,°). On the other hand, if h ^ 0 and J(l) = 0 then Ms h cannot contain a constant element, and this implies that ^&{yJ'h) does not contain any ß such that ^(|Z(C) denote the set of all probability measures on C which are invariant with respect to the transformations w -* wzk (w e C, k e S). Each rotationally invariant probability measure on C belongs to ^ Z (C). We define a measurable mapping : C -» M J 0 by (p(w) = (@4wz%eS. If a e 0>Z(C) then (wzk) for a-almost all w and all ke S. Applying Corollary (7.4) to the probability kernels from C to C which are induced by the mappings w ->• wzk, we see that focpis constant a-almost surely. Thus / is constant (p(a)-almost surely, and a second application of Corollary (7.4) shows that (a) e ex ^©(Q, #"). The last statement of Theorem (13.36) therefore implies that ßc* (a) e ex^@(yJ'°). The proof of the first assertion of the corollary is completed by noting that for each r > 0 there is some ar e ex^ z (C) with ccr(w e C: \w\ = r) = 1, namely the Haar measure on the closure of the set {rzk: k e S). (See Theorem 6.20 of Walters (1982), for example.) 2) To prove the converse we assume that J has no root in G, and we let fi e y&(yJ'°) be such that ß{al) < oo. We will show that fi = fic. We start with the observation that the infinite sums X J(j - i)(7;

(i G S)

jeS

exist almost surely and are square integrable. This is because 2N

E \JU - i)oj\ ' ) = /

E

\J(j-i)J{k-i)\n(\
j.keS

2

E UU)\

<

OO.

The homogeneous case

279

The same argument as in the proof of Theorem (13.22) therefore shows that the covariance function C of /i is an inverse of J in the sense of this theorem. Moreover, the constant mean m o f / i belongs to Mj 0 . Since J(l) ^ 0, we conclude that m = 0. Further, since C is homogeneous the Herglotz lemma (13.A9) ensures the existence of a measure a on G such that equation (13.38) holds. The argument following (13.38) then implies that a(dz) = J(z)" 1 dz. Now let v e ^ @ (Q, #") be such that ß = nc * v. Then V(CT,) = ^(CT,) — nc{at) = 0 and J/(z)- 1 dz = / i ( f f f ) - ^ ) 2 + vfe2) - v(a;)2

= M O - VM?

= JJ(z)- 1 dz + v(ai2) for all i. Hence v{a{ = 0 for all i) = 1 and n = nc.

a

As a matter of fact, the second assertion of the preceding corollary can be improved. If J has no root in G then, by a classical result of Wiener (1932), the absolute summability of J implies that the covariance function C at (13.37) is also absolutely summable, in that

X |C(0J)|
(A modern approach to Wiener's theorem using Gelfand's representation of commutative Banach algebras can be found in Loomis (1953), for example.) The absolute summability of C then implies that 0 is the only bounded element of Mj 0 . Indeed, if m e MJ0 is bounded then X \J(k-j)C(j,i)mk\

< oo

j,keS

and thus

»»<= I f i J(k- j)C(j,i)) mk /

keS \jeS

= 1 C(7,o(z^(/c-7> k ) jeS

\ksS

/

= 0 for all i e S. We thus conclude from Theorem (13.36) that there exists only one n e
280

Gaussian fields

Dobrushin (1980): If J has no root in G and J(i) ~ c\i\ a as \i\ -» oo with suitable numbers c ^ 0 and a > d then lim \j\aC(0,j) exists. The same interchange of summations as above then allows us to conclude that Mj 0 = {0}. Hence <${yJ'°) = {fic}- This result thus provides us with a non-trivial example of uniqueness, even without any temperedness condition as above. It also shows that Remark (13.39) fails when J has infinite range. Turning to the next corollary of Theorem (13.36), we assume that J has a root z e G. Step 2 in the proof of Remark (13.39) then shows that m = {0te{zl))ieS is a bounded non-zero element of Mj 0 . Applying Corollary (9.24) to the dissipative continuous symmetry group (T""), S H , we can thus conclude that (S(yJ'h) — 0 for all h e U when d :g 2 and J satisfies the decay condition (9.21) resp. (9.35). The same result can be derived directly from Theorem (13.36). In fact, when d = 1 we can replace (9.35) by a slightly weaker condition. (13.41) Corollary. Let d ^ 2, h e R, and J:Zd-+R is a constant L < oo such that

be as above. Suppose there

\i\d\J(i)\^Llogn

X 0<\i\S"

J(z) = 0 for some zeG then >g(yJ-h) = 0.

for all n^2.If

Proof. We will prove that J ^ z ) ^ 1 dz = oo. Let J be defined as in the proof of (13.39), and let q e ] — 1, Y]d be such that zq = z. In the case d = 2, we conclude from Step 3) in the proof of Lemma (9.33) that £ |i11 J"(i")| < oo. Hence J is differentiable with gradient 'e s g r a d J ( p ) = — n £ iJ(i)sin np • i. ieS

Since J ^ 0 and J(q) = 0, we have grad J(q) = 0. Consequently, in either dimension we can write J(P) — X -J(i)C cos np-i — cosnq• i + (d — l)(p — q)• i sinnq• i~\ ieS

£

X

2

\J(i)\\n(p-q)-i\d/d

m\P-q\-

+

X

\J(i)\l2 +

(d-l)\(p-q)-in

\i\>\p-q\~2

^nd\p-q\dL\og\p-qr2

+ \p-q\d

I

|./(0l[2|*r/2 + (rf-l)|»| 3/2 ]

i>l>|p-«|-2

^M\p-q\d\og\p-q\-1 for some constant 0 < M < oo and all p with 0 < \p - q\ S à = 2~xß. Thus

The homogeneous case

7(p)-ldp^M-1

J

dp\p - q\-d(\og\p -

J

281

qrr1

= M'1 (In)"-1 J d r ^ l o g r " 1 ) " 1 o 1 1 = M" (27t)"- ( - l o g log r- 1 )!* = oo. This completes the proof of the corollary,

a

Our final corollary characterizes the centered Gauss fields in ^B(yJ'°) in terms of their spectral measure. Recall that a centered Gauss field /i is shift-invariant if and only if its covariance function C is homogeneous. In this case, C admits a unique representation of the form (13.38) in terms of a finite measure a on G (cf. Proposition (13.A9)). a is called the spectral measure of/i. (13.42) Corollary. Let \ibe a shift-invariant centered Gauss field on Zd, and let J: Zd -> M be as before. Then /i e ^©(y7'0) if and only if its spectral measure a can be written in the form a(dz) = ./(z) -1 dz + a0(dz), where a 0 is supported on the set {J = 0}. Proof "only if". The proof of this is identical to Step 2) in the proof of Theorem (13.36). "if". Let nc be as in Theorem (13.36). The measure a0 is the spectral measure of a unique shift-invariant centered Gauss field v (cf. Proposition (13.A7)). The convolution \ic * v is centered Gaussian and has the same covariance as fi. Hence /i = \ic * v. Also, if i e S then v ( I J(j - i)oj 2 ) = I \

JeS

/

J(j - i)J(k - i) J z*-'a 0 (dz)

j,keS

= J J(z) 2 a 0 (dz) = 0. Thus v(Mj 0 ) = 1, and Theorem (13.36) shows that fj, e %(yJ'°).

a

We conclude this chapter with three examples. (13.43) Example. The harmonic oscillator. (Continuation of Example (13.29).) Let ß > 0 and JO)

-ß/2d ß 0

if|i| = l, if i = 0, otherwise.

Since J(l) = 0, MJi0 contains all constant configurations, and yJ'° exhibits the continuous symmetry co -> (coi + t)icS, t e R. By the way, yJ'° can be

282

Gaussian fields

written in the form yA•" = pAXA with pA(œ) = ZA(œ) *exp

( A e y , f f l e Q). Ii-Jl = l

This is an immediate consequence of Proposition (13.13) and shows that yJ,° coincides with the model of Section 6.3, except for the choice of the a priori measure. If d S 2 then %(yJ-0) = 0, because of Corollary (13.41). So let d ^ 3 and J be as in the proof of Remark (13.39). Then ^

d

J(P) = 2ß Z (! -

cos n

Vt)

e=\.

= Aß X sin2 npt/2
^ 4>S t PÏ = W\v\2 for all p e ] — 1, l] d . Since |p|~ 2 is integrable over ] — 1, l ] d when d ^ 3 we conclude that J" 1 is integrable. Theorem (13.36) thus implies that &B(yJ,0)r\ ex^(yJ'°) contains uncountably many Gibbs measures which break the continuous symmetry of yJ'°. (Recall that the same result has already been obtained by different arguments at (13.29).) o (13.44) Example. Long range interaction in one dimension. Let d = 1, ß > 0, a > 1, and J: Z ->• (R be such that J(l) = 0 and J(i) = -ß\i\~a when i ± 0. J is positive definite. This can be derived either directly as in Comment (13.14), or by using Proposition (13.A8) and the fact that 3(P) = ß I

l * r ( l - cosnpi)

is nonnegative. If a ^ 2 then Corollary (13.41) shows that ^(y 7,0 ) = 0. So let 1 < a < 2. Then J is integrable over ] — 1,1]. Indeed, if 0 < |p| ^ 1 then J(p)^2ß

|/r û sin 2 (7ipi/2)

X o<|i|si/|p|

All

2ß

o<

All

2ß\p\'

\pi\

X

J

All

3 < | i | S l / | p | |pi|-|p| 1/2 2 -

J dx \x

-1/2

ll>

2-a

l\p\

••iâi -i

c -i

2ß\p\

lP\a\pi

I

ipr 1 IM

dxx

2-a

The homogeneous case

283

Hence i

l

J J{pYlàp^M

J |p|1-adp
-l

-l

Theorem (13.36) thus gives the result that e\^(yJ'°) contains uncountably many shift-invariant Gauss fields which are related to each other by the continuous symmetry co -• (a>; 4- t)ieS (t e R) of yJ,°. o (13.45) Example. Long range interaction in two dimensions. Let d = 2, ß > 0, a > 2, and J: I2 -> U be given by J(i) = - j 8 | i r ° (»' ^ 0) and J(l) = 0. J is positive definite. From Corollary (13.41) we know that J - 1 fails to be integrable when a ^ 4. We will show here that J" 1 is integrable when 2 < a < 4. Consider J(p) = J(zp). Since J is bounded away from zero on the complement of each neighbourhood of the origin, we only need to find a lower bound for J(p) when |p| is small. We fix any 0 < ô < 1. For each p ^ Owe have, writing r = |p| and u = p/r, \p\2-°J(p)}zßr2-°

X

|i|--(l — cosrep-i)

ÄSr|i'|Sl,u-ia|i|/2

^

ieS •5ä|ri|Sl,u-iä|i|/2

^ M(p). We have used the fact that 1 — cos ret = 2sin 2 7tt/2 ^ 2t2 when \t\ ^ 1. If r = |p| is small, M(p) is approximately equal to the integral ß 2

J

{xeR2:Äg|x|gl,u-xa|x|/2}

ß J

*/3

= Tjdss

J

L

à

|x|2^dx

2-a

dts

-jt/3

= JBTI(1 - (54-")/3(4 - a) ^ 2M > 0. More precisely, since the function x -> \x\2~" is uniformly continuous on the annulus {ô ^ |-| ^ 1}, there exists an e > 0 such that M(p) ^ M whenever |p| ^ e. Since a < 4, the function p -• | p | 2 ~" is integrable over the disc {| • | ^ e}. This shows that J" 1 is integrable over ] — 1, l ] 2 , and Theorem (13.36) yields the same conclusion as in the preceding examples, o

284

13.A

Gaussian fields

Appendix. Some tools of Gaussian analysis

In this appendix we list some elementary facts about Gaussian random vectors and positive definite functions. Let n ^ 1 and U = (f/,) l â l â n be an U"-valued random vector on a probability space (Q, tF,n). U is said to be Gaussian (or, more precisely, to have a Gaussian or multivariate normal distribution) if there is a symmetric matrix C = (C(i,j))lûiJèn and a vector m = (mj)is,-gn e W such that the Fourier transform of U is given by (13.A1) n(exp[if-I/]) = e x p [ - i t - C f + if-m]

(teW).

Here i is the imaginary unit, and the dot denotes scalar product. The matrix C is necessarily nonnegative definite because the left side of (13.A1) is at most 1 in modulus. By Fatou's lemma, ßiU?) ^ 2 lim t;2n(l - costjl/j) = r ; ->0

d_

2-^a*q>(t) ^2 '

Oti

< oo

(i e S),

r=o

where
mi)(Uj

- m,.))

(1 g i, j g n).

Thus m is the mean and C the covariance matrix of U. If C is positive definite, then the distribution of U has the well-known Gaussian density x -> (2n)'"121det C| - 1 / 2 exp [-•£(x - m) • C - 1 (x - m)] with respect to Lebesgue measure on U". Gaussian random vectors enjoy the following simple but important property. (13.A3) Remark. Let U be Gaussian, 1 ^ k < n, U' = ( l ^ i s . s * ' a n d U" = (Uj)k<j£n- Then the following statements are equivalent. (i) U' and U" are uncorrelated, in that C(i,j) = 0 for all 1 ^ i ^ k < j ^ n. (ii) U' and 17" are independent. Proof. The implication (ii) => (i) is trivial. To prove the converse it is sufficient to note that (i) and (13.A1) imply that fi(e"'u'en"v")

=

for all t' e Rk and t" e R"~k.

fi(e,t'u')ß(e""u") a

One useful consequence of this remark is stated in the next proposition. (13.A4) Proposition. Le? U = (I7j)1g1-5„fee a Gaussian random vector and Û1 = H{U± \U2,.--, Un) the conditional expectation of Ul with respect to the a-algebra which is generated by U2, •.., U„. Then there are real numbers a,b2,---,b„ such that

Appendix. Some tools of Gaussian analysis &i =

a

+ S ht Ut

285

fi-a.S..

i= 2

Proof. Consider the real Hilbert space L2([i) of all square integrable functions on the underlying probability space (Q, !F, fi), and let Jf c L2(/i) be the linear span ofU2,...,U„ and the constant function 1. Let Ü1 be the orthogonal projection of Ul onto Jf. That is, L^ is the unique linear combination of 1, U2, ..., Un which satisfies fi(Ui - 0t) = 0 and ^ ( ( ^ - fy)!/,) = 0 for all 2 ^ i ^ n. The random vector (l^ — Ül,U2,...,U„) is a linear image of the extended vector (1, U), and U is Gaussian. A glance at (13.A1) thus shows that (C/j — Ü1,U2,...,U„) is Gaussian. But l^ — Ü1 and ([/ 2 ,..., [/„) are uncorrelated. Remark (13.A3) therefore implies that l^ — Ü1 and (U2,..., U„) are independent. Consequently, if A belongs to the cr-algebra generated by U2, ...,[/„ then

This means that 0l is a version of L^. G (13.A5) Proposition. Let (U(k))kil be a sequence of W-valued Gaussian random vectors U(k) with mean m(k) and covariance matrix C(k\ Suppose Uik) converges in distribution to a random vector U. Then the limits m = lim mw and k-* oo

C = lim Ök) exist, and U is Gaussian with mean m and covariance C. k-*cc

Proof. Looking at (13.A1) we see that the limit l i m e x p [ - i f C ( k ) t + zt-m (k) ]

(teW)

k-*oo

exists and is equal to the Fourier transform of U. Taking the modulus and letting t = (0,..., 1,0,...) be any basis vector, we find that the limits C{i, i) = lim C(k)(i,i) exist in [0, oo], 1 ^ i ^ n. As the Fourier transform of U is continuous at the origin, the C{i, i)'s must be finite. If we let t be the sum of two basis vectors, we obtain that C(i,j) = lim C(k){i,j) exists for all i =£ j . We further conclude that lim exp [it • w(fc)] exists and is continuous in t. k-*oo

This implies the existence of m = lim mik).

a

k-*oo

(13.A6) Corollary. Let (U„)nèl be a sequence of real random variables on a probability space (Q,J^,/i). Suppose Un converges in probability to a random variable Uœ. If U„ — Uk is Gaussian for all n, k ^ 1 then lim n(117,-14,12) = 0. n-*oo

286

Gaussian fields

Proof. For each n ^ 1, U„ — Uk converges in distribution to U„ — Uœ as k -> oo. Proposition (13.A5) thus shows that U„ — Uœ is Gaussian. In the limit n -> oo, Un — Um tends to the (Gaussian) random variable 0. A second application of Proposition (13.A5) thus shows that lim fi(U„ — Uœ) = 0 and k~*<X>

2

2

lim [>(([/„ - t / J ) - p{Un - l / J ] = 0. n->oo

This proves the corollary, D The next proposition ensures the existence of a centered Gauss field with a given covariance function. (13.A7) Proposition. Let S be a countably infinite set and C: S x S -> U a nonnegative definite symmetric function. Then there exists a unique centered Gaussian field /i c on S with covariance function C. Proof. For each A e £f we let vA be the centered Gaussian distribution on UA with covariance matrix CA = (C(i,j))iJeA. The vA's are consistent. For, if A c A then equation (13.A1) shows that the A-projection cA(vA) of vA is centered Gaussian with covariance CA. Hence cA(vA) = vA. By Kolmogorov's extension theorem there exists a unique probability measure nc on IRS such that <JA(HC) = VA f° r aH A. e y . /i c satisfies equation (13.2) with m = 0. D We conclude this appendix with two propositions which deal with the characterization of nonnegative definite functions on Zd in Fourier analytic terms. We let K denote the complex unit circle. (13.A8) Proposition. Let J': Zd -> M be an even, absolutely summable function and J its Fourier transform. J is positive definite if and only if J ^ 0 and J(z) ^ 0 for at least one z e Kd. Proof, "if". Let w; e C (i e Zd) be such that u{ ^ 0 for at least one but at most finitely many i, and define g(z)=

X ukzk

(zeKd).

keZd

Then E

i,jeZd

«,•/('•-/)«;= =

Z

i,j,keZd

UtUjJik) $d zk+J->dz K

2

\\g(z)\ J(z)dz.

Since J is continuous, J > 0 on a set of positive measure. On the other hand, \g\2 can only vanish on a null set. This is because there is an n e Zd such that g(z)z" is a polynomial. Consequently, for any choice of z 2 , . . . , zd there are at

Appendix. Some tools of Gaussian analysis

287

most finitely many z^s such that g(z) = 0. Combining these facts we see that the last integral is positive. "only if". The equation above shows that $\g(z)\2J(z)dz>0 for every nontrivial trigonometric polynomial g on Kd. Using the StoneWeierstrass theorem, we conclude that J"/(z)J(z)dz ^ 0 for all continuous functions / : Kd -> [0, oo[. This gives us that J(z) ^ 0 for almost all z e Kd. Since J is continuous, this implies that J ^ 0 everywhere, D Last but not least, we quote a well-known lemma of Herglotz. This lemma is the discrete counterpart of Bochner's famous theorem. (13.A9) Proposition. Let C:Zd->C be any even function. C is nonnegative definite if and only if there exists a finite measure a on Kd such that C(i) = J z;a(dz)

for all i e Zd.

Proof, "if". This is easy. "only if". For each finite A c Z ' w e define gA(z) = \Ar

I

C(;-!>'"•'•

(zeKd).

i,jeA

By hypothesis, gA(z) ^ 0 for all z. Writing gA(z)=

X C(fc)s-*|An(A + fc)|/|A| keTd

we see that Jz'"0 A (z)ds = C(i)|An(A + i)|/|A| for all i e Zd. Here A + i is the translate of A by the vector i. In particular, we have J" gA(z) dz = C(0) for all A. Now we let A run through a sequence of cubes. As Kd is compact, the measures gA(z) dz have a cluster point a in the weak topology. This a has the required property, a

Part III Shift-invariant Gibbs measures

Shift-invariant random fields on an integer lattice Zd are a well-studied area of Probability Theory. Within the Gibbsian theory, a similarly important role is played by the shift-invariant Gibbs measures on Zd. Actually, such Gibbs measures appeared already in various places in Parts I and II, and they will also be crucial for our investigations in Part IV. Here we shall look at shift-invariant Gibbs measures from a general, and in some sense global, point of view. We commence Chapter 14 by noting several classical facts concerning the (shift-) ergodic random fields. (These constitute the extreme boundary of the simplex of all shift-invariant random fields.) This chapter then provides some general results on ergodic Gibbs measures for shift-invariant specifications. These include the decomposition of shift-invariant Gibbs measures into ergodic components, as well as the approximation of ergodic Gibbs measures by averaged Gibbs distributions. In the two further chapters of this part we shall concentrate on shiftinvariant Gibbs measures for shift-invariant and absolutely summable potentials. In Chapter 15 we shall introduce and discuss the specific entropy and specific free energy of shift-invariant random fields, and the central result is the variational principle which characterizes the shift-invariant Gibbs measures as the minimizers of the specific free energy. We shall further see that the minimum free energy principle also determines the exponential decay rates of Gibbsian probabilities of untypical behaviour. Chapter 16 presents a geometric view of Gibbs measures. Namely, as a consequence of the variational principle, each shift-invariant Gibbs measure can be identified with a tangent hyperplane to a convex function (the "pressure") on the space of potentials. This identification opens the door to a convexity approach to both the existence of phase transitions, and the uniqueness problem for shift-invariant Gibbs measures. Whilst much of Chapter 15 can be read independently of Chapter 14, Chapter 16 depends on the preceding chapters 14 and 15. These in turn are based on Chapters 4, 5, 7 (and, of course, 1 and 2). The other chapters of Parts I and II do not enter into the main stream of this part.

Chapter 14 Ergodicity

Let S = Zd and ^©(Q, !F) be the set of all shift-invariant probability measures on (Q, F). Also, let y be a homogeneous specification and ^©(y) the set of all shift-invariant Gibbs measures for y. The objective of this chapter is to provide a number of basic facts about the sets ^®(Q, !F) and &&(y) as well as their extreme boundaries ex ^®(Q, F) and ex "^©(y). The elements of ex ^©(Q, F) are called ergodic. From an abstract point of view, the definitions of ^©(Q, F) and ^©(y) are very similar to that of @(y). As a consequence, most results of this chapter have a direct counterpart in Chapter 7. In fact, some fundamental results of Chapter 7 have been stated in sufficient generality to include the present cases. Hence, the reader should refresh his memory of Sections 7.1 and 7.3 before continuing. Section 14.1 is devoted to a study of the simplex ^©(Q, !F). We shall start with some standard characterizations of ergodic probability measures. Then we shall show that each \i e ^©(Q, F) admits a unique ergodic decomposition, in that \i is the barycenter of a unique probability weight on ex ^©(Q, IF). Moreover, we shall establish the remarkable fact that the ergodic measures are dense in ^©(Q, F) relative to the ^-topology. In Section 14.2 we shall look at @@(y). We shall see that %(y) is a face of the simplex ^©(Q, F). In particular, each \x e ^®(y) has a unique extreme decomposition, and this extreme decomposition coincides with the ergodic decomposition of \i. Finally, we shall prove that each ergodic Gibbs measure can be approximated by averaged Gibbs distributions in finite volumes with suitable boundary conditions. Several results of this chapter rely on the multidimensional ergodic theorem. For the sake of convenience, this is stated and proved in Appendix 14.A.

14.1

Ergodic random fields

Throughout this chapter we put S = Zd, the d-dimensional integer lattice, and we let (£, S) be an arbitrary measurable space. As before, we put (Q, F) = (E, ê)s. (Q, F) is acted on by the shift group 0 which was defined in Example (5.2) (1). We look at the convex set (14.1)

^ e (ft, F) = {M e ^(Q, JF): 0,(/i) = ß for all i e S}

of all shift-invariant random fields on Zd. ^©(Q, F) is always non-empty. We also consider the u-algebra

Ergodic random

(14.2)

S = {Ae&:0,A

=

fields

291

AîoTaX[ieS}

of all shift-invariant events. (14.3) Remark. (1) An & -measurable function / : Q -* M is ^-measurable if and only if / is invariant, in that / o 0. = / for all i e S. (2) For each \i e ^ e (Q, &), the
/i-a.s. for all i e S}

of all /^-almost surely invariant events is the /^-completion of J. Proof. (1) If / is invariant then {/ ^ c} e J for all c e U. Conversely, any ^-measurable / is the limit of ^-measurable step functions, and each such step function is invariant. (2) For given A e J(n) let B = (J BtA. Then B e J and fi(AAB)^

£ fi(AABtA) = 0.

a

is S

In Section 7.1 we have seen that the structure of a Gibbs simplex @(y) is governed by the tail c-field ST. The set ^@(Q, &) is linked to the
ôt(A\(o) = 1A(0,Û>)

(ieS,Ae^,œe

Q).

Using the notation of Corollary (7.4), we then have ^©(Q, &) = ^ n a n d ^(/i) = ^ n (/i). Combining this corollary with Remark (14.3)(2) and mimicking the proof of Theorem (7.7), we arrive at the following result. (14.5) Theorem, (a) A probability measure p. e ^@(Q, &) is extreme in ^@(Q, &) if and only if \i is trivial on the invariant a-algebra J'. (b) Suppose fi e ^©(Q, 3?) and v e ë?{Çl, &) is absolutely continuous relative to fi. Then v e ^®(Q, &) if and only ifv = ffi for some J"-measurable function f (c) Each ß e ^@(Q, &) is uniquely determined (within ^©(Q, &)) by its restriction to J'. (d) Distinct probability measures \i, v e ex ^@(Q, J^) are mutually singular on J, in that there exists an Ae J such that n(A) = 1 and v(A) = 0. Needless to say, Theorem (14.5) remains true when the shift-group 0 is replaced by any countable subgroup of T. The following definition is standard. (14.6) Definition. A probability measure \i e ^@(Q, &) is said to be ergodic (with respect to the shift-group 0 ) if \i is trivial on J. In the language of mathematical physics, any such \i is often called a pure state.

292

Ergodicity

Ergodicity can be characterized by a mixing property which resembles that in Proposition (7.9). Let us agree to call a set A e Sf a cube if d

A = Z d n n [ m ^ + n] i= l

with m = (ml,...,

md) e Zd and n ^ 0.

(14.7) Proposition. Let /i e ^®(Q, IF) and (A„) n>1 be any sequence of cubes in S such that |AJ -» oo as n -» oo. Then the following statements are equivalent. (i) /i is ergodic. (ii) For all Ae^, lim sup IAJ- 1 X

n(AndtB)-n(A)n(B)

0.

ieA„

(iii) For arbitrary cylinder events A and B, limlAJ" 1 n-*oo

X ß(ADdiB) = n(AMB). ieAn

Proof, (i) implies (ii). If A, B e ^ then

i e A_

1 Jd/* IAJ- Z

^fd/x IAJ-1 I

l^ofl.-^) l x o0.- M (A)

By virtue of the mean ergodic theorem (14.A5), the last integral tends to zero as n -» oo. (ii) implies (iii). This is trivial. (iii) implies (i). For fixed A e ^ we let ^ denote the set of all B e ^ such that limlAJ" 1 X M^n0 J B) = /i(A)/i(B). n->ao

ieAr

It is easily seen that 2>A is a Dynkin system. Similarly, the set of all A e #" with ^4 = ^ is a Dynkin system. Since any Dynkin system containing the cylinder events is equal to SF, we conclude from (iii) that 3>A = 3F for all A e !F. Now let AB J. Then Ae2>A, and this implies that /iC4) = n(A)2. Hence /x(y4) = 0 or 1. • It is not difficult to see that condition (ii) of Proposition (7.9) implies condition (iii) of Proposition (14.7). This observation is sharpened by the following result. In its proof and later on, we shall use the notation

Ergodic random fields

(14.8)

293

A+j={i+j:ieA}

for the translate of a set A c S by a vector j e S. (14.9) Proposition. Let /z e 0»e(Q, ^ ) . Then J <^ ST

/i-a.s.,

in t/iaf /or eac/i Ae J there exists a set B e 3" such that p.(A A B) = 0. /n particular, if /x is trivial on 3~ then /x is ergodic. Proof. Let A e J^ be given. Then we can find a sequence (ß„)„äi of cylinder events such that fi(A A B„) ^ 2~" for all n ^ 1. (This is a well-known corollary to Carathéodory's extension theorem. It can also be proved by checking that the set of all A with this property is a Dynkin system which contains all cylinder events.) For any i e S and n ^ l w e have H(A A 6tBn) = fi(dtA A ßtBn) = n{A AB n ) < 2"". Let A„ e y be such that B„ e 3FKn. Without loss we can assume that (A„)„ àl is a cofinal sequence. For each n M we choose any i(n) e S such that A„ fl(A„ + i(n)) = 0, and we set Bn = BmBn. Then Bn e ^An+m <= ^ „ and p(A A B„) g 2"". Therefore, if B = (~) (J Bn then ß e ST and

M^4AB)^/x(n U ^ A 5 n ) ^ l i m Z 2 " - = 0. This proves the proposition,

o

As is evident from the proof above, Proposition (14.9) remains valid when the full shift-group 0 is replaced by any infinite subgroup of 0 . Next we will show that ^@(Q, #") is completely determined by its extreme boundary (provided (E, S) is standard Borel). More precisely, each // e ^@(Q, 3F) can be represented as a mixture of ergodic probability measures. This ergodic decomposition of/i is governed by a (^ e (Q, $F\ ^-kernel in the sense of Definition (7.21). In accordance with Section 7.3, e x ^ ^ Q , ^ " ) is endowed with the c-algebra s(ex ^@(Q, #")) which is generated by the evaluation mappings v -> v(^4), ^4 e J r . (14.10) Theorem. Let (£, ê) be a standard Borel space. Then there exists a {SP&(£1, !F\ jykernel %. Consequently, there is an affine bijection as'.fi^w^ from 0>&(Q., &) onto 0>(ex 0>&(Çl, &), ^(ex 0»e(Q, &))) such that V=

1

v

w^(dv)

for all n e ^@(Q, J^). w^ is the image of p. under the mapping m -> nm. In particular, for any A e !F and c e R we have

294

Ergodicity

w„(v G ex^ 0 (Q, J^): v(A) g c) = MA] J) ^ c). Proo/. In view of Remark (4.A3), (Q, J*) is a standard Borel space. Theorem (4.All) thus ensures that J^ has a countable core (€. We let (A„)„>1 be an increasing sequence of cubes in S such that |A„| -» GO, and we look at the invariant set ( a e Ü : lim |A„r x

Qo= (.

£

n-*ao

l ^ e o ) exists for all A e
ieA„

J

= . I

n^ao

ieA„

'

J

According to Definition (4.A9) of a core, there is a well-defined mapping œ -> nm = 7t( • |co) from Q 0 into ^>(Q, J*) such that 7r(X|ca)= lim \A„\~l n->oo

^l^co)

for all A e^.

>'eA„

This mapping can be extended to all of Q by choosing any v0 e ^®(Q, J5") and putting 71™ = 7t(-|co) = v0 when œ eQ\Q0. The resulting function n: J* x Q -> [0,1] is a probability kernel from J to J*\ Indeed, if A e
{n- e 0>&(Q, &)} =

H

M M ) = *"(^)}

A e
is ./-measurable and has /^-probability one. This is a consequence of the obvious fact that ß(9iA\) /i-a.s. for all A and i. We have thus shown that n is a (^©(Q, &), ./)-kernel. To complete the proof we now need only apply Proposition (7.22) in the same way as in the proof of Theorem (7.26). • As is seen by a glance at the preceding proof, Theorem (14.10) remains true when the shift-group 0 is replaced by any finitely generated abelian subgroup of T. Let us state a corollary which is similar to (7.28). Suppose x e T is a transformation which commutes with 0 . In the notation (5.14) this means that T o 0 = 0 o T. (Examples of such transformations have been given in (5.17).) Also, let [x e 0»0(Q, &). It is then clear that x(n) e 0»0(Q, 3F\ Moreover, if [x is ergodic then so is T(/I). Consequently, each probability measure w on ex ^"©(Q, J^) has a well-defined t-image T(W) = w(v: T(V) e •)•

Ergodic random

fields

295

(14.11) Corollary. Let (E, S) be a standard Borel space and x e T be such that x o O = @ o T. Let u> be as in Theorem (14.10). Then u> commutes with x, in that wI(/1) = x(wß) for all /x e ^®(Q, &). Proof. Same as Corollary (7.28).

o

Theorem (14.10) can be summarized in a short phrase: ^@(Q, J*) is a simplex. This simplex, however, cannot be visualized in the naive way. This follows from the next theorem which states that ex^ 0 (Q, J^) is dense in ^@(Q, OF) relative to the topology of local convergence. Of course, this result depends on the chosen topology. For example, assertion (14.5)(d) implies that any two distinct ergodic measures have total variation distance 2. Therefore, if ^®(£2, J*) is endowed with the total variation metric then ex^®(Q,&) is certainly not dense in ^ ( Q , J*) (except when |£| = 1). Recall from the paragraph below (5.12) that a?&{ÇL,&) is closed in the y -topology. (14.12) Theorem. Relative to the y-topology, ^@(Q, J^) has a dense extreme boundary. Proof. Fix any /x e ^ 0 (Q, &). For each n ^ l w e define A(n) = [-w, ri]d fl S,

and

v„ = |A(n)r

X 0,.(/O. jeA(n)

Let / e if and A e ^ be such that / e j§?A. Then k(/)-/xCDI = |A(n)r

Z

[/i„(/o0,)-M/ooy)]

^ 211/11 \{j e A(fi): (A - j)\A(n) * 0}|/|A(w)| because f°6j6j?A_j. The last expression tends to zero as n->oo. Thus tx = lim v„ in the JS?-topology. n-*GO

On the other hand, we have v„ E ex i^0(Q, ^ ) for all n ^ 1. Indeed, since ö (2n+i)i(Mn) = A*« for all ! e S it is easily checked that v„ is shift-invariant. We thus only need to show that v„ is ergodic. So let A e J be given. According to Proposition (14.9), there is a set B e 9~ such that v„(A A B) = 0. In particular, n„(AAB) = 0. Since ^ e / , w e conclude that 0,-(/x„)(,4) = /x„(,4) = ix„(B) for all j e A(n). Thus v„(/l) = xx„(B). But Kolmogorov's 0-1 law (cf. (7.14)) implies that tx„(B) = 0 or 1. Hence v„(A) = 0 or 1. o (14.13) Comment. Suppose £ is a compact metric space and ê its Borel aalgebra. It is well-known that in this case ^@(Q, &) is compact and metrizable

296

Ergodicity

in the weak topology. From Remark (4.3)(3) we know that the weak toplogy is coarser than the if-topology. Theorem (14.12) therefore implies that ex ^©(Q, &) is weakly dense in éP&(Q, &). Moreover, ex ^©(Q. #") is a dense Gs set in ^ @ (Q, J^). In other words, ^ @ (Q, J^)\ex ^ 0 (Q, #") is of first Baire category, i.e., a countable union of nowhere dense sets. To see this we let d(-, •) denote any metric for the weak topology. Then we can write ^ @ (Q,
with Kn = {{ii + v)/2: n, v e 0>@(Q, &), d(n, v) ^ 1/n}. In view of the compactness of ^©(Q, &), Kn is easily seen to be closed. Hence ex ^®(Q, J^) is the countable intersection of the open sets ^©(Q, ^)\K„, n ^ 1. A set of first Baire category should be considered as being exceptional. We thus arrive at the surprising conclusion that a "typical" shift-invariant probability measure is ergodic. There is a further result which should be mentioned in connection with this. With reference to a paper of Poulsen (1961), a compact metrizable simplex with a dense extreme boundary is sometimes called a Poulsen simplex. Lindenstrauss et al. (1978) have shown that all Poulsen simplices are affmely homeomorphic to each other, o

14.2

Ergodic Gibbs measures

Let y be a shift-invariant specification. (See Example (5.8) for the definition.) We will consider the set (14.14)

%{y)±9(y)n0>B(Q,&)

of all shift-invariant Gibbs measures for y. As we have seen in (11.46), %(y) can be empty even when <8(y) ^ 0. Sufficient conditions for %(y) to be nonempty have been given in Section 5.2. By definition, a probability measure /z belongs to %(y) if and only if fi is preserved by all probability kernels in the countable set n = {yA: A e y j U {ôt: i e S}. 0i was defined in (14.4).) In this case, the a-algebra Jn(n) of all /z-almost surely n-invariant events is the /z-completion of J. Indeed, Remarks (7.6) and (14.3)(2) imply that Jn(fi) is the intersection of the /z-completion of $~ and the \icompletion oi J. But Proposition (14.9) asserts that the latter completion is contained in the former. We thus obtain the following counterpart to Theorems (7.7) and (14.5).

Ergodic Gibbs measures

297

(14.15) Theorem. Let y be a shift-invariant specification. (a) A Gibbs measure p e &&(y) is extreme in ^@(y) if and only if /i is ergodic. Thus ex%(y) = %(y)nex0>&(Q,^). (b) If fie ^ e (y) and v e ^@(0,8F) is absolutely continuous relative to fi then v e %(y). (c) %(y) is a face of 0>&(Q, &). I.e., iffi,ve 0>@(Q, #") and 0 < s < 1 are such that s/i + (1 — s)v e ^©(y) then n,ve &&(y). Proof. Assertion (a) follows from Corollary (7.4) and the remarks above. Combining Theorems (14.5)(b) and (7.7)(b) with Proposition (14.9), we obtain (b). Statement (c) is an immediate consequence of (b). D We note that assertion (c) implies (a). We also note that the theorem above remains valid when the shift-group 0 is replaced by any countable subgroup of T which contains a (non-trivial) shift. For in this case Proposition (14.9) still holds. As a last remark we point out that statements (c) and (d) of Theorem (14.5) remain true when ^©(O, #") is replaced by ^@(y). In view of (14.14) and (14.15)(c), one might ask whether %{y) is also a face of ^(y). For example, this happens under the conditions of Theorem (10.35). In general, however, ^©(y) fails to be a face of ^(y). There are ergodic Gibbs measures which are not trivial on 2T, and we now give a simple example. (14.16) Example. The Ising antiferromagnet in one dimension at zero temperature. Let S = Z, £ = { —1,1}, and X be counting measure. Define a shiftinvariant ^-specification y as in Example (10.3) by means of the functions , x (1 i f x ^ v , ^• ( x ' y ) = l 0 ïîx = y,

(X'yeE'JeV-

From Section 3.2, Case 2, we know that y describes the zero temperature limit of the Ising antiferromagnet without external field. Let (5+_ and <5_+ be the Dirac measures at the alternating configurations a>+~ an oT + which are defined by +_ , f 1 if fis even, co = coi++i =_+ e ex^(y) and (c>+_ + <5_+)/2 e ex^@(y). Thus ex^ @ (y)\ex^(y)^0. o A further, more interesting example with the same property is provided by the two-dimensional Ising antiferromagnet at low but positive temperatures; cf. the discussion at the end of Section 6.2. Also, in (16.30) we shall obtain a general theorem on the existence of ergodic but non-extreme Gibbs measures. Thus, an extreme element fi of ^®(y) is always a pure state in the sense of

298

Ergodicity

Definition (14.6) but not necessarily a phase in the sense of Comment (7.8). In spite of this, any such ß is often called a pure phase. We emphasize that Theorem (14.15) relies primarily on Proposition (14.9). In addition, this proposition implies that the ergodic components of a shiftinvariant Gibbs measure are also Gibbs measures. More precisely, if (E, S) is standard Borel then assertion (14.15)(c) can be sharpened as follows. (14.17) Theorem. Suppose (E,S) is a standard Borel space and y a shiftinvariant specification with ^©(y) # 0. Let ß e ^©(y) and wß be the unique probability measure on ex^©(Q, J^) which represents fi; cf. Theorem (14.10). Then wß is supported on ex ^©(y), Consequently, fi has a unique extreme decomposition within ^©(y), namely \i =

\

v wß(dv).

Proof Let /i e ^©(y) be given. In view of Theorem (14.10), the representing weight wß on ex^©(Q, J*) is defined in terms of/i via a (^©(Q, J*), ^-kernel n. We thus need only show that n' e ^©(y) /i-a.s.. (In other words, any (^©(Q, 3F\ ^-kernel is also a (%(y), ^-kernel.) Since TV e ^©(Q, 3F) /i-a.s., it is sufficient to check that if e &(y) /i-a.s.. Let <€ be any countable generator of 3F which is stable under finite intersections. Such a <ê exists because (Q, 3F') is standard Borel. For each A e <€ and A e y we can write, using Proposition (14.9) for the third and fifth equality, •K-yK{Ä) = fi(yA(A\ -)\J) =

n(n{Am\S)

= VL(vL(n(A\rA)\F)\J) =

n(n(A\f)\S)

= n(A\J>) = n'(A) /i-a.s.. Thus H(n e 9(y)) = fi(nyA(A) = n\A) for allAeV and the proof is complete.

and A e Sf) = 1,

•

(14.18) Corollary. Let {E,S) be a standard Borel space, y a shift-invariant specification, and N ^ 1 an integer. Then |ex^©(y)| 2: N if and only if ^©(y) contains at least N linearly independent elements. Proof. Just as Corollary (7.29).

a

Theorem (14.17) and the preceding corollary still hold when the shift-group 0 is replaced by any finitely generated abelian subgroup of T which contains a non-trivial shift. This follows from the fact that Theorems (14.5), (14.10), and (14.15) as well as Proposition (14.9) remain valid under this replacement. In (7.12) we have seen that each extreme Gibbs measure is a limit of

Ergodic Gibbs measures

299

finite volume Gibbs distributions with suitable boundary conditions. What about ergodic Gibbs measures? The ergodic theorem (14.A8) provides us with the following preliminary answer: U fie ex^@(Q,^) and (A„)„ èl is any increasing sequence of cubes with |AJ -> oo then

M/) = Hm IAJ-1 X fißM for /i-almost all a> and any bounded measurable function /. Thus ^=lim|A„r1 n -> oo

X \«B

for/i-a.a. to

ieAn

in any topology which is generated by countably many evaluation mappings v -> v(/). (When E is compact metrizable, the weak topology has this property.) The above approximation result is unsatisfactory when \i is an ergodic Gibbs measure for y. In this case, the approximating measures should be related to y. We will give such an approximation result in the final theorem of this section. In it we will show that each fi e ex ^@(y) is the limit of averaged finite volume Gibbs distributions with suitable boundary configurations. For the proof we shall need the following refinement of the backward martingale convergence theorem. (14.19) Lemma. Let (Q, J^,/i) be a probability space, (/„)„ è i a sequence of measurable real functions on Q, and (^„)„ à l a decreasing sequence of asubalgebras of J*\ Suppose (0 l/J = 9 for aH n and some integrable function g, and (») /oo = u m fn exists /i-a.s.. Then f] sén I = lim / i ( / j i j n^.1

/

n-a.s

n—>oo

Proof We claim that /M/«>

H < ) Û lim inf /i(/„ | s/n) n^ 1

/

/i-a.s..

n-*oo

Once this is proved, the lemma will follow by the same argument which is used to derive the dominated convergence theorem from Fatou's lemma. For each N ^ 1 we put gN = inf /„. Since 1^1 ^g,gN is integrable. The backward martingale convergence theorem gives P\9N \

Since gN \ f

Pi *** ) = lim Möwl-O ^ liminf/i(/„K„) «èl

/

n-KX)

/i-a.s..

n->oo

as N Î oo, the result follows in the limit N -> oo. D

300

Ergodicity

(14.20) Theorem. Let y be a shift-invariant specification, /j e ex ^ 0 (y), and A„ = [ — n, n] d fi S for alln^. 1. Then the following conclusions hold in the limit n -> oo.

(a)

TA^IAJ-1 ; Z

f°e\^n(f)

ß-a.s.

for all bounded measurable functions f on Q. (£>) / / E is a compact metric space and S its Borel a-algebra then lAnT1 Z yA„+i(-\diœ)->H weakly for ß-a.a. co. ieA„

(c) / / y is a ^.-specification for some X e Jt{E, S) then |A n | _1 YJ yA„+i(' |ö;Co) -> fi in the ^-topology for [i-a.a. to. ieA„

Proof (a) For given / and n, we put

/„ = iA„r z /°0,, ieA„

The ergodic theorem (14.A8) ensures that ß(f) = lim /„

ß-a.s..

n-»oo

On the other hand, since /x e @(y) we have

Assertion (a) thus follows from Lemma (14.19). (b) In view of equation (5.5), statement (b) is an immediate consequence of (a). (Cf. the proof of Theorem (7.12)(b).) (c) We can assume without loss that X e äP(E,$), cf. Remark (1.28)(3). Let p be a /(.-modification with y = pX., and let A e y be fixed. We set A'„ = {i e A„: A„ + i is A}, and we consider the function p^C,(o) = \K\~1

X V+«\A(PAB+JICA(Ö^W)

(C,û>efi)

on Q 2 . By definition, we have (14.21)

\A'„r

z

yA„ +i (/i0 i «) = i m ) / ( O P A , M

for all /ej£? A . We will show that p"A(-,co) converges in L x (/l s )-norm for /z-almost all co when n tends to infinity. To this end we introduce the probability measure v = Xs x /x on (Q, J^) 2 , the measurable function pA(£, co) = pA(CAcoS\A) on Q 2 , and the measurable transformations 0,-(£, co) = (£, ö;Co) of Q 2 . (Here £, co e Q and j e S.) We claim that (14.22)

pi = vflAJ- 1

X p A ° 0,.|^A x ^

v-a.s.

Ergodic Gibbs measures

301

for all n. In order to prove this we pick any n and i e A'„, and we let A e ^A and B e 9~Kn be given. Then J

v(dC,dco)A(Ari+0\A(pAn+i|CA(ö1-co)s\A)

AxB

= \ fi(dco)yAn+i(A\6ico) B

= J fi(dco)yAn+i(A\co) etB

= »(An9iB)

=

nyA(An9iB)

= \ Kdoj) \ P(dOßA(C,co) etB

=

A

I v(dC,dco)pAo9i(i:,co). AxB

The shift-invariance of /* has been used in the second and the last step. Since each term in the sum defining pA is measurable with respect to J ^ x 3~An, the equation above proves that (14.22) holds as claimed. Next we look at the right side of (14.22). As v is invariant under (ö;) ieS , the ergodic theorem (14.A8) implies that (14.23)

limlAJ" 1 £ p A ° ö ; = v(p A |J0

v-a.s..

Here J is the cr-algebra of all (^)iGS-invariant events in J^ x $F. Letting pA(0 = ^(dr,)pA(i:,r])

(CeQ),

we conclude from the ergodicity of fi that (14.24)

v(pA\J)((,co) = pA(()

forv-a.a.(C,
Indeed, let g be any bounded ./-measurable function. Applying Remark (14.3)(1) twice we see that, for any £ e Q,, g(C, •) is ./-measurable and thereby constant /i-almost surely. We thus obtain « A )

= 1 ^ S (d0 J / « | KdcoMZ, OJ)PA(C, V) = Jv(dC,dü))flf(C,ü))pÄ(C).

This proves (14.24). We now combine equations (14.22) to (14.24) with Lemma (14.19). Since the function ((, co) -> pA(() is measurable with respect to (~) J ^ x STAn, we get the result that lim p£{Ç, co) = pA(()

for v-a.a. (£, to).

n-*oo

Finally, we note that the functions pA( •, co) and pA are probability densities relative to Xs. Using Fatou's lemma in the same way as in the proof of Theorem (7.12)(c), we therefore obtain that lim A s (|^(-,co) - p A | ) = 0 n-*<x>

302

Ergodicity

for all co in a set QA e J* with yu(QA) = 1. As n e ^(y), pA is the probability density of yu|J^A relative to A S |JV Consequently, the left side of (14.21) tends to jx{f) for all / e ifA and co e QA. Since | An\A|,|/| A„| -> 0 and A is arbitrary, assertion (c) follows, a Theorems (14.17) and (14.20) have a corollary which is similar to (7.30). Suppose y is a quasilocal shift-invariant specification. We let %jim(y) denote the set of all JX e ^>(Q, #") which are such that /i=lim|A„r1 n -> oo

X y^+i(-\Qi
for a suitable to e Q. Here A„ = [ — n, n\d D S, and the convergence is understood in the sense of the if-topology. Each JX e %>hm(y) is called an averaged limiting Gibbs measure. It is easily checked that ^&Am{y) <= ^©(Q, J^). By an argument of Example (5.20)(1), it even follows that ^&Aim{y) <= @0(y). (The last inclusion even holds when ^ 0 lim(y) is defined to be the set of all cluster points of averaged Gibbs distributions with possibly varying boundary conditions.) (14.25) Corollary. Let (E,$) be a standard Borel space, X e Jt{E,ê), and y a shift-invariant quasilocal ^-specification. Then ^&(y) is the closed convex hull of ^e,iim(y) (in the &-topology). Proof. By virtue of Theorems (14.17) and (14.20), the proof of Corollary (7.30) applies unchanged. •

14.A

Appendix. The multidimensional ergodic theorem

Here we will prove the ergodic theorem which has been used in this chapter and will also be applied later on. Let (Q, J*\yu) be a probability space, d ^ 1, and 0 = (9i)iezd be a group of measurable yu-preserving transformations of Q which satisfy 0; o Oj = Qi+j for all i, j e Zd. Let / c j ^ b e the cr-algebra of all ©-invariant events. Also, let (A„)„â 1 be any sequence of cubes in Zd such that |A„| -> oo as n -> oo. For a given measurable function / : Q -> M we put (14.A1) RJ=\An\~1

I

fo6t

(n£l).

ie An

We investigate the limiting behaviour of Rnf as n -> oo. We shall start by proving mean square convergence. Then we shall obtain convergence in mean, and finally we shall prove almost sure convergence. To prove convergence in L2(/i) we shall apply the following Hilbert space lemma. (14.A2) Lemma. Let Jf be a Hilbert space with norm \\ • || and <€ be a nonempty closed convex subset of Jf. Then there exists a unique element f e <€ such

Appendix. The multidimensional ergodic theorem

303

that 11/11 = min9e<^ \\g\\. Moreover, if ( / J „ È 1 is any sequence in <6 such that lim ll/JI = 11/11 then lim \\f„ - f\\ = 0. n-*oo

n->oo

Froo/ Define c = inf ||g||. Take any sequence (/„) n è l i n ^ with lim ||/„|| =c. ge*

rt->oo

By hypothesis, || • || satisfies the parallelogram law which can be written in the form 11/ - 0ll2/4 = (U/H2 + ||2ll2)/2 - ||(/ + g)/2\\2

(fge

3f).

Consequently, if e > 0 is arbitrarily fixed and m, n ^ 1 are so large that ||/ m || 2 S c2 + s and H/JI2 ^ c2 + e then ||/„ - fJ2/4 ^ s because (fm + /„)/2 e ( €. This shows that (/„)„ èl is a Cauchy sequence. Let / be its limit. By continuity, ||/|| = c. As <& is closed,/ e #. If g e <€ is an arbitrary element with ||gf|| = c then the parallelogram law again implies that ||/ — g\\ = 0 . Hence / is unique, D The theorem below is usually referred to as the L 2 ergodic theorem. (14.A3) Theorem. For each measurable f with /i(|/| 2 ) < oo, \im ß(\RJ

- n(f\J)\2)

= 0.

Proof. We consider the Hilbert space L2(/i) of all (equivalence classes of) square integrable functions with norm ||g|| 2 = ^(lôf|2)1/2, g e L2(ß). Let <€ c L2(/i) be the closed convex hull of the set {f o Q/.j eZd}. Define c — inf || öf || 2. We will show that (14.A4) lim ||R„/|| 2 = c. n->oo

This will prove the theorem. Indeed, Lemma (14.A2) then will imply that lim | | R „ / - / | | 2 = 0, n-*oo

where / i s the unique element of <€ such that || / 1 | 2 = c. / i s nothing other than the conditional expectation \i(f\J). For let i e Zd be given. Since / o 9t e <& and | | / o 0.||2 = | | / 1 | 2 = c, Lemma (14.A2)_shows that / o 0. = / i n L2(n). In view of Remark (14.3) (2), this implies t h a t / h a s an ./-measurable representative. Further, if g is any J -measurable functon in L2(/i) then M(f - f)9)\ = Km \K(R„f - f)g)\ £\\g\\2 lim \\Rnf-f\\2

= 0.

n->oo

This proves t h a t / = \i{f\J\ We now turn to the proof of (14.A4). Let e > 0 be given. By the definition

304

Ergodicity

of <€, we can find a finite set A e Zd and nonnegative numbers tt, ie A, with YJ ti = 1 such that the function ie A

g-Zhfodj is close to /, in that \\g — f\\2 ^ e. Clearly, \\g\\2 S c + e. Moreover, for each n > 1 we can write

\\Rnf-R,9h

Y

=

tjlRJ-RH(foOj)

j'eA

^ I

fyl|Ä ll /-Ä„(/oEl y )

j'eA

= iA„r j'eA z o I / o 0 , - I /o0 ( ieA n

^ ll/WAJ-

1

ieA„+j

I tJ-|(A„+7)AAJ. j'eA

The last expression is less than e when n is sufficiently large. For such n we have C^||/Ull2^!l*,,0ll2+ll*»/-*.,0ll2 ^ II0II2 + e ^ c + 2e. This proves (14.A4) and thereby the theorem,

o

As an immediate consequence, we obtain the mean ergodic theorem. (14.A5) Corollary. Suppose that /x(|/|) < 00. Then Km p(\RHf - n(f\S)\)

= 0.

Proof. In view of Theorem (14.A3) and Jensen's inequality, the conclusion of the corollary holds whenever / e L2(/i). Since L2(/i) is dense in Lx(/i), this result can be extended to a l l / e L1(/i) by a straightforward application of the inequalities H(\Rnf - Rng\) ^ H(\f - g\ and H(\n(f\J) f,geLl(n).

- fi(g\J)\) ^ fi(\f - g\),

o

Our next objective is the individual ergodic theorem which ensures the almost sure convergence of Rnf. We still assume that (A„)„S1 is a sequence of cubes with |A„| -> 00. In addition, we now impose the condition that (A„) nàl be increasing. (But (A„)„ èl is not required to exhaust the lattice Zd.) The following maximal inequality is the key to the individual ergodic theorem.

Appendix. The multidimensional ergodic theorem

305

(14.A6) Lemma. Suppose that /i(\f\) < oo and c > 0. Then

J sup |K„/| > c W w D / c Proof. Since \R„f\ S Rn\f\, w e c a n assume without loss that / ^ 0. We also can and do assume that O e A , and thus 0 e A„ for all n. (Otherwise we translate all A„'s by a suitable vector and use the 0-invariance of fi.) Let N ^ 1 be an integer, A any cube in Zd, and a> e Q a fixed configuration. We define An = <^ sup # „ / > c and A^ =

{ieA:9iWeAN}.

For each i e A,,, there is some n(i, co) 5j N such that R„^œ)f{di<xî) > c. With the notation Ui(a = An(i>0)) + i, the last inequality can be rewritten as (14.A7)

X f(ôjœ) > c\UuJ.

To exploit this inequality we intend to cover a positive percentage of Ara with disjoint members of the family (UU(0)ieAœ. For each i e Am we let Vim be the cube in Zd which is concentric with UU(0 and satisfies \Vl>(0\ = 3d\Uit(0\. More precisely, if r ^ 1 and j e Zd are such that Uum = ([0, r [d fl Zd) + ; then VU(0 = ([ — r,2r[dr\ Zd) + j . We define a finite subset Wm = {ilt...,ik] of Am recursively as follows. Let i1 e Am be such that n(i1,a>) = max{n(i,œ): i e Aa}. If i1, e ..., ie are already defined and Aa\ [j VmAm # 0 then we take any v + 1 e
m = l

Ara\ U Vco,im such that e n(ie+1,co) = max ^ n(i, œ): i e A m \ (J F ^ L

m= l

Since iet/i, f f l c ^ ^ this recursion stops after finitely many steps. The resulting set Wœ is such that

U

Vi.œ^K-

Moreover, it is easily seen that the sets Ui(a, i e Wm, are pairwise disjoint. We let Um denote their union. Then 3d\UJ=

X 3"\UiJ=

I

\Viia\^\AJ,

and a summation of (14.A7) over all i e Wa gives X fidjco) > c\UJ.

306

Ergodicity

Since Um c A + AN = {i + j : i e A,j e AN} and / ^ 0, we arrive at the key inequality

£

f(dj0))>c3-d\AJ.

The rest of the proof is easy. As

IAJ = I U(0M ieA

an integration over co with respect to /i yields |A + A J V | / z ( / ) > C 3 - d | A | ^ J V ) . Letting first A run through any cofinal sequence of cubes and then N tend to infinity, we finally obtain the desired maximal inequality, a The following is the multidimensional individual ergodic theorem. As before, we let (A„)„ àl be an increasing sequence of cubes in Zd such that |A„| -> oo as n -> oo. R„f is defined by (14.A1). (14.A8) Theorem. For any measurable f with n(\f\) < oo, lim R„f = n(f\J)

n-as..

n-*ao

Proof. Let e > 0 be given. We first choose a bounded measurable function g with ji(\f — g\) ^ e23~d. Then we apply the mean ergodic theorem (14.A5) to obtain an integer N ^ 1 such that H(\RNg-ti{g\J)\)£e23-i. By Lemma (14.A6), AM s u p | Ä n ( / - 0 ) l > e j ^ £ and J sup \Rn(RNg - ii(g\J))\ > e) g e. We also have n(\n(f — g\ e) ^ e. Moreover, limsup|R„(RN0-0)| glA^I" 1 £ jcA,

limsuplAJ- 1 n->oo

£

|0o0.|

ie(A n +j')AA„

= 0

because g is bounded. Since R„/j,(g\J) = n(g\S) /x-a.s. for all n, we conclude that

Appendix. The multidimensional ergodic theorem

lim&up

307

\Rnf-n(f\S)\

n-^co

^\imsup\R„(f-g)\ n-^oo

+ lim sup \R„(RNg - v(g\s))\ + W-

g\s)\.

Combining this with the estimates above we obtain J lim sup \Rnf-n(f\J)\

> 3É ) ^ 3a.

The proof is completed by letting e tend to zero.

•

A final remark is in order. In the above we have always assumed that each A„ is a cube in Zd. This was just for simplicity. It is not difficult to extend all proofs of this appendix to sequences (A„)„èl of a more general shape.

Chapter 15 The specific free energy and its minimization

A cornerstone of the Gibbsian theory is the variational principle for homogeneous Gibbs measures. It asserts that the simplex ^e() of all homogeneous Gibbs measures for a shift-invariant potential O coincides with the class of all homogeneous random fields which minimize the specific free energy, viz. the difference of the specific (internal) energy relative to O and the specific entropy. The minimality of the free energy is precisely the classical equilibrium condition for a physical system at constant temperature. The variational principle thus supports the common belief that the Gibbs measures provide a proper description of physical systems in thermodynamic equilibrium. In fact, the variational principle for Gibbs measures is an extension to infinite homogeneous systems of an elementary variational principle for finite systems. Let us describe the latter in order to get some feeling for the subject. Suppose S is a finite set. For simplicity, we will also assume that the state space E is finite. Thus Q = Es is finite. Let O be any potential and H = £
the associated Hamiltonian. The unique Gibbs measure for
(co e Q).

Here Z = £ exp[ — H(a>)~\ is the partition function. For each probability coefi

vector /j on fl we let /x(if)= X n(œ)H(œ)

and

co eS2

Jt(n) = - £ M) co e Q

be the energy and the entropy of fi, respectively. The difference fi(H) — J^{n) is called the free energy of \i. The variational principle for finite systems now reads as follows: For all probability vectors /i on Q, the inequality

li{H)-tf{n)^

-logZ

holds true. Equality holds if and only if/i = v. Indeed, for each ft we can write, applying Jensen's inequality to the convex function q>(x) = xlogx, fi(H)-JiT(fi)

+ logZ=

X /x(
= X vM
Relative entropy

309

^ (p[ Z V MM W )/ V M meil

= «p(D = 0.

Since q> is strictly convex, equality holds if and only if the function a> -» fi(a>)/v(a>) is constant, and the latter means that fi = v. Let us note that the second expression above is just the relative entropy of fi with respect to v. This observation will be the starting point of our approach to the variational principle for infinite systems. The variational principle will be stated and proved in Section 15.4. In the first section we shall deal with some basic properties of relative entropy. These will be used in Section 15.2 to prove the existence of the specific entropy A per site relative to a given a priori measure on the state space. A will turn out to be an affine, upper semicontinuous functional on the simplex of all homogeneous random fields. For standard Borel state spaces, we shall derive the surprising fact that A can be represented as the integral of a suitable measurable function. Then, in Section 15.3, we shall turn to the specific free energy of a homogeneous random field relative to a homogeneous potential . In particular, we shall obtain the first half of the variational principle: The specific free energy attains its minimal value on the set &@(
15.1

Relative entropy

Throughout this first section we let (Q, 3P) be an arbitrary measurable space and fi, v any two finite measures on (Q., J^). (15.1) Definition. Suppose se is a cr-subalgebra of J*\ Define \v(f^ log/«,) 'oo

if n « v on se, otherwise.

Here f^ is any Radon-Nikodym density of y\sé relative to v\sé. Jf^(/^|v) is called the relative entropy of /x with respect to v on se. Jf^(/^|v) is also known under the names Kullback-Leibler information, informational divergence, and information gain.

310

The specific free energy and its minimization

In the definition above, we adopt the usual convention that OlogO = 0. Since the function x -» x log x on [0, oo [ is bounded from below, the integral v(fjsf logfrf) is always well-defined (possibly + oo). If se is generated by a finite J^-measurable partition n of Q then the relative entropy is given by the formula *Am=

Z

KA)logn(A)/v{A).

Aen

We also note that (15.2)

^(a/i\bv)

= a J f ^ | v ) + a/i(Q)\oga/b

for all a, b > 0. Consequently, we can (and will) assume henceforth that fi and v are probability measures. Under this assumption, we can write (15.3)

Jr j/ (^|v) = v ( ^ o / j / )

whenever fi = f^v on se for some .sZ-measurable function f^ ^ 0. Here x// stands for the function (15.4)

(A(x)= 1 - x + xlogx

(x^0).

This function is nonnegative and strictly convex. Moreover, \J/ attains its minimum 0 at x = 1 only. These properties of \)/ immediately imply the following basic facts. (15.5) Proposition. Let /j,ve ^(Q, fF) and se be a a-subalgebra of 3F. Then the following conclusions hold.

(a)^Mv)^0. (b) Jf^(fi\v) = 0 if and only iffi = v on se'. (c) J^^(fi\v) is an increasing function of se. (d) 2f?M{\i\ v) is a convex function of the pair (^, v). Proof, (a) This follows from (15.3) because i// ^ 0. It will also follow from assertion (c) because ^n,0}(^|v) = 0. (b) If Jfjf(n\v) = 0 then fi = f^v on se for some f^ ^ 0, and (15.3) shows that \\i o fa — 0 v-almost surely. Hence f^ — 1 v-almost surely, and thereby fi = v on se. The converse is obvious. (c) Let sé-t c sé2 be two er-subalgebras of $F. Without loss we can assume that fi = / ^ v on sé2 for some t G/ 2 " m e a s u r a r j le function fs#i ^ 0. Then [i = fsiv on stfi_, where f^i = v(fJ^2\s/1). Using (15.3) and Jensen's inequality for conditional expectations, we obtain ^J/1(^|v) =

v(^ov(/j/2|^1))

^vWiAo/^j^))

- ^2Mv).

Relative entropy

311

(d) Let n, n', v, v' e <^(Q, &), 0 < s < 1, ß = sfi + (1 - s)/x', and v = sv + (1 — s)v'. There is no loss in assuming that /x = f^v and /i' = f^V on J / , where / ^ , fa ^ 0 are .«/-measurable. By the Radon-Nikodym theorem, there exist ^/-measurable functions g^, g'^ ^ 0 such that v = g^v and v' = g'^v on j / . Since sg^ + (1 — s)g'^ = 1, we conclude from the convexity of i// that

•XTAm = vty(s9*f* + (1 - s ) ^ / ^ ) ) ^ nsg^Wf*)

+ (1 -

= sJTMv) + (1 The proof is thus complete,

sMMti)) sWAn'W)-

o

The next proposition provides us with a continuity property of relative entropy. This property will be applied in the next section but will not enter into the proof of the variational principle. Therefore, it can be skipped at a first reading. (15.6) Proposition. Let JJL, v e ^ Q , ^ ) , O^AeD be an increasing net of asubalgebras of 3F, and si = a I (J srfa J the smallest a-algebra that contains all s/a's. Then ^xeD ' WM?)

= um - 0 | v ) = sup JtT^(fi\v). aeD

IED

Proof Proposition (15.5)(c) immediately shows that JPAl*\v) ^

SU

P -#^.0*1 v) = l i m -*rf.M v ) - c-

aeD

iED

We thus only need to prove that J O ( ^ l v ) ä= c. Clearly, we can assume that c < oo. Then /i is v-continuous on each jrfa. Hence there exists a net {fa)aeD of ^-measurable functions fa ^ 0 such that /x = / a v on j / a . Plainly, {fa)asD is a martingale relative to v. We will show that this martingale converges in L1(v)-norm to some ^/-measurable function/ ^ 0. First of all, we note that (/ a ) a e D is uniformly v-integrable. This follows from the estimate v(/ a l { / . ä r } ) ^ G o g r r ^ a i o g / . l ^ . g , } ) ^ (c + l)/logr which holds for all a e D and r > 1. (The last inequality follows from the fact that x log x ^ — 1 when x 5: 0.) Next we claim that ( / , ) „ „ is a Cauchy-net in L*(v). For suppose the contrary. Then there exists an 6 > 0 and an increasing sequence (a„)„ èl in D such that v(|/ an+i — / a J ) ^ e for all n ^ 1. This is impossible because (X„)n&i is a uniformly integrable martingale and thereby convergent in L*(v). Since L*(v) is complete, we conclude that there exists an ^/-measurable

312

The specific free energy and its minimization

function / ^ 0 such that lim v(|/ a — / | ) = 0. Evidently, /j = / v o n se. Using oteD

these facts, it is now easy to derive the inequality J^^(ß\v) ^ c. For let N be any positive integer. Consider the functions g = N A ( / l o g / ) and ga = N A (/„log/,). Since fa converges in probability to / ga converges in probability to g. As — 1 ^ gx, g ^ N, this implies that lim v(\gx — g\) = 0. Hence l e »

v(g) = lim v(ga) ^ lim Jf^(/i|v) = c. CCGD

CCGD

Letting N tend to infinity we arrive at the desired inequality, D (15.7) Corollary. Let ß, v e ^(Q, 3F} and se be a a-subalgebra of !F. Then jT j / (/i|v)= sup TEG r i j ^

X M^)logM^)/v(^). .4 GTT

//ere 11^ stands for the collection of all finite se'-measurable partitions of Q. Proof. For each n e n ^ we let sJn be the finite er-algebra which is generated byre.With the usual partial order on 11^, (stf„)„sllj, is an increasing net. Since srf = a{sJn: n e IL^), the corollary follows from (15.6). D

15.2

Specific entropy

After the general preliminaries of the preceding section we return now to our standard setting. Thus (£, S) is supposed to be an arbitrary measurable space, S = Zd the d-dimensional integer lattice, and (Q, !F) = (£, S)s the associated configuration space. For each A c S and any two finite measures fi, v on (Q, J^A) we let (15.8)

jTA(n\v) *

jrrA(n\v)

be the relative entropy of ß with respect to v on 3FA. Let us assume that (£, S) is equipped with a finite a priori measure X. X will be held fixed throughout the following. For each A e ^ w e let o~^{XK) denote the finite measure on (Q, J \ ) which is given by ol\XK)(oKe

A) = P(A)

(AeêA).

Clearly, if X e &(E,&) then alx{XK) = XS\&A. (15.9) Definition. For each fi e ^(Q, &) and A e ^ , the quantity

is called the entropy of ß in A relative to X.

Specific entropy

313

Note that the A-dependence of JfA(fi) is dropped in our notation. If E is finite and X is counting measure on E then •*AM

= - I

M*A = OlOg/^A = 0 ^ 0 .

Thus in this case the entropy is given by Shannon's well-known formula. Looking at equation (15.2), we also see that multiplication of X by a factor b > 0 amounts to an addition of the constant | A| log b to J^A{-). Proposition (15.5)(a) thus shows that J^A(-) ^ |A|logA(£) for all A, and we can assume without loss that X is a probability measure whenever this is convenient. Suppose now we are given a shift-invariant random field \i. Let (A„)„ äl be any sequence of cubes with | A„| -> oo. (Recall that a cube was defined in the paragraph before (14.7).) We intend to show that the limit A(H) = lim l A J - ^ ^ e

[-oo.logA(£)]

n-*oo

exists and is independent of the choice of the sequence (A„) n à l . It is this limit which we will call the specific entropy of \i. The existence of A(/J) will be an easy consequence of the strong subadditivity property below. We put J^ifi) = 0 when \x e 0>(Q, &). (15.10) Proposition. For each \x e ^»(Q, #") and all A, A e Sf, 3fA(n) + JfA(n) ^ 3fAnA(n) + ^AUA(^)Proof. We can assume that Xe^(E, S) and ^AUA(/^) > — °°- Then /i = on r /AUA/*-S -^AUA f° some # AUA -measurable function fAUA ^ 0. Let fA = s ^(/AUAI-^A) be the /l -density of /J on J ^ , and let fA and / AnA be defined similarly. (If A D A = 0 we put / AnA = 1.) In view of Proposition (15.5)(c), all these functions have /i-integrable logarithms. Consider the measure v = nXA\A. Since v(/ A > 0) = /i(/ A > 0) = 1, the function /AUA|A = /AUA//A is well-defined v-almost surely. It is easily checked that ß = /AUAIAV on J \ U A . Similarly, we have /i = /A,AnAv on ^ A , where / A|AnA = /A//AHA- Therefore we can write, using Proposition (15.5)(c) again, •*AM

- ^ A U A M = /^(logAuAiA) = ^AUAMV) ^JifA(fi\v) = fi(logfMAnA)

The proof is thus complete, D Since ^(p) = 0, the preceding proposition implies in particular that JifA(fi) + J^A(fi) ^ ^ U A ( / 1 ) whenever A, A e £f are disjoint. On the other hand, if/i is shift-invariant then jfA+i(fi) = JfA(fi) for all A e y and i e S. The function a: A -> JfA(fi) thus satisfies the hypotheses (i) and (ii) of the following

314

The specific free energy and its minimization

well-known lemma on the limiting behaviour of homogeneous subadditive functions. We let y D denote the system of all rectangular boxes of the form

A = Z'fl ft [mt,nt] k= l

with mk, nk e Z, mk ^ nk. (15.11) Lemma. Let a: £fu -> [—oo, oo[ be a function which satisfies the conditions (i) a(A + i) = a(A) for all A e ifu and i e S, and (if) a(A) + a(A) ^ a(A U A) whenever A, A e t?a are such that A U A E ^ and A fl A = 0. Also, let (A„)„g ! be a sequence of cubes such that | A„| -> oo as n -> oo. Then lim |AJ- 1 a(AJ = inf |Ar x a(A) in [ — oo, oo [. Proof. We choose any real c with c > a = inf |A| -1 a(A), and let A e y D be such that |A| 1a(A) < c. Each cube A„ may De split into a number N„ of disjoint translates of A and a remainder which is contained in a boundary layer of A„. We assume that Nn is chosen as large as possible. Then lim |A„|/iVJA| = 1. Applying the subadditivity property (ii) successively (in the right order) and using (i) we obtain the estimate fl(A„) ^ iV„a(A) + (|A| - iV„|A|)a({0}), n ^ 1. Hence a ^ lim sup I A J ^ É ^ A J = lim sup N^1 lAp^A,,) n-*oo

n-»oo

^|A|-1a(A)
exists in [ — oo,\ogX(E)~\ and satisfies the equation A(pi)= inf lAI" 1 .^/!).

Specific entropy

315

(15.13) Definition. For each /i e ^®(Q, ê?\ the quantity A{ß) above is called the specific entropy per site (or the mean entropy or entropy rate) of /i relative to the a priori measure X. In the rest of this section we shall establish some properties of the specific entropy functional /!(•). Whilst Proposition (15.14) will enter into the proof of Proposition (16.11), Proposition (15.16) will only be used in Theorem (15.20) and Example (15.41), and Theorem (15.20) is only included here because it is interesting in its own right. On a first reading it might thus be preferable to jump to Section 15.3. (One can return when the need arises.) (15.14) Proposition. The function /£(•) on ^@(Q, !F) is affine and upper semicontinuous (relative to the ££-topology). Moreover, if (E, S) is a standard Bor el space then the sets {/!(•) ^ c] are compact (c e U). Proof. As before, we can assume that X e 3P(E, ê). 1) To begin, we shall prove that /!(•) is affine. By Proposition (15.5)(d), the functions J^(-) are concave. Hence /£(•) is concave. We thus only need to show that /!(•) is convex, too. So let /i l 5 ß2 e ^ @ (Q, J^), 0 < s < 1, and ß = sß1 + (1 — s)ß2. We must check that A(ß) ^ sA(ßl) + (1 — s)A{ß2). Clearly, we can assume that i(ß) > — oo. In this case, ß is absolutely continuous relative to Xs on each J \ . Since ß1 and ß2 are /^-continuous, we conclude that for each A e y and k = 1 or 2 there exists an J^-measurable function X\ = 0 s u c n t n a t ßk = fAXs on J \ . We can thus write, using the monotonicity of the logarithm, jfA(ß) = -MlOgCs/A1 + (1 - 4 / A 2 ] ) S - WlOgEsA 1 ]) - (1 - S)/X2(l0g[(l = sJfA(ßl)

S)fn)

+ (1 - s)jfA(ß2) - slogs - (1 - s)log(l - s).

Dividing by |A| and letting A run through a sequence of cubes we obtain the desired inequality. 2) Next we will prove that A( • ) is upper semicontinuous. In view of the last equation of Theorem (15.12), it is sufficient to verify that each JfA(-) is upper semicontinuous. This is an immediate consequence of Corollary (15.7). Alternatively, we can use the following argument which does not rely on Proposition (15.6). Let A e if, c e U, and /i e ^ 0 (Q, J^) with JfA(/i) < c be given. We will show that ß has an open neighbourhood U which is contained in the set {jfA(-) < c}. To this end we note that there exists an J^-measurable function g > 0 such that log g is bounded and

-ß(logg) 0 is sufficiently small. On the other hand, if ß fails to be 2 s -continuous on $FA then we choose a set Ae^A with ß(A) > 0 = XS(A) and define g = expCl^l - c)//z(X)].

316

The specific free energy and its minimization

Now we put U = {v G P9{0,&):

-v(logg)

< c + 1 - As(^)}.

1/ contains ft and is open because log ge i£. Moreover, if v G U and v = f^Xs on J \ for some ^-measurable / A =^ 0 then JfA(v)=-v(log0)-v(log/A/0)
Wkl9))

Sc. Here \j/ is given by (15.4). Hence U c {JfA(-) < c}. 3) To prove the last statement we fix a number c S 0 and look at the set K = {fie 0>@{n,^): Â(fi) ^ c}. We will show that K is compact in the <£topology. From Step 2) above we know that K is closed. By virtue of Proposition (4.9) it is therefore sufficient to show that each net in K is locally equicontinuous. So let A G yu and (Am)mèl be a sequence in J*\ with Am J. 0. For each fi G K we have ^ ( / i ) ^ c|A|. In particular, each fie K is Xscontinuous on J ^ with a density / / . For given e > 0 we let <5 > 0 be such that eloge/<5 5: 1 — c|A|. If m is so large that Xs(Am) S
+ H(Amn{f£

> e/ô})

(logs/ô)-ns(lArnn{f:>E/ô]fOogfjï) (loge/ô)-'(I-JfA(fi))

S 2e for ail fie K. The next to last inequality follows from the inequality x log x ^ — 1. The proof is thus complete. • We note that the proof of semicontinuity can easily be adapted to give the following result: If £ is a complete separable metric space with Borel tr-algebra ê then A(-) is upper semicontinuous with respect to the weak topology on ^ @ (Q, £F). (Just replace g by a continuous function which is sufficiently close to g in L1 (ft + Xs).) Next we shall use Proposition (15.6) to show that â(fi) can be represented as the conditional entropy of a single spin, conditioned on all spins in its "lexicographic past". (Cf. equation (15.18) below.) As a by-product, we shall also find that the system £fu can be replaced by Sf in Theorem (15.12). We introduce the lexicographic order on S = Zd by writing i < j whenever there exists some 1 S k S à such that ik < jk and i( = j( when 1 S t < k. Here i = (i1,...,id),j = (ji,...,jd) e S. We shall use the notation (15.15)

V(j) = {ie S:i<j

or

i=j}

for the lexicographic past of a site 7" e S. We also put V*(j) =

V(j)\{j}.

Specific entropy

317

(15.16) Proposition. For each fi e 3P&{&., fF) and] e S, Ä(n)= inf \A\-1jT&(li)=

-jfrru)(ii\iü{J]).

Proof. We can assume that X e 0>(E,ë) and j = 0. We put v = nX^. Theorem (15.12), we only need to verify the inequalities (15.17)

inf l A p 1 . * ^ ) £ -Jrv{0)(ii\v)

By

Z a(vi\

To prove the first inequality we can assume that JrfF(0)(^|v) < oo. Then for each A c V(0) there exists an J^-measurable function gA such that fi = gAv on 3F^. Let A e y be arbitrary and define

A = FI 9(vu)n\)-j ° 9-jUsing the shift-invariance of fi and the lexicographic order of A, it is easily checked that fi = fAXs on !FA. Thus |A|- 1 jr A (^)= - l A p 1 X

ß(}oggiVU)nu-j)

= -IAI" 1 X ^(0)n,A-;)(Hv) ^

-JfF(0)(/i|v)

because of Proposition (15.5)(c). Turning to the proof of the second inequality in (15.17), we assume that Â{pL) > —eG. Together with Theorem (15.12) and Proposition (15.5)(c), this implies that JfA(/i) > - o o for all A e ^ . Take any A, A e Sf with A c V(Q). Then we can write l A I " 1 ^ ^ ) = IAI"1 X [-*W(,>M - ^Af1K.(i)(^)] ieA =

2 J L^(A-«)nK(0)(W ~~ ^ A - o n r ' f o t C / 1 ) ]

|A|

i e A

= - l A p 1 X -#(A-onK(o)Mv) ieA

^ -IAI"

1

X

^A-onF,o)(^|v)

i e A: A —Î3 A

^ - | A r 1 | { i e A : A + i c A}|JfA(/i|v). The third equality was established in the proof of Proposition (15.10), and the inequalities follow from Proposition (15.5) (a) and (c). Letting A run through a sequence of cubes with |A| -> oo, we conclude that 4(/i) S — 3fA(n\v). Finally, we let A increase to V(0). Proposition (15.6) then implies that â{\x) ^ ~ ^F(0)(^lv)- The proof is thus complete, D The expression jfF(0)(/i|/i/l{0}) in Proposition (15.16) can be rewritten in a more appealing manner. Suppose j^vm(fi\fiX{0]) < oo. Then /i = g(nX{0]) for

318

The specific free energy and its minimization

some Jv (0) -measurable function g. In less condensed form, this means that = qa(A) = J X(dx)g(xœs\{0})

li(o0 e A\^v,(0))(œ)

A

for ^i-almost all œ and each A e S. We also can write JifVi0)(n\nX{0}) = nÀ{0](glogg) =

Hi^Al'W)

We thus end up with the formula (15.18)

4(n) = -ii(jr,(f\t))-

That is, Â(H) is equal to the conditional entropy of a0 given the lexicographic past JV(O)- In particular, let S = Z, E be any finite set, and X counting measure on E. Then (15.19)

40*)= -ii\Y

li(o0 = x\&ï-œM)\ogii(p0

= x | ^ - 0 0 , 0 [ )J.

We emphasize that the probability kernel q in equation (15.18) depends on fi. It is therefore surprising that the specific entropy can be represented as the expectation value of a measurable function which does not depend on fi. In particular, this result sharpens the conclusion of Proposition (15.14) that &(•) is affine. Recall from (14.2) that J stands for the <7-algebra of all shift-invariant events. (15.20) Theorem. Suppose (E, S) is a standard Borel space. Then there exists a function h: Q -> [ — oo, log X(E)~\ which is measurable with respect to J T)F and satisfies the equation â(n) = n(h) for all fi e $P&{£1, F). In particular, if fi e ^©(Q, #") admits a representation H=

|

v w(dv)

in terms of some weight w e 0>{0>&{Q, &), e{0>@{Q., F))) then A(n)=

\

4(v)w(dv).

^»«(fi..^)

Proof. 1) We define h = â(w), where n is the (^©(Q, F), ,/)-kernel which was constructed in the proof of Theorem (14.10). As we have noticed there, we can achieve that %' is measurable relative to F DF V\ JFV(0). Hence h is measurable relative to F 0 F f) FV{0), provided we can prove that Â(-) is measurable with respect to ^(^©(Q,F)). This will follow once we have shown that each JfA(-) is measurable with respect to ^(^@(Q, !F)). Since (E, ê) is standard Borel, each !FK is countably generated. Consequently, if we set se = FK then the supremum in Corollary (15.7) can be replaced by the supremum over a countable subcollection of 11^. This implies the required measurability.

Specific energy and free energy

319

2) Now we let ji e ^©(Q, fF) be given. We will prove that Â(JI) — ji(h). To this end we can assume that >f(/i) > — oo or fi(h) > — oo. Proposition (15.16) then shows that J^fV{0)(fi\fiX^) < oo or j^fV{0)(n'\n'X^) < oo fi-a.s.. In each case there exists an J*y(0)-measurable function g such that /i = g(fiX^) on J*y(0). This is obvious in the first case. In the second case we have %"' « n^X^ on JV(o) for /i-almost all co. Since p = fin, this implies that ft « fiX^ on JV(o>The existence of g thus follows from the Radon-Nikodym theorem. Next we will show that nm = g(nmX^) on JV(o> f° r /^-almost all co. Since J*V(0) i s countably generated, we need only prove that n'(B) = n'X^(glB) /i-a.s. for any B e JV (0) . But for each A e J (1 &~ (1 #"K(0) we have H{lAn{B)) = / i ( l ^ ( ß | , / ) ) = M ^ n S ) = A^.{o}(0Unji) = M M { O } ( 0 1 B ) ) = MUM^{O}(01JI)I^))

= H{iAn-X{0]{glB)). As 71' is , / (12T (I Jv (0) -measurable, this gives us the desired result. We now combine the above results with Proposition (15.16) to obtain the final identity >%) = -A*O°g0) = -Awrflogg) = -A*(-*r(0)("'M{0})) = A*C0-

The second assertion of the theorem follows immediately,

15.3

o

Specific energy and free energy

In the preceding section we have shown that the limit -4(Az)=lim|AB|-1jrA>|As) exists whenever \i e ^»0(Q, SF\ X e 0>{E, S), and {A„)„èl is a sequence of cubes with | A„| -> GO. Of course, the product measure Xs can be considered to be the unique Gibbs measure for the potential d> = 0. It is thus natural to ask what happens to the above limit when Xs is replaced by a Gibbs measure for a non-trivial potential $. This is the question which we will discuss in this section. We shall see that the proposed replacement amounts to the addition of two terms to — 4(fi). The first expression is a bilinear form in n and $ and is called the specific energy. The second term is a function of only and is called the pressure. As in Example (5.8), we let äß& denote the Banach space of all shift-invariant potentials $ with finite norm

320

(15.21)

The specific free energy and its minimization

||0||o= Z I I O J .

For each $ £ J 9 w e introduce the function (15.22)

/.A

£ \Ar®A. A30

C l e a r l y , / , e f and ||/.|| £ || oo and {oin)n^l any sequence of configurations. Then fi(fj

= lim | A n r V ( H * J = lim | A n r V ( < , K , ^ v ü ) -

(15.24) Definition. /i(/ 0 ) is called the specific (internal) energy per site of \i relative to
A ) = iArv(z/«.°0 for all A e y , it is sufficient to show that the expression (15.25)

sup meCl

Z / • ° 9-i - ^ ( ^ » s u )

ieA

is of order ^(|A|) when A runs through (A„) nàl . For each A e £f and œ e Q we can write J3AK<05\A)=I Z ieA i'e/lcA

i^r1«^

Z

^>A«S\A)

/lnA#0,/l\A^0

and

Z/*°0_,.= z I i^r<^+Z

ieA

I E A i€^lcA

Z

i € A A 3 i: /1\A ^ 0

i^r1^

Consequently, the expression in (15.25) is bounded by

-(A,0) = 2 Z

Z

ieA

110),

A3i:A\Ait
But for each A e y we have r(A,
Z

Z

i e A : A + i
g2|A|

Z

H^II+2

Z j'eA:A+t<£A

Z H^ll A^i

l|OJ+2||O||0|{ieA:A + i^A}|.

For each e > 0 there exists some A e y such that the next to last term is at

Specific energy and free energy

321

most e | A |. The last term has order
n-^oo

(provided (A„)„fcl is increasing). This follows from the preceding estimate of (15.25) together with the ergodic theorem (14.A8). (2) The mapping (p, <J>) -> p(fq,) on ^©(Q, #") x ^ @ is affine in /i and linear in O. If ^®(Q, #") is endowed with the i£-topology and â&& with the normtopology, this mapping is jointly continuous in p and uniformly in p.) Moreover, in the next chapter we shall see that this mapping establishes a duality of ^ @ (Q, !F) and ^@. For this reason, we shall henceforth use the notation (15.27)

£

\A\-^(OA)

AsO

for the specific energy, o We shall now assume again that {E,i) is equipped with a finite a priori measure X. The finiteness of X ensures that each $ e ^ @ is A-admissible. The following lemma provides a key to the question which was posed at the beginning of this section. (15.28) Lemma. Consider any p e ^@(Ü, #") and O e J 9 . Let (v„)„èl and (v„)„ai be two sequences in ^(Q, !F\ and let (A„)„ âl be a sequence of cubes with \A„\ -> oo. Under these conditions, the limit

lim l A J - ^ O x k t f J exists if and only if the limit

lim l A J - ^ ^ l v ^ J n->ao

exists, and in this case both limits coincide. Proof. We can assume that X e &{E, S). We fix any A e if and v, v e ^(Q, &). On &A, vyA is As-continuous with positive density p A , v = \v(dœ)pA(aAœs\A). Similarly, vyA is As-continuous on J \ with density pA-\ From this we conclude that either J^A{p\vyX) = oo = J ^ ^ v y ^ ) , or there exists an ^-measurable function fA ^ 0 such that p = / A A S on J*A. In the second case we have (15.29)

JeA(p\vyX) =

p(logfJp^)

= ^A(p\vy^)

+

p(logp^/pX'v).

322

The specific free energy and its minimization

Let r(A,) be defined as in the proof of Theorem (15.23). From that proof we know that r(A, (S>) — #{\ A|) when A runs through (A„)„ s ,. It is easily seen that sup |/tffo A f svv ) - A?(»7Aû)s\A)| û r(A,<&) and therefore e -2KA,*) g

P

> A Ç S X A ) / P A > A « ; S X A ) g e 2 * A -«>

for all Ç,œeQ. Hence l|logpr7pril^2r(A,(D). Inserting this estimate into (15.29) we obtain the lemma,

o

(15.30) Theorem. Let e ^ 0 a n d ( A n ) „ â l be a sequence of cubes with \ A„ | ->• oo. T/ien t/ie following conclusions hold. (a) For any sequence ((o")n^ ^ in Q, the limit (15.31)

P(cD) = l i m | A n r 1 l o g Z ^ > ' 1 ) n->ao

exists and depends only on <1> (and X). (b) For each \i e ^ © ( ß , ^ ) and v <E ^ ( $ ) , f«e limit A((i\v)=

lim I A J - ' J T A ^ A I I V )

exists and is equal to the quantity (15.32) *(/*|) = P(O) + (AI, O) -

4(p).

Proof. From Theorem (4.23) and Corollary (5.16) we know that there exists some ß e ^@(<S>). Putting v„ = ß and v„ = ôœ„ in Lemma (15.28), we see that (15.33)

lim|AJ- 1 jr A >|y? i i (-|( ü -)) = 0. n->ao

On the other hand, for each A e ^ and co e Q we can write

(15.34) jrA{M(- \œ)) = -jtrA(ß) + M(fl?Kû)S\A)) + logzJM (cf. equation (15.29)). Together with Theorems (15.12) and (15.23), equations (15.33) and (15.34) imply that the limit in (15.31) exists in ] — oo, oo]. In fact, this limit must be finite. This follows from the easily verified fact that | AI"11| log Z j || S ll^llo + I log 1(E) | for all A e ^.Assertion (a) is thus proved. Turning to the proof of (b), we let \i e ^@(Q, #") be arbitrary. Combining Theorems (15.12) and (15.23) with assertion (a), we can conclude from (15.34) that the limit on the left side of (15.33) exists and is equal to the expression in (15.32). Assertion (b) thus follows from Lemma (15.28), as applied to v„ = v and v„ = ôa„. a

The variational principle

(15.35) Corollary. For all ® e @& and pe &9(a,&) p e %(<&) then Â(p\Q>) = 0.

323

we have /f(/i|
Proof. Combine Theorem (15.30)(b) with Proposition (15.5)(a).

a

The preceding corollary is the first half of the variational principle: The function A{-10) at (15.32) attains its minimum 0 on ^ ( O ) . In the next section we shall prove the other half: Each p which minimizes â{-\<$) is a Gibbs measure for $. Corollary (15.35) also clarifies the significance of the function P(O) in (15.31): — P(O) is the common value of the specific free energy of all p e %(). In the lattice gas interpretation of our mathematical setting, P(
15.4

The variational principle

Here we shall prove the converse of Corollary (15.35): If /f(/i|$) = 0 then p. e %($>). Surprisingly, there are no difficulties in proving this interesting half of the variational principle in much greater generality. (15.37) Theorem. Let y be a quasilocal specification, v e ^®(y), and (A„)„ âl a sequence of cubes with | AJ -» oo. Suppose p e ^@(Q, &) is such that liminf|A n r 1 t 3f An (/i|v) = 0. n-><x>

Then p e %{y). Proof To begin, we notice that JfA(p\v) < oo for all Ae ,¥. This follows from Proposition (15.5)(c) and the shift-invariance of p and v. Consequently, for each Ae ,9" there exists an ^-measurable function fA ^ 0 such that p = fAv on J v Let us fix an arbitrary set A e £f. We shall prove, in three steps, that WA = P- As we shall see in the third step, we only need to show that there exist arbitrarily large sets A e 9> for which v(|/AVV — / J ) is arbitrarily small. This will be verified in the first two steps. Step 1. For each ô > 0 and each cube C D A there exists a set A e SP with A => C such that

324

The specific free energy and its minimization

JfA(n\v) - -*MAMV) ^ 6. To show this we let n ^ 1 be such that |A„| ^ \C\ and \An\^JfAri(ß\v) ô/2d\C\. Next we choose an integer m ^ 1 such that

^

m d |C| g | A J g(2m)"|C|. Finally, we choose md lattice sites i(l), ..., i(md) e S in such a way that the translates C(/c) = C + i(k), 1 ^ k ^ md, are pairwise disjoint subsets of A„. For each 1 S k ^ md we put W(k) = C(l) U • • • U C(k) and A(/c) = A + i(k). Then we can write, using Proposition (15.5)(c) in the first and third step,

jt=i

^ m"d £ [^V w (/i|v) - ^V (tACW (/i|v)] =

m^Jfw,(md)(/i|v)

gm-"jr A >|v)

^'ICIIAJ-^O/lv) Consequently, there exists an index /c such that

The claim of Step 1 thus follows by putting A = W(k) — i(k) and using the shift-invariance of \i and v. Step 2. For each e > 0 there exists some ô > 0 such that v(|/ A — / A \ A |) ^ e whenever A c A e ^ and J^(/i|v) — JfA\A(ß\v) ^ (5. For let \\i be the function at (15.4). Then (15.38)

JfA(/i|v) - JfAXA(/i|v) = /i(log/ A // MA ) = v(/ A log/ A // AXA ) =

V

(/A\A"A(/A//A\A))-

(Note that / A = 0 v-a.s. on {/AXA = 0} because / M A = v(/A|J^AXA) v-a.s..) A glance at the graph of ifr shows that there is a number 0 < r < oo such that |x — 1| S r&ix) + e/2 for all x ^ 0. Inserting this inequality into (15.38) we complete the proof of Step 2 by putting ô = e/2r. (In fact, we might invoke an inequality of Csiszâr (1967) to obtain the stronger result that the expression in (15.38) is not less than v(|/ A -/ AXA |) 2 /2.) Step 3. To prove that /iyA = /i we fix any g e $£ and e > 0. Since y is assumed to be quasilocal, there exists a ^-measurable function g e $£ such that \l\9 ~ #11 < £- Let C => A be a cube such that g e !£c and g e J5?C\A- Choose 5 in terms of e as in Step 2, and define A in terms of C and ô as in Step 1. Then

The variational principle IWA(0) - n(a)\ ^ MITAA - S\) + \K§)

-

325

V

(/A\A9)\

+ v(/ M A |g - T A 0|) + \v(fA\A(yAg - g))\ + \\g\\ V(IA\A - AD + |v(/A0) - n(g)\. ar

e

tne

Since g e ^AXA >d 0 =^A> second and the last term on the right are zero. The fourth term vanishes because v e ^(y) and fA\A is 5^-measurable. Due to the choice of g, the first and the third term are each at most e. The only non-trivial term is the fifth one. This term is not larger than ||g||e because of our choice of A. As e was arbitrary, we conclude that fiyA(g) = n(g)- The proof is thus complete, a Combining the theorem above with Example (2.25), Theorem (15.30)(b), and Corollary (15.35) we arrive at the following variational principle. (15.39) Theorem. For each ) is the set on which the specific free energy functional <• ,0> — &{•) attains its minimum — P(G>). Let us illustrate Theorem (15.39) by looking at the special case of finite state Markov chains. (15.40) Example. Suppose that S = Z, E is a finite set, and X is counting measure on E. Let Q be a positive stochastic matrix on E and /i Q e ^@(Q, F) the unique homogeneous Markov chain with transition matrix Q. We will show that fiQ can be characterized by a variational principle. First of all, we recall that { JXQ} = <&(y), where y is defined in terms of Q as in Comment (3.8) (1). By Theorem (2.30) and Corollaries (2.32) and (5.9), there exists a nearestneighbour potential $ e f 0 such that y = y®. Hence {/iQ} = ^(O). Theorem (15.39) therefore implies that £{n\0) = 0 if and only if \i = /j,Q. In fact, an explicit computation of A(- |G>) will provide a direct proof of this variational principle, as we shall now show. For convenience we will assume that O is a gas potential with a vacuum state a e E. Theorem (2.35)(b) guarantees that this can be done without loss. We let a>" denote the constant configuration with value a. Then we can write, using the notation (10.1), P(0) = - l i m n~nogy[Un](a[Un]

= a\coa)

n-*oo

= - l i m n'1 logQ{a,a)n+1/Qn+1 {a,a) n-*ao

= - l o g £(tf, a). The last identity follows from the fact that Q"{a,a) tends to a positive limit (cf. Theorem (3.A3)). On the other hand, if n ^ 1 and œ e Q are such that

326

The specific free energy and its minimization

o»s\[i. B ] =

a

t h e n

H*Un](co) = H?Un](ti) - H*Un](co°) logp*Un](coa)/p*,n](co)

=

= logf\

[Q(a,a)/Q(a>t,a>t+1).-]

i=0

Hence =

n{\ogQ{a,a)IQ{o_uo0))

for all n e ^ @ (Q, J^). This, together with equation (15.19), yields the identity *(/i|O) =

/i(-logÖ(ff-i,ff0) +

Z K
=

x\&i-aoM)) J

= J /i(da)pf,M<70 = • ^-„.o^MIßfa-i. •))• This identity provides an easy proof of the variational principle for /i Q : By Proposition (15.5), ^(/x|0) = 0 if and only if n{a0 = x|^_ œ > 0 [ ) = QOT-^X) /i-a.s. for all x e E. Since /i is shift-invariant, the latter condition just means that fi = HQ.

O

The next chapter is devoted to a number of consequences of the variational principle. Here we will just mention that the variational principle (15.39) sheds a new light on several results of Sections 14.2 and 4.4. Firstly, we have shown in (14.15) that each Gibbs simplex %(0) is a face of ^@(Q, 3F). This is an immediate consequence of Theorem (15.39) and the fact that the functional A{-10) is affine (cf. Proposition (15.14)). Secondly, in the case of a standard Borel state space it has been shown in (14.17) that the extreme decomposition in ^=)(0) of a given \i e ^©(O) coincides with the ergodic decomposition of \i. This follows directly from the variational principle together with Theorem (15.20). (Note, however, that Theorem (15.20) is much deeper than Theorem (14.17).) Next, we have seen in (4.23)(c) that the graph {(0,/i): O e i , / i e ^(O)} of the Gibbs correspondence is closed. Plainly, this implies that the set

{((D,//):
Large deviations and equivalence of ensembles

327

last assertion of Proposition (15.14). For, %(M) c L

PB(0,F):

A(n) ^ P(0) - 2 sup ||O|| 0 l

(

<SeM

)

because |<-,0>>| ^ ||O|| 0 and |P(
%\F) = j© £ ^ 0 : inf 4n\®) = ol, and Remark (15.26)(2) and Proposition (16.1) imply that inf Â(H\-) is conMeF tinuous.

15.5

Large deviations and equivalence of ensembles

The final topic of this chapter is the role played by the specific free energy for the large deviation probabilities of Gibbs measures. Recall that the ergodic theorems of Section 14.A deal with spatial averages of the form

ÄA/ = |A|- 1 J ] / o 0 _ ; ieA

in the limit when A runs through a sequence of cubes with |A| ->• oo. (In contrast to Section 14.A, we use here the backward shifts #_, rather than the forward shifts 6>,, so that RAf depends on the spins around A itself instead of its reflection —A.) It will be convenient to modify RA as follows. (15.41) Definition. Let A be a cube in 5 = Zd of side length p e N, i.e., A = S(1 n ^ i D 7 1 * ' mk+p[ with m e S. For each configuration co e Q, let co°A be the periodic continuation of coA, which at site i e S takes the value &>/(,), the spin of co at the unique site j (i) € A with ik = j (i)k mod p for all 1 ^ k ^ d. The probability measure

(15.42) °R»A e:\A\~1

J2S^<

ieA

on (Q, ^) is then called the periodic empirical field of co in A. Since °/?A depends on coA only, we will use the same notation when co e EA.

328

The specific free energy and its minimization

The advantage of the periodization is that each °RCA> is translation invariant, so that °R% e ^&(Q, J O for every cube A and all co. On the other hand, the next remark shows that this change makes no real difference when A becomes large. Throughout this section we write | A| —> oo for the infinite volume limit along an arbitrarily chosen sequence of cubes. Note also that the probability kernels °RA : (co, A)-»°/?^(A) can be applied as usual to functions / on Q. (15.43)Remarks. ( l ) F o r e a c h / G %', \\°RAf-RAf\\ -* Owhen|A| -* oo. To see this, let A e ^ be such that / G JZ?A- Then / ( 0 _ ; < Ü ) = /(0_,-Û> A ) for all i G A with A + i c A and all co e Q. Hence \\°RAf — RA/W is not larger than 21|/|| |{* G A : A + / ç£ A}|/|A|, which tends to zero as |A| -> oo. (2) For each /x G ex &@(Q, ß), the random measures °RA '• co —> °R^ in £?@(Ç2, ß) converge to /x in /x-probability when | A| —> oo. That is, for every neighbourhood U G -e(£P&(Q, ß)) of/x we have (15.44) ß(°RA G U) -* 1 a s | A | - ^ o o . Indeed, by (4.2) we can assume that U consists of all v G ^@(^2, ß) with \v(fj) - p(fj)\ < s for all 1 £ j ^ k, where k G N, fu ..., fk G ££ and s > 0 are arbitrarily chosen. The claim then follows from the first remark together with the ergodic theorem (14.A5). (3) Let O G ^ 0 and °// A = //* A be the Hamiltonian in A with periodic boundary conditions; recall its definition in Example (4.20)(2). It is then easily checked that °//* can be expressed in terms of °RA via the simple formula °H*

= \A\<°RA,<Ï>>.

o

The following large deviation principle implies that the convergence in (15.44) is exponentially fast when ß is Gibbsian for some O and U is a neighbourhood of the whole set ^©(O). Equivalently, a behaviour of °RA that is untypical for the whole class of Gibbs measures can only occur with a probability decaying exponentially fast to zero with speed | A |. The rate of this exponential decay is determined by the excess free energy functional /?(-|0). Throughout this section we assume that (E, S) is standard Borel. For each set C c ^ © ( ß , ß) we let C (resp. C°) denote the closure (resp. the interior) of C in the Jz?-topology. (15.45) Theorem. For each® G 3ë®andC G - e ( ^ 0 ( ß , ß)), the inequalities (15.46) limsuplAr'logsupyjfCÄA G C\co) ^ lAH-oo

(oeQ

1

(15.47) lim inf | Al" log inf y?(°RA G C\co) ^ |A|-*oo

oeQ

- inf A(v\4>), veC

- inf rf(v|<&) veC

hold. Inparticular, (15.46) and(15.47) remain true when the measures y£(- \co) are replaced by any JJL G £f(0).

Large deviations and equivalence of ensembles

329

Note that the infimum on the right side of ( 15.46) is attained whenever it is finite. This is because the level sets {/C(-|) ^ c] are compact; see Proposition (15.14) and recall that <•, > is bounded and continuous. In particular, the variational principle (15.39) implies that inf ye c Â (u|0) > 0 when %() n C = 0. We remark further that the inequalities (15.46) and (15.47) remain true when the Gibbs distributions y*(-\co) with configurational boundary conditions to e Œ are replaced by the Gibbs distributions °y * with periodic boundary conditions. The latter are given by > * = exp[-°//*]ÀA/°Z*, where °//* is as in Remark (15.43)(3) above and °Z* the appropriate normalizing constant. In fact, since °//* is a function of °RA, the proof of Theorem (15.45) then becomes slightly simpler because there is no need to consider the energy errors (15.25). We postpone the proof of the theorem and consider first a classical special case. For simplicity we confine ourselves to the case of periodic boundary conditions. Let us set up the stage. We fix some O e 88® as well as k potentials * ' , . . . , * * e SSQ, which are combined to a vector-valued potential * = ( * ' , . . . , tyk). Let t • * e <%& be the usual inner product of * with a vector t e Rk. We shall look at the probabilities that the Revalued Hamiltonians °//* = | A| (°RA , *> with periodic boundary conditions take prescribed values. (15.48) Corollary. In the setup above, we have for any Borel set B c Rk

limsuplAr'logV^IAr^eß)

^ - inf 7*(x|
|A|->oo

xcB

lim inf \A\~l \og°y%(\A\-1

°H* e B)

^

|A|-»oo

- inf 7*(x|0). x<=B°

Here, Mx\0) (15.49)

= =

inf

Â(v\<&)

sup[f -x - P ( 0 -t

• * ) ] + P(O)

(el

is a convex function ofx e Rk which has compact level sets {7* (• | O) fi c} and satisfies U,(-|- P ( 0 — t • * ) , which is convex by (16.1) below. Deferring again the proof we mention an even more elementary case. (15.50) Example. with * ( 0 ) = f(a0),

Let $ ë J 0 be arbitrary and * e â§® the self-potential where f : E -> R is any bounded measurable function.

330

The specific free energy and its minimization

Large deviations and equivalence of ensembles

331

332

The specific free energy and its minimization

Step 2. Here we establish a weaker form of (15.46), which is obtained by replacing C on its right-hand side with the closed convex hull ex C of C. The basic observation is that the measures Y^^RA belong to ex C. Indeed, for any choice of cylinder events A\,..., At and £ > 0, a convex combination v of measures in C with max]<_,-°^A(^J)\ < £ c a n be found as follows. Divide [0, 1]* into finitely many Borel sets h of diameter less than s, let Bi be the set of all œ e EA with°/?^ e C a n d ( ° ^ ( A i ) , . °R%(Ak)) e h, 0)1 pick some a>( in every non-empty Be, and define v — J2( YA\W^I) °^A We note further that the measures y^M and y ^ ^ A in &&(&, ß) are asymptotically equivalent in the .if-topology, in that lyACi,(/)-yAi^A(/)l^o if / e =5f and |A| —> oo. This is analogous to Remark (15.43)(1). Now, by the identities in Step 1 we need to show that (15.53) liminf[-/l(y A c ) + < y c ° i ? A î 0 > |A|->-oo "

-

i

+ P(0)]^mcA

l

J

inf

^

^

vecxC

where co e Q is allowed to vary arbitrarily with A. Suppose this inequality fails. There is then a constant c < mc and a sequence of A's and &>'s satisfying

- * ( y £ j + + pm s c. Since |<-, > | ^ || <ï>||0, the last property implies that all y£. belong to a set of the form {A(-) ^ a} which, by Proposition (15.14), is compact. So the y^ admit a cluster point ß. As noted above, \i is then also a cluster point of the sequence Y^J^RAHence ß e cxC. Moreover, A(ß\<&) ^ c because A is upper semicontinuous and <-, O) is continuous. Hence mc ^ c contrary to our assumption. Step 3. It remains to show that (15.54) m c 4

inf 4(v|0) = inf 4(v|4>). vecxC

vec

This relies on the fact that A (-|0) is affine and its level sets L^C={A (-|<ï>) ^ c} are compact. We use some convex analysis. To meet the standard hypotheses we need to change the topology on ^&(Q, ß). By Corollary (4.A13) we can choose a countable core <€ of & with <€ c ß°. The <îf-topology on ^ 0 ( Œ , ß) is then defined as the coarsest topology making all maps v —> v(A) with A e l continuous. It is coarser than the .if-topology and, by Comment (4.A10)(1), turns ^©(Q, ß) into a compact metrizable set. As the level sets L<j>.c are compact in the S£-topology, they are so in the 'if-topology, so that A(-\Q>) is still lower semicontinuous in the latter topology. We conclude further that both

Large deviations and equivalence of ensembles

333

topologies coincide on each L® c, and thus on {A(•{<&) < oo}. We write C for the closure of C in the ^-topology. Suppose now mc is finite (since otherwise (15.54) is trivial) and let \x e ex C be such that A(ß\$>) — mc- Proposition 1.2 and Lemma 9.7 of Phelps (1966) then show that there exists a probability measure wM on (the ^f-Borel sets of) £P®(Q, JF) which is supported on C and satisfies

K/*l<*>)

[ *(v\ ®)vrJdv).

So, vyß is in fact concentrated on C D {/f (-|0) < oo}, and C can be replaced by C here. Hence mc > inf ve c 4(v|4>), and (15.54) follows. • Proof of the lower bound (15.47). We still use the abbreviations introduced in the proof of the upper bound. It is sufficient to show that (15.55) liminf |A|-' logyMoJ(°RA

G C) ^ -/t(v|
|A|->oo

for every v e C° with 4(v|4>) < oo and arbitrary boundary conditions co depending on A. Let£ > 0 be given. By Proposition (15.52) and the continuity of <•, 0> one can find a measure v G C° D e x ^ 0 ( Œ , ^") with 4 ( v | 0 ) < 4(v|<ï>) -f £. On the other hand, using (15.25) and (15.31) we obtain logKA|Xi?A e C) = logX A ( e - w A^A^A) A

1(ORA6C))

_ logZ*(o>)

R

£ log^(,-' l<° -*> l,oRAeC)) - | A | ( P ( * ) + e ) if A is large enough. To estimate this further we introduce the event AA = {°/?A e C°, <°RA, $> < + £ , lAI-'log / A <

-A(v)+ej.

Here, / A is the Radon-Nikodym density of ÛIJ^A relative to a A "'(^ A ), which exists because A(v) is finite. Then we can write lAp'logKAiXtfA e C) ^ l A r ' l o g ^ / " 1 1AA) -<Î5, 0 > - P ( 0 ) - 2 £ ^

1

|Ar logv(AA)-/f(v|0)-4£

when A is sufficiently large. Now, the point is that AA is typical for v, in that v(A A ) —> 1 as |A| —> oo. This follows from the ergodicity of v together with (15.44), the continuity of <•, 0>, and the theorem of McMillan. The latter asserts that vdlAr'log/A + ^ v ) ! ) - » ^

as|A|^oo;

334

The specific free energy and its minimization

see Theorem 9.2.4 of Krengel (1985) or the references in the Notes on Section 15.2. So, letting first |A| -> oo and then e ^ O w e arrive at (15.55). • Proof of Corollary (15.48). We apply the so-called contraction principle to the continuous mapping ey : v -> from ^ © ( ß , &) to Rk; for a general version see Dembo and Zeitouni (1998), Theorem 4.2.1. The level sets of y*(-|*) are the e«j,-images of the level sets of A(-\<$>) and therefore compact. Likewise, the variational principle (15.39) shows that 7* (• | O) = 0 precisely on e* (^e(O)). The claimed inequalities are obtained by recalling the comment on °Y1 below Theorem (15.45), inserting C = e~\B) into (15.46) and (15.47), and noting that C C e~^{B) and C° D e^{B°) by continuity. It remains to verify (15.49). Let x G R* be such that e*(v) = x for some v € £ * e ( ß . •?) with A{y\-f • * ) + *•* - P ( 0 - ^ - * ) + P ( 0 ) . Since A (v| — ? • * ) > 0, an optimization over y and f shows that / * (x |O) exceeds the right-hand side of (15.49). Suppose this inequality is strict. Then there exists a number c < Jy(x\) such that t -x S P(<&-t • * ) - P ( 0 ) + c for all ! e K * . We shall see in (16.1) that the pressure P is convex. The HahnBanach theorem thus tells us that the linear functional t • V —>• t • x can be extended to a functional L on 3§® with L ^ P($ — •) — P{&) + c. Invoking Theorem (16.13) below (and taking care of the minus sign in the argument of P) we can conclude that L — (v, •) for some v e éP®(Q, &) with A (v) =

inf [ P ( 0 - * ) + L ( 0 - * ) ] ^ - c + P(O) + L(O).

Hence = x and ^ (vj) ^ c, so that 7*(x|0) ^ c in contradiction to the choice of c. • The proof of the upper bound (15.46) has the advantage of giving us some control of th the conditional measures y^w in the limit as |A| -> oo. Suppose C is such that (15.56) mc = inf *(v|) < oo. veC"

veC

In view of (15.47), the conditional probabilities y^w are then eventually welldefined, and the identities in the proof of (15.46) together with (15.47) give lim sup [(y£lw°RA, O) - A(y^J

+ P(O)]

|A|-

S - lim inf | A p 1 log yA]a>(°RA G C) ^ mc . \A\—KyD

336

The specific free energy and its minimization

Large deviations and equivalence of ensembles

337

Chapter 16 Convex geometry and the phase diagram

One of the principal problems of Statistical Mechanics is the investigation of the sets { 0 e # e : | e x # 8 ( 0 ) | = Ar} (1 ^ N ^ oo) in the Banach space ää@ of all shift-invariant absolutely summable potentials. These sets form a partition of @)@ which is called the phase diagram. Whilst the true nature of the phase diagram is still unknown, we can make some statements on its general structure, and this is the subject of this chapter. We shall establish two different kinds of results on the phase diagram. First, we shall be concerned with potentials < P e f 9 that admit at least N distinct pure phases which can be distinguished by means of N — 1 prescribed local observables. Such potentials will be shown to exist in every closed convex subcone of <%@ which is related to the given observables in a suitable way. Moreover, such potentials are not isolated. Using the same method, we shall also be able to establish the existence of potentials which exhibit a breaking of shift-invariance or a breaking of a continuous symmetry. All this will be the subject of Section 16.3. In Section 16.4 (which can be read independently of Section 16.3) we will study a different feature of the phase diagram. Namely, we shall consider all those potentials which admit only one shift-invariant Gibbs measure. These potentials will be shown to be generic. Moreover, the associated unique ergodic Gibbs measures will turn out to be dense in the set of all shift-invariant random fields with finite specific entropy. The above-mentioned results depend on two key facts: the convexity of the pressure, and the variational principle. The latter implies that each shiftinvariant random field \i with finite specific entropy can be identified with a unique closed hyperplane in &@ x M which is a tangent or an asymptote to the graph of the pressure P. Moreover, ^ is a Gibbs measure for a potential
The pressure and its tangent functionals

Throughout this chapter we let S = Zd be the integer lattice of arbitrary dimension d^.1. Also, we let (E, S) be any measurable space which is equipped

The pressure and its tangent functionals

339

with a finite a priori measure X. First of all, we shall look at the pressure P which was defined in (15.31) and (15.36). P is a real function on the Banach space ffl& of all shift-invariant absolutely summable potentials with norm (15.21); cf. Example (5.8). (16.1) Proposition. The pressure P: 3§& -> M is convex and satisfies the inequality |P(
(
Proof Fix any +*(û,) = l o g ^ K W X ) 1 " » S slogZ*(co) + (1 -

s)\ogZ]ï(œ),

by Holder's inequality. In view of equation (15.31), we see that P is convex. On the other hand, we have

Z * M = iA{hltijrv\cD) ^Z*(ui)exp||flJ-*|| gZ*»exp(|A|||O-¥||0). Taking the logarithm, dividing by | A |, letting A run through a cofinal sequence of cubes, and interchanging the roles of : n e ^>@{Q,^)} which follows from Corollary (15.35).) n The convexity of the pressure is a key result which underlies all further results of this chapter. As a first consequence of convexity, the one-sided directional derivatives (16.2)

ôTP())/f = sup (P())/t «to

t
a£P(
oo

exist and satisfy the inequality 3^P(0. More generally, P is said to be differentiable in all directions of a subset <€ oi@@ if ô^P( of all is a dense Gô set in &@. In other words, 8ß&\ß is a set of first Baire category.

340

Convex geometry and the phase diagram

Proof. By (16.1) and (16.2), the functions *P Q) = (~) Gy, where

dyP() are continuous. Hence

"fe(f

GT = {
V

=

D

U

"Jl

s>0

G

T,n,S-

Here Gv,njS = {n>s are open. Consequently, ® is the countable intersection of the open sets G^ „ = (J G^ „ s, f e ^ n ^ l . s>0

Let us show that the sets Gy „ are dense. The Baire category theorem then implies that<2)is dense. (See Reed and Simon (1972), Theorem III.8, for example.) Fix any Y and n and suppose G>j. „ were not dense. Then there is some <50 e ^@ and e > 0 such that <£ G Vn whenever ||0 — O 0 || 0 < e. In particular, if \t\ < ô = e/||T|| 0 then 0 + f¥ <£ G v „. Hence, the convex function t -> P(o + f ^ ) would be nowhere differentiate on the interval ] — <5, <5 [. This is impossible, G The preceding proposition shows that the restriction of the pressure to any separable subspace of ^ 0 is differentiate on a dense Gs set. In particular, if E is finite then ^@ has a countable dense subset, namely the set of all finite range potentials with rational values. Thus, in this case, the pressure is differentiable on a dense Ga subset of 3ä@. We shall now consider linear functionals on <M% which are related to the pressure as follows. (16.4) (a) L^P (b)

Definition. Suppose L: @S@ -> IR is linear. L is said to be P-bounded if there exists a constant c < oo such that + c. L is called a tangent functional to P at e ^ 0 if L(*P) ^ P(
for all f e J e . The set of all tangent functionals to P at $ will be denoted by ÖP(O). Clearly, a linear functional L: ^ 0 -> IR is P-bounded if and only if the quantity (16.5)

i ( L ) = inf [POP) - LOP)]

is finite. (In Section 16.2 we shall see that A(L) is the specific entropy of a shift-invariant random field which is associated with L.) L is tangent to P at some . In particular, every

The pressure and its tangent functionals

341

tangent functional is P-bounded. Moreover, each P-bounded functional L can be identified with the hyperplane HL = {(»F, t) e ^ @ x R: t = L(¥) + 4(L)} in J'Q x R. Two cases are possible: Either HL is disjoint from the graph GP = { p F , t ) e # e

x

R:f = P(*)}

of P. Since J/L and GP have distance zero, we then might think of HL as an asymptote to GP. Or HL is a tangent to GP at some point (0, P()). This case occurs if and only if L is a tangent functional to P at O. We add two further remarks. (16.6) Remarks. (1) If + t*F) - P( 0. In particular, every linear subspace of 38% is a convex cone. If C is closed then C contains its vertex 0. For any given O 0 e @@ we write O 0 + C = { 0. TTien tnere exists a potential $ e O 0 + C and a functional L e ÖP(O) sucn t/iat

342

(16.8)

Convex geometry and the phase diagram

||0 - O°|| 0 g 8- 1 [P(0°) - L 0 (O°) - 4(L 0 )]

and (16.9)

L(T)^L0(T)-8||T!|0

for allV

eC.

(If C is a linear subspace of ^ @ then the last inequality just means that ||L-L0|!C4

sup

|L0F) - L0OF)| ^ e.)

Proo/. There is no loss in assuming that L 0 = 0. Otherwise we replace P by the convex continuous function P — L0 and the resulting L by L + L 0 . 1) As a first step, we shall construct the potential O. For each O e J 9 we set C(O) = {¥ G O + C: P(»F) £ P(O) - e || V - O|| 0 }. We will show that there exists some O e $ ° + C which satisfies (16.8) and is such that C() = {}. First of all, we note that each C(O) is closed and convex. Moreover, the C(0)'s are nested, in that C(T) a C(O) when »F e C(O). For if ¥ ' e C(T) then *F,GxF + Cc:(I. + C + C = (I> + Cand PPF')^P0F)-e||¥'-¥||o ^ P(O)-e||^-O||0-e||V'-V||0 ^P(O)-e||V-O||0. Starting from the given potential O 0 , we define a sequence (O") nè0 recursively by choosing O n+1 e C(C>") in such a way that P($" +1 ) <

inf

PPF) + 82"".

4» e C(4>")

(O n ) nS1 is a Cauchy sequence. For if n > m then O" e C(m) and therefore e||0"

_ o»|| 0 g p(o«) _ p(0") < e 2~ (m ~ 1) .

Since ^ @ is complete, the sequence (0") n ä l converges to a limit O e &@. By construction, O e C(") - P(¥) < e2- ( "- 1) for all n. Hence C(O) = {O}. 2) In the second step we shall use the separating hyperplane theorem to establish the existence of L. (For a statement and proof of this theorem we refer to Reed and Simon (1972), Theorem V.4(a), for example.) We look at the sets

A geometric view of Gibbs measures

343

A = {PF,t)e# e x R:r>P(»F)} and B = {ÇV,t)eâlB

xU:x¥e®

+ C,t ^ P(0>) - e || v F - O|| 0 }.

Since P is convex and continuous, its epigraph A is convex and open. Similarly, since O + C is convex and the mapping ¥ -> HT — O|| 0 is convex, the set B is also convex. Moreover, A and B are disjoint. For suppose that (¥, t) e A H J3. Then ¥ e C(0>), whence ¥ = 0> and P(0>) < t ^ P(0>). This is impossible. Consequently, the sets A and B can be separated by a closed hyperplane. More precisely, there exists a continuous linear functional L on ^ @ x M and a number c e IR such that L(¥, t) ^ c when (¥, t) e /I and L(¥, t) ^ c when (¥, t) e B. Clearly, L can be written in the form LQ¥, t) = L(«P) + at

p F e 3B9, t e U),

where L: && -> R is a continuous linear functional and a e IR. It is easy to see that a < 0. Rescaling L and c if necessary we can thus assume that a = — 1. Then L(¥) - t ^ c for all ¥ e J ^ and all t > P(¥). Hence L ^ P + c. On the other hand, for each ¥ e 0> + C we obtain, putting t = P(0>) - e||*F - ). In other words, the infimum in (16.5) is attained at $. Hence L e 3P($). We also conclude that L ( ¥ - 0>) + e || ¥ - O|| 0 ^ 0 for all ¥ e O + C. Since this result is equivalent to (16.9) when L 0 = 0, the proof is complete. •

16.2

A geometric view of Gibbs measures

In this section we shall derive the interesting fact that each shift-invariant Gibbs measure can be identified with a tangent to the pressure. Moreover, each shift-invariant random field with finite specific entropy will turn out to be identifiable with a P-bounded linear functional on @)&. This identification depends on the variational principle as well as the duality of 38& and ^®(Q, 3F) which is established by the specific energy <•,•>; recall (15.27) and (15.23). To begin with, we introduce the set

of all shift-invariant random fields with finite specific entropy (relative to the a priori measure X). If E is finite and A is counting measure then ^®(Q, &) = &>&(Q,&). We also consider the linear space 3ß% of all linear functionals L:@&->M. The specific energy <•,•> establishes an affine mapping / : ^@(Q, &) - • 38% by

344

Convex geometry and the phase diagram

J(H)= - < / V >

(fie &*<&,&)).

We will show that^' is a bijection from ^®(Q, #") onto the set of all P-bounded linear functionals on ^ @ . The simple lemma below implies that J(/j.) determines all integrals /j.(f) (f e if) and thus \i. In other words, j is injective. (16.10) Lemma. Let f be a bounded local function on Q. Choose any A e y with f e S£K and define a potential O = 0 ( A , / ) by fl.

f / ° fy if A = A — i for some i e S, [0 otherwise.

Then O e # e , ||O|| 0 = |A| ||/||, and <^, = n(f) for all n e 0>%{Q.,&). Proof. Obvious,

a

Next we turn to the identification of the /-image of ^©(fi, 3F\ To this end we investigate the quantity ML) which was defined in (16.5). By definition, a function L e @&% is P-bounded if and only if A(L) > — oo. This occurs if and only if L e/'(^e(Q, J*)), as we will now show. The main ingredients of the proof are the first half of the variational principle which is stated as Corollary (15.35), as well as the properties of specific entropy which appear in (15.14). (16.11) Proposition. Suppose {E,S) is a standard Borel space, and let L e 0&% be given. If L = j(ß) for a {necessarily unique) ß e ^©(Q, #") then A(L) = A(n) > — oo. In the alternative case, A(L) = — oo. Proof. We extend the specific entropy to a function on 0&% by putting ML) = \A{ß) \ — oo

if L =

^ for otherwise

S

°me

ß e

^®(n'

^

when L e 0&%. The statement of the proposition is then equivalent to the identity A = A. The proof of this identity consists of three steps. 1) We start from the equation P(0)=

sup

[-+ /!(/!)]

(
which is an immediate consequence of Corollary (15.35). This equation can be rewritten in the form P(O) = sup [L(O) + A(L)~\

(
That is, the pressure P is the Legendre-Fenchel transform of the function — A. On the other hand, the claimed identity A = A is equivalent to the converse statement that — A is the Legendre-Fenchel transform of P, in that

A geometric view of Gibbs measures

-Â{L) = sup [L(O) - P(O)]

345

(L £ 3S%\

In the next step we establish the essential ingredient which allows us to deduce this equation from the previous one. 2) The function A: 38% -> [ — oo, oo[ is concave and upper semicontinuous relative to the weak* topology on 38%. (By definition, the weak* topology is the locally convex topology on 38% which is generated by the seminorms L -> |L(0)|, O e 3#@.) The concavity of A is an immediate consequence of the concavity of the specific entropy; cf. Proposition (15.14) or, even simpler, Proposition (15.5)(d). To prove the upper semicontinuity of A we need to show that the sets {L e 38%: A(L) ^c) (ce U) are closed. In fact, these sets are even compact. For they are the ^'-images of the sets {/j, e ^©(Q,3r): A{p) ^ c} which, by Proposition (15.14), are compact (in the if-topology). The compactness of the image sets thus follows from the easily verified fact that J- is continuous relative to the topologies chosen. 3) In view of Step 2 above, the identity A = A follows from a general duality theorem for Fenchel transforms; see Ekeland and Temam (1976), Proposition 4.1, for example. For the convenience of the reader we provide a proof. First of all, we conclude from Step 1 that P(O) — L(O) ^ A(L) for all O and L. Hence A^A. Suppose there exists some L 0 e 3S% with A(L0) < A(L0). Consider the locally convex space 38% x U. and its subset C = {(L,t)e<%% x

U:t^A{L)}.

C / 0 because ^®(Q, &) / 0. In view of Step 2, C is convex and closed. By assumption, (L0,A(L0)) $ C. The sets C and {(L0,A(L0))} can thus be strictly separated by a closed hyperplane; cf. Reed and Simon (1972), Theorem V.4(c). This hyperplane is described by a continuous linear functional on 38%. Moreover, a well-known theorem of Banach asserts that each continuous linear functional on 3ä% is of the form L -» L(O) for some O e 38&; see Reed and Simon (1973), Theorem IV.20, for example. Consequently, there exists some $ e l 9 and real numbers a, c such that L(O) + at < c for all (L,t) e C, and L0( c. It is easy to see that a > 0. We thus can assume that a = 1. (Otherwise we replace O and c by a " 1 0 and c/a.) Then L(O) + /|(L) < c for all L and thus P(O) ^ c. On the other hand, P(O) ^ L0(O) + J(L 0 ) > c. This contradiction completes the proof, n The above proposition combines two significant results. The first of these can be stated as follows: The pressure P and the specific entropy A are not only linked by the variational equation -P(
min

[-4(/x)]

(
which follows from Corollary (15.35), but also by the converse identity

346

Convex geometry and the phase diagram

4(fi) = inf [ + P(0)]

(/x G 0»e(Q, ^ ) ) -

That is, — P and 4 are conjugate concave functions relative to the <•»')duality of @& and 0>@(Q.,^). In terms of the quantity 4(/x|
min

*(/x|-) = 0, inf *(-|O) = 0.

The second result which is contained in Proposition (16.11) is the following one: For each L G 3$%, £(L) is finite if and only if L = j(n) for some /x G ^e(Q, J* ). So we have the following theorem. (16.13) Theorem. Suppose (E,S) is a standard Bor el space. The mapping /': /x -» — then establishes a one-to-one correspondence between ^Q(Q, J^) and the set of all F'-bounded linear junctionals on 3&@. The theorem above also admits an alternate proof: Starting from a P-bounded L, one can define a linear form /x on j£? by setting /x(/) = — L(O) when is associated to / via Lemma (16.10). The P-boundedness of L then implies that \i is uniquely defined, positive, normalized, shift-invariant, and ^-continuous. The Daniell-Stone theorem (cf. Bauer (1981), Theorem 39.4) thus shows that H G ^e(Q, #"), and the construction of /x implies that^'(/x) = L. This argument is carried out in Israel (1979), Theorem II.1.2. So far, we have only used that half of the variational principle which is stated as Corollary (15.35). We shall now use the other half to obtain the following geometric characterization of shift-invariant Gibbs measures. (16.14) Theorem. Suppose (E,ê) is a standard Borel space. For each establishes a one-to-one correspondence between ^ e (0) and the set 3P(<1>) of all tangent Junctionals to P at = P( ^P(). Then ^ P 0 P ) - P ( O ) for all ¥ G # e . Hence *(/x|0)^ inf A{ß\V).

A geometric view of Gibbs measures

347

In view of (16.12), this implies that a(ß\Q>) = 0. Theorem (15.39) thus shows that /x e ^©(Œ). Combining this result with Theorem (16.13) we conclude that f (%($>)) = dP(Q>), and the proof is complete, o We conclude this section with some simple consequences of the preceding theorem. The first result states that the pressure is strictly convex on suitable subspaces of ^@. For each a e # ( £ , l ) we let 38® denote the space of all a-normalized potentials in ^©. We know from (2.35) that two distinct elements of ^® can never be equivalent. (16.15) Corollary. Suppose E is a complete separable metric space, S the Bor el a-algebra, and À is everywhere dense. (a) Let $, 4* e J 0 be continuous, in that all <5/s and T^'s are continuous. Then <5 ~ *P if and only if the pressure P is affine on the interval [$, *P] = {tO + (l - t ) » F : 0 ^ t g 1}. (b) For each a. e 3?{E, $), the pressure is strictly convex on the space of all continuous potentials in 0&®. Proof (a) By Theorem (2.34), O ~ ¥ if and only if %(Q) (1 %Ç¥) # 0. In view of Theorem (16.14), the latter means that there exists a tangent functional L to P which touches P at 0> and ¥ simultaneously. Such an L exists if and only ifP is affine on [ 0 , ¥ ] . (b) This follows from (a) and Theorem (2.35). o As an application of the corollary above, we can state that each A # 0} of marginal distributions of an arbitrary /x e %( - »P> = then Q> ~ *F. Proof. Theorem (16.14) implies that 2P&Q + £¥) - P(O) - P(¥) = P ( 0 + £(¥ - O)) - P(O) + PÇV + i(0> - ¥)) - POP)

= o. Since P is convex, this inequality can only hold when P is affine on [
348

Convex geometry and the phase diagram

Our final corollary combines Theorem (16.14) with Dobrushin's uniqueness theorem. As in (8.36) we let J?@ denote the space of all O £ i 0 with finite norm

lll®lll= I

\A\\\<S>A\\,

ABO

and we look at the region 2 = { O e l 0 : HI(D|| < 1}. We know from (8.7) and (8.8), together with (5.17)(1) or (5.20)(3), that S(0) = S e (0) = { ^ } whenever £> e 3 (and (E, S) is standard Borel). Recall the notation (15.22). (16.17) Corollary. Suppose (E, S) is standard Borel. Then the pressure P is twice continuously differentiate on S in all directions of J?@. Its derivatives are given by the formulas

and dsdt

ieS

Here 0> e 3> and *F, ¥ e Proo/. Fix any (D e Q) and ¥ , »F e J@. Theorem (16.14) shows that dP(Q>) = {-<>}• Together with Remark (16.6)(2), this implies that d$P(<S>) = dq,P((!>) = — to(fq,). Putting g = —fq, in Corollary (8.37) we thus obtain the result, a

16.3

Phase transitions with prescribed order parameters

In this section, we will use the material of the preceding sections to establish the existence of phase transitions. More precisely, we shall start from a finite system {fi,---,fN} of local observables, and seek potentials $ e && which admit a family {fiy.j e J} of distinct Gibbs measures which can be discriminated by means of the associated expectation values {/^(/i),..., fij(fN)}. In the language of Statistical Physics, this condition on £> is usually expressed by saying that {/ x ,..., fN} is an order parameter for cD. Under weak hypotheses on {/ l5 ... ,fN) we shall find that such potentials $ exist. We shall also obtain some information on the general type of these potentials, but we shall be unable to specify any of these. This is because Proposition (16.7) provides

Phase transitions with prescribed order parameters

349

rather incomplete information on the touching points of the approximating tangent functionals. To begin, we introduce a concept of discrimination which underlies Theorem (16.20) below. Throughout the following we put / 0 = 1. (16.18) Definition. Let <4 {fx, • • • ,fN}. (16.19) Comment. Suppose (E, S) is standard Borel and $ admits a discriminating system {fl9.. .,fN}. Corollary (14.18) then shows that |ex %(<S>)\ ^ N + 1. In fact, the measures p0,...,pN for which (p.m(f„))0èm,nèN has full rank can then be chosen to be extreme in %($). For one can modify the proof of Corollary (7.30) to show that the set {(p(fn))0^nâN- ß e %(®)} <= ^N+1 is the closed convex hull of {{p(f„))oènâN: \i e ex%(<&)}. Consequently, these sets span the same linear subspace of UN+1. o We now give a characterization of potentials that exhibit a phase transition with a prescribed discriminating system {fi,...,fN}. We set N

fa=

Z Qnfn n= l

when a = (ai,...,aN)eUN.LetJ

be defined by (14.2).

(16.20) Lemma. Let
(ONI/,,...,/,}.

(ii) There exists some p. e%(<&) such that p(p(fa\ J)2) > p(fa)2 for all a e UN with \a\ = 1. Proof (i) implies (ii). Let p0,...,

pN e %(0) be such that (nm(fn))oèm

nèN

has

N

full rank. Define p = (TV + l)" 1 £ pm. For each o e R " w e let fa be a version oîp(fa\J). Since M i ? ) — M / J 2 is the variance of fa with respect to p, we only need to show that none of the functions fa is constant /^-almost surely. Suppose the contrary. Then there exists a vector a = (at,..., aN) e UN with \a\ = 1 and a number a 0 e M with ji(/ a = - a 0 ) = 1. By Theorem (14.5)(b), there exist ./-measurable functions gm ^ 0 with pm = gmp, 0 <; m ^ N. Thus /i m (/ a ) = ^9 m fa) = Ä9mfa) = - a0 for all m. Consequently, £ a„pm(fn) = 0 for all m. n= 0

This is impossible because of our choice of p0,...,

pN.

350

Convex geometry and the phase diagram

(ii) implies (i). For any fixed versions f„ of n{f„\J) we let «s/œ = o-(/i,... ,fN) be the smallest cr-algebra relative to which / j , . . . , fN are measurable. Plainly, «s/œ c «/. Moreover, «s/œ is countably generated. Thus we can find an increasing sequence («s/k)kâ j of finite subalgebras of «s/œ such that «s/œ is generated by (J «s/k. For each fe ;> 1, the atoms of sék constitute an ./-measurable fcai

partition of Q which will be denoted by nk. We claim that there exists an integer k ^ 1 such that the matrix has rank AT + 1. For otherwise there exists a sequence of vectors b(k) = (b$\...,fcj*>) e RN+1 with |ft(*>| = 1 such that

t WWAL) = 0 n=0

for all fc ^ 1 and A e nk (and thus all ^4 e «s/t). By compactness, the sequence (b (k) ) käl has a cluster point b e [R^ 1 with \b\ = 1. Since (^4) k â l is increasing, we conclude that

t KnixJn) = o n= 0 N

for all ^ e (J ,s/k and therefore all A e ^ œ . Since £ b„f„ is ^/ œ -measurable, we get the result that £ b„/„ = — b0 /x-almost surely. The vector b' =

(b1,...,

n= l

bjv) is non-zero because otherwise b = 0. Therefore, if a = b'/l^'l then n{fa2) = ß(fa)2, in contradiction to hypothesis (ii). This proves the claim. Now let k ^ 1 be such that Mk has rank N + 1. By a well-known theorem of linear algebra, there exist sets A0, ..., AN e nk such that the vectors vm = (ß(^A„L))o^neN (° ^™ 0 for all m. So the conditional probabilities nm = n(- \Am) are well-defined. Theorems (14.5)(b) and (14.15)(b) imply that ßme%(<5)) for all m. Since (/*m(/J)osm,™siv = (/'(4)^ 1 »m)osmp is a matrix of full rank, we conclude that ®<{fi,-•-,/»}• ° In order to establish the existence of potentials <5 with a given discriminating system {fi,.-.,fN} we shall combine the theorems of Section 16.2 with Proposition (16.7) on the approximation of P-bounded functionals by tangents to P. To this end we need to introduce certain subspaces of âS@. Let / t , . . . , fN e SC be given. We fix any cube A0 e Sf with / i , . . . , fN e SCAo, and we put A; = A0 U (A0 — i) when i # 0. Clearly, if i ^ ±j then At and A,are not a translate of each other. Also, if i ^ 0 and 1 ^ m, n ^ N then fmfn°6i e =^Aj- Therefore we can define a linear subspace @@(fi,..-,fN) of @S@ as follows. A potential <5 e ^ @ is an element of ^©(/j,...,f N )i(and only if <5 satisfies the three conditions below.

Phase transitions with prescribed order parameters

351

(i) 0 A o is a linear combination of the functions /„, 1 ^ n ^ N. (ii) For each i # 0, 0 A . is a linear combination of the functions fm /„°0,-, 1 :g m, n ^ iV. (iii) If yl e y is not a translate of some A; then <&A = 0. It is easily seen that @>@{fi,..-,fN) is closed relative to ||-|| 0 . More precisely, @&(fi,---,fN) is the closed linear span of the potentials 0 ( A °' / n ) and $(Ai,/m/n°8i) (j ^ o, 1 ^ m, n <; iV) which appear in Lemma (16.10). Recall the definition of ^@(Q, OF) at the beginning of Section 16.2. (16.21) Theorem. Let (E, S) be a standard Bor el space, N ^ 1, / i , ..., fN e JS?, and âS&{fi,. •. ,fN) be as above. (a) Suppose there exist measures ß0,..., ßN e ^©(Q, 3?) such that the matrix nas u (Mm(/n))osm,nsiv f ^ rank. Then for each O 0 e ^@ there exists a potential ®e° + é@(f,,..., fN) with 0 there exists some S > 0 with the following property: Each of the affine spaces $ 0 + ^e(/i,...,/iv) (®° e 3b%, \\<&° - THo < (3) contains a potential d> such that | ß(fa)2 for all a e UN with \a\ = 1. For in case (b) we know from Lemma (16.20) that ^©OP) contains such a ß. In case (a), a glance at the proof of the same N

lemma shows that we can take ji — (/V+l) - 1 Yl ßmm=0

Since the unit sphere in UN is compact and the mappings a -> ß(ß(fa | J*)2) — ß(fa)2 and a -> ||/fl|| are continuous, we can find an e e ]0,1 [ such that M(Mai^) 2 )-M(/ a ) 2 >4 £ |A 0 in/ f l ii 2 for all a e UN with |a| = 1. In case (b) we also require that e ^ e 0 , and we put Ö = e2/4. Now let O 0 e ^@ be given. In case (b) we assume that IJO0 — *P||0 < 5. We look at the linear functional L 0 = — on Sè%. By Theorem (16.13), L 0 is P-bounded. We apply Proposition (16.7) to L 0 and the closed convex cone C = ^@(f1,...,fN), and we use Proposition (16.11) and Theorem (16.14) to interpret the conclusion of (16.7). We conclude that there exist a potential ° + ^ 0 ( / i , . . . , f N ) and a Gibbs measure v e %($) such that |d> - O°|| 0 S rt(Ml$0) and -^ - - e | | ¥ | | 0 for all ¥ e <%@(f!,...,fN). The last inequality implies that O <§ {/ 1 ; ...,/#}. In order to prove this

352

Convex geometry and the phase diagram

claim we fix an arbitrary aeUN with \a\ = 1. We let T ° , ¥ ° , and ¥< (i ^ 0) be the potentials which are defined by Lemma (16.10) in terms of the functions v{fa)fa, n(fa)fa, - / a / a ° 0 ; and the sets A 0 , A 0 , A,., respectively. All these potentials belong to @&{fi,...,fN), and we have the estimates li^llo v ||¥°||o S |A 0 | l!/J!2 and ||¥'|| 0 rg |A ; | H/J 2 ^ 2|A 0 | ||/J| 2 , i # 0 . Hence (16.22)

v(fafaoei) - v(fa)2 = - < v , ^ + T°> ^ - < M , T i + T0>-3£|A0|||/a||2 = Malawi)

-

v(fa)Kfa)

~ ^\A0\

\\fa\\2

= Ma/ a °ö 1 .)--3e|A 0 ||!/ a || 2 ^ Kfafa°0t) - <^°y

-

4s\A0\\\fJ2

= M/ u / a °ö i )-M/ a ) 2 -4e|A 0 ||!/ a || 2 for all i # 0. We average the above inequality over all i # 0 in a cube A and let j A | tend to infinity. The mean ergodic theorem (14.A3) then gives (16.23)

v(v(ftt\J)2) - v(fa)2 = v(fa v{fa\S)) - v(fa)2 = Urn |Al"1 A

X

[va/ao0;)-v(/a)2]

O^ieA

^KKfa\*)2)-Kfa)2-te\*0\\\fa\\2

>0. Lemma (16.20) thus shows that °)^2|! 0 -T|| 0 <25. Hence ||0 - €>°||0 < 25/e ^ e 0 /2 and thereby ||0 - T|| 0 < (e0/2) + <5 ^ e0. The proof is thus complete, D (16.24) Comments. (1) In general, the phase transition potential 0 be any real number, and let ^(f1,...,fN) denote the space of all potentials A = 0 except when diam^ ^ R or A is a translate of A 0 . Statements (a) and (b) of the theorem remain true when 3fi&{fY ,...,fN)is replaced by &®(fx ,...,fN). Indeed, 08®{fr,...,fN) is a closed convex cone. The same argument as before thus shows that the inequality (16.22) holds for all i i= 0 with diam A; ^ R. Hence (16.22) holds for all except finitely many i. Since the value of the limit in (16.23) does not depend on these finitely many exceptions, we still get the result that

Phase transitions with prescribed order parameters

353

the first expression in (16.23) does not vanish. This is all what is needed to complete the proof. (2) Consider the case TV = 1 in Theorem (16.21). So we are given just one function f e S£. In this case, Theorem (16.21) remains true when @)&{f) is replaced by the closed convex cone ^©(/) of all O e @&{f) which are such that O A is a nonpositive multiple of ffoOi when i ^ 0. To see this it is sufficient to note that in the case N = 1 the potentials T, (i ^ 0) of (16.22) belong to &&(/)• Combining this observation with comment (1) above we even see that âS@(f) can be replaced by J£(/)H^@(f) for all R > 0. Also, if / ^ 0 then the potentials T 0 , T 0 in (16.22) belong to the closed convex cone ^e> + (/) °f all ® 6 ^®{f) f° r which 0 A o is a nonnegative multiple of/. Thus in this case Theorem (16.21) still holds when &@(f) is replaced by 3t@ + (f) H # £ ( / ) for any R > 0. (3) Consider the special case when / l 5 ..., fN are J^-measurable. Thus fn — f* ° °o f° r suitable bounded measurable functions /„* on E and all 1 ^ n ^ N. @)&{fi,...,fN) then consists of pair potentials, provided we put A0 = {0}. Moreover, the hypothesis of statement (16.21)(a) is then equivalent to the simple requirement that the functions 1, ff, . . . , / / are linearly independent modulo 2-null sets. (By definition, this requirement means that Hf* + c) > 0 for all a e UN\{0] and all c e U.) For suppose the last condition holds. Then we can imitate the proof of the implication (ii)=>(i) of Lemma (16.20) to obtain the existence of disjoint sets A0, ..., ANeS such that the matrix {X{\Amf*))oèmnûN has full rank. We put ßm = X{- \Am)s, O^m^N. Then nm e>^(Q,#") because A(\im) = Jtr{0){fiJ = log X(Am) > - o o . Also,A*m(/B) = HlAJn*)ß(AJ for all m,n. Thus has (^m(/»))osm,»äJv full rankConversely, suppose that X(f* j= c) = 0 for some a e UN\{0} and some ceU. Then the hypothesis of assertion (16.21)(a) cannot hold. For otherwise we can use the proof of the implication (i) => (ii) of Lemma (16.20) to construct a random field fi e <^(Q,#") with fi(fa2) > n(fa)2. But Jf {0} (^) ^ Â(\i) > - c o and thus {E, ê) such that <x(/„*) = 0 for all 1 ^ n ^ N. Then each potential in 88%{f^,. ..,fN) is an a-normalized pair potential (provided we put Ao = {0}). Combining this fact with comment (1) above we arrive at the following special version of statement (16.21)(b): Let *P be an a-normalized pair potential with ^ <^ {f1,...,fN}. Also, let £0 > 0 be given and ô > 0 be sufficiently small. Then the following conclusion holds. Given any a-normalized pair potential O 0 e 3S@ of a finite range R such that ||O0 — Tllo < 5, there exists an a-normalized pair potential
354

Convex geometry and the phase diagram

that (i) - 4*||0 < e 0 ,and

(iii)

®<{fu...,fN}.

The point of this result is that any two distinct a-normalized potentials can never be equivalent. (See Theorem (2.35)(a).) Property (i) above thus implies that the potentials O which correspond to distinct O 0 's are certainly not equivalent. On the other hand, there are infinitely many degrees of freedom for the choice of O 0 . Therefore we can conclude that there exists an infinitedimensional entity of pairwise non-equivalent pair potentials with discriminating system {/i,...,/*}. o (16.25) Examples. (1) Let E = {0,...,N}, X be counting measure, and /„ = l{ffo=„}, 1 <^n z% N. ^@{fY,...,fN) then coincides with the space of all shiftinvariant absolutely summable pair potentials which are gas potentials with vacuum state 0. By Comment (16.24)(3), the hypothesis of assertion (16.21)(a) holds. (In fact, we can put nm = ôm, the Dirac measure at the constant configuration with value m.) We thus conclude that for each O 0 e 3ß% there exists a gas pair potential O 1 e @@ with vacuum state 0 such that |ex &&(®° + O x )| ^ N + 1. In the case N = 1 we can even achieve that O 1 e ^© + (/i). In the lattice gas language, this means that the pair interaction between distinct particles is attractive, whilst the self-potential tries to push the particles out of the system. Applying Comment (16.24)(4), we also see that the set of all gas pair potentials O with |ex^ 0 (®)| ^ N + 1 forms something like an infinitedimensional manifold. (2) Suppose E c= M is bounded, even, and measurable. Let S be the restriction of the Borel er-algebra to E and A be finite and non-degenerate. We put / = <70. The cone ^® (/) then consists of all pair potentials O which can be written in the form ^

- J(i - jK-er,- ifA = {i,j}, i # j , — hai if A = {i}, 0 otherwise.

Here h e M is an "external field", and J: S -> [0, oo [ is even and absolutely summable. The nonnegativity of J means that O is ferromagnetic. Each such O is normalized by every even probability measure a e ^(E, ê) such a s a = {ôx + cLx)/2, x e E; cf. Example (2.38). Comment (16.24)(3) ensures that the hypothesis of Theorem (16.21) (a) is satisfied. Consequently, there exists some O of the above form such that ^©(®) contains two measures n~,n+ of different magnetization (i.e., /j.-{cr0) < n+(a0)). Moreover, the associated function J can be required to take prescribed values on any finite subset of S, and O belongs to an infinite-dimensional ensemble of potentials with the same properties. This follows again from the comments above, o

Phase transitions with prescribed order parameters

355

In the rest of this section we shall present two variants of Theorem (16.21) which take advantage of the presence of symmetries. The first variant concerns the breaking of the rotational symmetry in long range Heisenberg models. For given N 5: 1 we let E = {x e UN: \x\ = 1} be the unit sphere in UN and X the surface measure. (In particular, if N = 1 then E = { — 1,1} and X is counting measure.) We shall consider potentials
i ^ = {U}>'^> I 0 otherwise. Here J: S\{0} -> [0, oo [ is an arbitrary even function, and the dot denotes the usual inner product. Each such O is called a ferromagnetic Heisenberg potential. Clearly, any such O is preserved by all spin rotations (cf. Example (5.2) (3)).

(16.26)

*A

=

(16.27) Theorem. Let E be the unit sphere in UN and X as above. Given any ô > 0 and R < oo, there exists a potential O e 0S& of the form (16.26) with the following properties: (i) J{i) = 0 whenO < \i\ < R. (ii) ||) contains a family {/J-u)ueE °f random fields \iu which are magnetized in the direction u, in that ßu{ 0. In particular, \ex^@{(t>)\ ^ |£|. Proof. 1) Let 0 ^ h e UN be given. (Later on, we shall require that \h\ be sufficiently small.) We put Zh = \X{dy)ehy, and we let ah denote the probability measure on E with density fh{x) = Qh'x/Zh relative to X. The product measure vh = af belongs to 0>&{Çl,&). In fact, Proposition (14.7) (iii) shows that vh is ergodic. The magnetization vh{a0) is a positive multiple of h. For, 2Zhvh{a0 • h) = \X{dx)x • h{exh - e~xh) > 0 because t sinh t > 0 for all t ^ 0, and a simple symmetry argument shows that V ;,K • k) = 0 whenever k e UN is orthogonal to h. The specific entropy of vh is given by - Â{vh) = ah{\ogfh) = vh{a0 • h) - log Zh. In particular, 4{vh) > — oo. 2) We choose any e > 0 with e < \vh{<j0)\2/2, and we apply Proposition (16.7) to O 0 = 0, L 0 = — , and the closed convex cone C of all ferromagnetic Heisenberg potentials O e @& which satisfy property (i) of the theorem. We obtain a potential $ E C and, by Theorem (16.14), a Gibbs measure fj. e %{<$>) such that ||0>||0 ^ e"14(vÄ|0) and -^--£||Y||0 for all *F e C. In the next step we shall use this inequality to verify condition

356

Convex geometry and the phase diagram

(iii), and in Step 4) we shall apply the former inequality to prove property (ii). 3) For given i with \i\ ^ R we let 4" e C be the potential of the form (16.26) with J = l{j,_,-}. Inserting 4" into the last inequality and using the shiftinvariance of n and vh we find that Mo"o • °";) ^ vh(
H(o0-n(o0\S)) Vfc(ffo-Vfc(ffol-^))-2e

= |v,(a0)|2-2£

>0. The last equality follows from the fact that vh is ergodic. By Theorem (14.10), there exists a (^©(Q,!F),^-kernel n. Thus n(\n'(o0)\2) > 0. From Theorem (14.17) we know that n' e ex ^0() ju-almost surely. Consequently, there exists some co e Q with 7rra e ex^0(O>) and \nm{a0)\2 > 0. We put c = |7tu((T0)|, i; = 7rra((70)/c e £, and yUy = n'°. For arbitrary u e £ we let yUu be the image of nv under any rotation which maps v to u. Remark (5.10) and the observations before Corollary (14.11) then ensure that ßu e ex^@(). Clearly, ßu(cr0) — eu for all u. This completes the proof of (iii). 4) In order to verify property (ii) we exploit the estimate ||O|| 0 ^ £_1/€(vh|0). We shall prove that h and s can be chosen in such a way that £_1/C(vJ0) < N + ö. According to our condition on e we only need to find an h such that 2â(vh\0)\vh(a0)\-2 < N + Ö. We write x — (x1,.. .,xN) when x e £. Then lim \vh(a0)\/\h\ = lim \vh(a0-h)\/\h\2 = lim Jx-Äex'*A(dx)/|Ä|2A(£) |k|->0

= lim jx^e*1""1 - l)A(dx)/|Ä|A(£) 1*1-»o

= l/N. The second equality follows from the relation lim Zh = 2(£), the third from |A|-0 _

the rotational invariance of X, and the last from the identity N j xjA(dx) = J X *»*(d*) = A(E). n= l

On the other hand, we have *(vJ0) = logA(E)-4(vJ = |v Ä ( < 7 0 )||Ä|-logZ h M(£).

Phase transitions with prescribed order parameters

357

Writing ZJX(E) = 1 + \h\2X{E)-' J A(dj)[e^l"l - 1 - y, \h\M\h\2 we see that lim Â(vh\G)l\h\2 = l/N - 1/2N = 1/2JV. Hence lim 2/f(vh|0)|vh(a0)r2 = N. The proof is thus complete,

a

We note that a Heisenberg potential of the form (16.26) does not include a self-potential part that corresponds to the action of an external field. Property (iii) of the theorem above can thus be summarized by saying that
4 of the proof above. It is also worthwile to recall the results of Chapter 9: If d = 1 and N ^ 1 or d — 2 and N ^ 2 then property (iii) can only be satisfied when <5 has infinite range and a rather slow decay. It might seem surprising that the calculus of shift-invariant Gibbs measures can also be used to establish the existence of potentials which show a breaking of their shift-invariance. The following lemma explains why this is possible. (16.28) Lemma. Suppose (E, ) such that (16.29)

lim sup /*(//o0,) > liminf /i(//o0j).

(ii) There exists a measure v e ex %()\ex &((&) with v(v(f\&~) ^ v(/)) > 0. (iii) There exists a measure v e ex ^( (ii) => (iii) hold. Proof, (i) implies (ii). In view of the ergodic decomposition theorem (14.17) and the dominated convergence theorem, there exists v e ex %(<$) such that lim v(ffoßt) does not exist. We pick any A e ^ with fe J^A, and we set |ij->oo,ie J A

n = [ - " , n\d fi S, n ^ 1. For each n ^ 1 and i e I with (A - i) fl A„ = 0 we

358

Convex geometry and the phase diagram

have v(//°0i) - v(/) 2 = v(/o0,. ( v ( / | ^ A J - v(/))). Hence O
^||/|| limsupv(|v(/|^)-v(/)|) n-»oo

= 11/11 v ( | v ( / | ^ ) - v ( / ) | ) . The last identity follows from the backward martingale convergence theorem. We conclude that v(f\&~) ^ v(/) with positive probability. In particular, v is not trivial on 2T and thereby not extreme in ^{Q>). (ii) implies (iii). Let v be as in statement (ii). Suppose that v(foQj\3~) = v(f\3T) v-a.s. for all; e S. Then the ergodic theorem (14.A8) shows that v(/m=lim|AJ-1

X v(/o0 ; |^)

= v| limlAJ" 1 "X = v(/)

" e"j sr f°

v-almost surely,

in contradiction to our hypothesis. Therefore we can find j e S such that v(v(foBj\^)¥= v{f\2T))>0. By Theorem (7.26), there exists a ( ^ ( 0 ) , ^ > kernel n. The set

has positive v-probability and is therefore non-empty. This proves statement (iii). D We are now ready to state a second variant of Theorem (16.21). We fix any function / e jSf and a symmetric infinite subset / of S. We choose any cube A 0 e y with / e i? Ao , and we let A; (i ^ 0) and &@(f) be defined as in the paragraph before (16.21). We introduce the linear space âa®{f) of all potentials G âS&(f) which are such that 0 A j = 0 when 0 ^ i $ I. Let us agree to call a random field p f-I-oscillating if p satisfies (16.29). (16.30) Theorem. Suppose (E, S) is a standard Borel space, and let f and I be as above. Suppose further that there exists an j-1-oscillating p. G ^®(Q, 3F\ Then for each <1>0 e &% there exists a potential O e O 0 + &®(f) which admits an f-I-oscillating shift-invariant Gibbs measure. In particular, ex ^0()\ex (3(^>) ^ 0 and ex £(0)\£ e (0) ^ 0. Proof. By hypothesis, we can find some p. G ^®(Q, J*) and an e > 0 such that the difference of both sides of inequality (16.29) exceeds 2e|A 0 | ||/|| 2 . Using Proposition (16.7) and Theorem (16.14), we obtain $> e
Ubiquity of pure phases

359

Gibbs measure v e ^©(<1>) such that |-<M,¥>| ^£|!¥||o for all ¥ e # £ ( / ) . Hence |v(//oöi)-M//°öi)|^2e|A0|||/||2 for all i e I. By our choice of e, this inequality implies that v is /-/-oscillating. The last sentence follows from Lemma (16.28). • In analogy with Theorem (16.21)(b) one can also show that the potentials $ e ^ 8 which admit an /-/-oscillating v e %(<&) are non-isolated. Mutatis mutandis, Comment (16.24)(4) applies as well. We leave this to the reader. (16.31) Example. Let E = { —1,1}, X be counting measure, / = aQ, R > 0, and / = {{iy,..., id) e S: 11\ | ^ R, i2 = • • • = id = 0}. There exists some f-Ioscillating \i e ^©(Q, 3F\ For let fi0 be the unique shift-invariant Markov chain with state space E, parameter set Z, and transition matrix (° £)• (In the notation of Example (14.16), n0 = (<5+_ + <5_+)/2.) Identifying Q with (£ z ) z "~\ we can define fi = /ig""1. Then n e ^ ( Ü , ^ ) = ^©(Q,^) and 1 ^((TQO-;) = (— l)' for all i e I. Theorem (16.30) thus yields the following result: Given any potential <ï>° e 3S@, there exists a potential $ e l e which exhibits breaking of its translational invariance and coincides with <ï>° except on /-pairs, in that Q>A = <1>° unless A = {i,j} and i — j el. o

16.4

Ubiquity of pure phases

This section is devoted to the phenomenon that ergodic Gibbs measures are ubiquitous. Our first result will show that a "typical" potential $ e ^© admits only one shift-invariant Gibbs measure / v By Theorem (14.15)(a), ^ is necessarily ergodic. Then we shall prove that the collection of all these JJL&S is dense in the set ^©(ü, 3F) of all shift-invariant random fields with finite specific entropy. This proof will be broken up into three stages. First we shall show that the /i^'s are dense in the set of all ergodic Gibbs measures. These in turn will then be shown to be dense in the set of all shift-invariant Gibbs measures. Finally we shall see that [J {^©():
360

Convex geometry and the phase diagram

is a dense Gô set in 3ß%. In other words, the potentials 1 farm a set of first Baire category and can thus be viewed as being exceptional. Proof Since S is countably generated, there exists a countable generator se of J* which consists of cylinder events and is stable under finite intersections. For each A e se we choose a potential in â$& which is associated to 1A according to Lemma (16.10), and we let # denote the countable collection of all these potentials. We also fix an arbitrary $ e f 0 . We claim that | ^©(O) | = 1 if and only if P is differentiable at 3> in all directions of (€. In view of (16.3), this claim will prove the proposition. Suppose first that )%{<S>)\ = 1. Theorem (16.14) and Remark (16.6)(2) then imply that P is differentiable at )| = 1. n The preceding proposition can be viewed to be a weak form of the famous Gibbs phase rule, which can be phrased as follows: For each N ^ 1 and any finite-dimensional subspace M of 3ß@ which contains at most one representative of each equivalence class, the set {
%m = (M whenever <1> belongs to the uniqueness region °ll&. The next two theorems will show that the set {(O, p^)\ $ e $r©} is dense in the graph of the correspondence %: %($). Of course, the space 3S@ x ^ 0 (Q, J*) will be topologized with the product of the ]| • ||0-topology on ffl& and the i?-topology on ^@(Q, êF\ (16.33) Theorem. Suppose (£, S) is a standard Borel space. Then the set {(<&, p^)\ O e ^©} is dense in {(O,^):Oe# e ,/xeex0 e (O)}. Proof. We let C denote the closure of {( s W@}. By Theorem (4.23)(c) (or the remarks at the end of Section 15.4), C is contained in the graph {(3>, p.): cD s @@, p e %(<&)}. Consequently, if cD e 3ß@ is fixed then the O-section

Ubiquity of pure phases

361

C(G>) = {p e ^ ( Q , ^ ) : (®,p) e C} is a closed subset of %{0). We need to show that ex %(<£) c C(S>). This will be done in two steps. Step 1. Given any /i e "^©(S*) a n d / e if, there exists some v e C(O) such that v(/) ^ ju(/). Indeed, let *F e ^ @ be any potential which is associated to / via Lemma (16.10). By Proposition (16.32), there exists a sequence (0")„ à l in ^l& such that ||O — n" lv P — O"||0 < n~2 for all n ^ 1. The last condition implies that -v(/)| S P(®") - P(O) ^ - < / v , 0 " - <&>• Hence /i(/) ^ / v - ( / ) + 2/n for all n ^ 1. By Theorem (4.23) (b), the sequence (/vOnäi has a cluster point v. This v has the required properties. Step 2. Let p e ex ^ 0 (O) be given. We will show that p e C(O). Since p is ergodic, we conclude from Theorem (14.10) that p(n' = p) = wM({/i}) = 1 for some (^®(Q, #"), ^-kernel 7i. By hypothesis, there exists a metric d on E which generates 1 as n -> oo. We set 0„)k = (1 - kd{-,K„)) v 0, fe,n ^ 1. Then #„, k l lK>i as fc| oo, for all n. It is easily checked that each gnk is uniformly continuous. Remark (2.21) thus shows that gnJi e &. Hence ||ginjt - /„ jk || < 1/fc for suitable functions fn>k e if. According to Step 1, there exist measures vnk e C(«D) such that v„ k(f„ k) ^ p(f„yk). Since C(«D) is a closed subset of the compact set %{Q>), the sequence K . A ä i has a cluster point v„ e C(«D), n ^ 1. Clearly, vn(gn,k)^limmfvn<s(g„^ co

^ lim p(g„s) /-co

= MK„) for all k and n. Hence v„(K„) ^ /x(K„) for all n and thus v„(7f = p) -• 1 as n -> co. This in turn implies that v„ -> p as n -> oo. For if g e if then \vn(g)~K9)\Svn(\n\g)-p(g)\) S2\\g\\vn(n-^p)^0 because v„ = v„n. Since C(«D) is closed, we conclude that p e C(«D). D Next we combine Propositions (15.52) and (16.7) to show that the ergodic Gibbs measures are dense in the set of all shift-invariant Gibbs measures.

362

Convex geometry and the phase diagram

(16.34) Theorem. Suppose (E, é>) is standard Borel. Then the set {(«D,M):$eÄ 0 ,^ex^((J))} is dense in the graph

of the Gibbsian correspondence ^ 0 : O -» ^©(O). Proof. We fix any <5 e ^ @ and fi e ^ 0 (O). By Proposition (16.34), there exists a sequence (^„)nä i in ex ^©(Q, #") such that fi„-> fi and /l(^„) -> /l(^) as n -> oo. In particular, /l(^„|0) -> Â{H\Q>) = 0. Passing to a subsequence if necessary we can therefore assume that A(n„\<$>) < n~2 for all n 2; 1. For each n 2> 1, we apply Proposition (16.7) to the cone C = @@, the P-bounded functional L 0 = — <^„, • >, the potential <5° = <5, and e = 1/n. Together with Theorem (16.14), we obtain a potential O" e ^ @ with (16.35)

||4>" - O||o ^/ii«(Ai I I |0)< l / n

and a Gibbs measure v„ e ^©(O") such that (16.36)

|-<^,T>|^||T||0/n

for all f e l 9 . To exploit the last inequality we fix any A e ^ and / e ifA. We consider the potentials which are associated to the local functions ff°Ot and the sets AU (A — i) in the sense of Lemma (16.10). Inserting these potentials into (16.36) we obtain that |v„(//oei)-^(//oei)|^2|A|||/||2/n for all i e S. Just as in the proof of Theorem (16.21), the mean ergodic theorem (14.A3) now implies that |v>n(/|^)2)-^(/)2|^2|A|||/||>. (Note that fi„ is ergodic). Therefore, if % is a (^©(Q, #"), J)-kernel then |v„(7r-(/)2)-^(/)2|^2|A|||/||>. Inequality (16.36) also yields the estimate k(/)-AU/)|g|A|||/H/n. We thus conclude that

v„([X(/)-M/)]2) ^2^(/)2 + 2|A|||/||>-2vn(/K(/) ^4|A|||/||2/n for all n > 1.

Ubiquity of pure phases

363

Now we let {Ak: k ^ 1} be a countable generator of #" which consists of cylinder events and is stable under finite intersections. For each k we choose a set Ake£f with Ak e J \ k , and we define fk = 2"k \AJ\Aft|. Then

v»( t I W / t ) - ^ ( / » ) ] 2 ) ^ 4 / n for all n ^ 1. From Theorem (14.17) we know that v„{if e e x ^ O " ) ) = 1. Consequently, for each n ^ 1 there exists some v„ e ex ^0(") such that (16.37)

X [v„(/k)-M„(/k)]2^4/n.

Since ^ has a subnet (V)aeß which converges towards some v. But (16.37) implies that v(Ak) = Hm v„a(Ak) = lim nnj,Ak) = n{Ak) xeD

OLSD

for all k ^ 1. Hence v = n and therefore ( U e o which converges to a given point (a can be chosen to belong to the affine space <5) + ^(fx,.. .,fN). Here R > 0 is arbitrary, and ^ ® ( / l 5 . . . , / N ) is defined as in Comment (16.24)(1). Indeed, the order parameter {/i,...,/ w } can replace the sequence {fk: k ^ 1} in (16.37), whence (16.36) is only needed when ¥ e ^@(/ 1 ; ... ,fN). It is therefore sufficient to apply Proposition (16.7) to the cone ^ | ( / i , • • • ,/jv) instead of 08&. Theorems (16.33) and (16.34) immediately imply the following corollary. (16.38) Corollary. / / (E,S) is standard Borel then {(O,/^): )}. In particular, the set {/v <5) e <%@} of all unique ergodic Gibbs measures is dense in the set <&@(3S@) = (J {%(<&): $ e 3§&} of all shift-invariant Gibbs measures. (16.39) Comment. Corollary (16.38) can be sharpened as follows: For each 0 e @)& and ß e ^ ( O ) there exists a sequence (rather than a net) (Q>")nè0 m <%@ such that O" -» (5) and / v -» ß as n -» oo. For let {^4k:fe^ 1} be a countable generator of J* which consists of cylinder events and is stable under finite intersections. Corollary (16.38) then implies that for each n ^ 1 there exists some
By Comment (4.14)(1), the sequence (/v>)„ è i is locally equicontinuous. In view of Proposition (4.15), we thus can assume that (/v>)„Ê1 converges to a

364

Convex geometry and the phase diagram

limit v. (Otherwise we can pass to a subsequence.) Clearly, v(Ak) = /J.{Ak) for all k ^ 1. Hence v = /x and thereby (<ï>, ^) = lim ($", n^„). o n-*co

The next and final corollary completes our proof of the ubiquity of unique ergodic Gibbs measures. Since {^.j,: cD e W&} is a subset of ex^@(Q, 3F\ this corollary provides a remarkable refinement of Theorem (14.12). (16.40) Corollary. / / (E,$) is standard Bor el then {[i^. <S) e W@} is dense in

Proof. By Corollary (16.38), we only need to show that &@(@@) is dense in ^e(Q, #"). Let n e ^©(Q, 3F) be given. Pick any cofinal sequence (A„)„Ä1 in y. For each n ^ 1, we obtain from Proposition (16.7) and Theorems (16.13) and (16.14) a potential -O*,¥>|gpl0/n|AJ for all *F e ^ 0 . In view of Lemma (16.10), this inequality implies that sup | v „ ( ^ ) - ^ ) | ^ l / n Ae

^„

for all n 2: 1. Hence v„ -> fj. as n -> oo.

a

Part IV Phase transitions in reflection positive models

The phenomenon of non-uniqueness of Gibbs measures has already been studied in several places throughout this book: In Chapter 6 we examined three specific models which exhibit a breaking of one or several discrete symmetries. In Chapters 11 and 12, we encountered several Markovian examples of phase transition. (In some of these examples, the phase transition was not, or only to some extent, accompanied by a breaking of symmetries.) The breakdown of a continuous symmetry group was established in some Gaussian models of Chapter 13. Finally, in Chapter 16 we derived a (non-constructive) existence result concerning phase transitions with prescribed order parameters. In this part we shall provide quite a number of further examples of phase transition. Specifically, we shall deal with shift-invariant Gibbs specifications on a lattice Zd. In Chapters 18 and 19 we shall assume that d 2: 2. The spins will be allowed to take values in an arbitrary space, but their interaction will be assumed to have at most the range yfd. Our analysis of such systems will take advantage of some concepts of percolation theory, i.e., we shall be concerned with the existence and uniqueness of infinite clusters in certain random subgraphs of the lattice. This geometrical analysis will culminate in some general theorems on the existence of phase transitions. On the one hand, we shall provide stability conditions under which a degeneracy of ground states gives rise to a phase transition at low temperatures. On the other hand, we shall show how a conflict of energy and entropy can lead to a phase transition at a distinguished value of temperature. These results can be easily applied to various specific models, as we shall indeed show in various examples. The final chapter, 20, will be devoted to a discussion of a particular highlight in the theory of phase transitions: the proof of spontaneous magnetization in systems with continuous symmetries. Specifically, we shall consider systems of Revalued spins whose interaction is invariant under the group of all rotations of UN. Under some hypotheses on the interaction, the rotational symmetry of these systems can be shown to break down at low temperatures. The proof uses some techniques of harmonic analysis. A common tool for both the geometric methods of Chapters 18 and 19 and the Fourier analytic techniques of Chapter 20 is provided by a certain inequality which refines the Cauchy-Schwarz inequality and is usually referred to as the chessboard estimate. It involves the correlations of homogeneous

366

Part IV

random fields on discrete tori and relies on a definiteness condition which is called reflection positivity. This condition will be studied in Chapter 17. The chapters of this part are linked together according to the scheme ,48-19 17N 20 To read these chapters it is sufficient to know Chapters 1, 2, 4 and 5 and the main results of Chapters 7 and 14. (In Subsection 19.3.2 we shall also use some facts from Chapters 15 and 16, but one can do without them if one is willing to accept less complete results.) A deeper understanding also requires some familiarity with the essential ideas of Section 6.2 (for Chapters 18 and 19) and Sections 9.2 and 13.3 (for Chapter 20). The remaining sections of Parts I to III are not related to the contents of this part.

Chapter 17 Reflection positivity

The objective of this chapter is to establish a certain key inequality which underlies the results of Chapters 18 to 20. This inequality is a refinement of the well-known Cauchy-Schwarz inequality. It applies to random fields /i which have a torus A for their parameter set, and exhibit the following two properties. (i) fi is preserved by the rotations of the torus A. This property will be called A-periodicity. (ii) ß is reflection positive. This property means that a certain bilinear form which is defined in terms of JX and a reflection of A is nonnegative definite. Of course, the latter property will provide us with a Cauchy-Schwarz inequality. Repeated use of this inequality, combined with the application of suitable rotations of A, will give us the desired key inequality. (See Theorem (17.11).) The main idea is contained in a little combinatorial lemma which is called the chessboard estimate. All this will be the contents of Section 17.1. Section 17.2 will deal with sufficient conditions for property (ii) to hold. More precisely, we shall look at Gibbs distributions on a cube A in Zd with respect to shift-invariant potentials. To guarantee property (i) we shall impose periodic boundary conditions. We then shall distinguish between reflections in planes through lattice sites and reflections in planes between lattice sites. In each case we shall find a class of potentials which are reflection positive in that the associated periodic Gibbs distributions are reflection positive.

17.1

The chessboard estimate

Let d, N ^ 1 be any two integers. We consider the d-dimensional cube (17.1)

A=

A(N)^]-N,NYnZd

= {i = {i1,...,id)eZd:

-N

< ik ^ N for 1 ^ k ^ d}.

A is thought of as a torus. Accordingly, we equip A with the addition modulo A which comes from the identification of A with the factor group Zd/2NZd. More precisely, if i, j e A then the notation (17.2)

a = i+j

mod A

means that a is the unique element of A which satisfies ak = ik + jk mod 2N for

368

Reflection positivity

all 1 ^ k ^ d. Similarly, for each A c A and i e A w e write A = A + i mod A when A = {a e A: a = i + j mod A for some j e A}. Next we let (£, ê) be an arbitrary measurable space. We shall be concerned with finite measures on the product space (£ A , iA) which exhibit a certain behaviour with regard to two particular classes of transformations of £ A . The first of these classes consists of the rotations (or periodic shifts) 0A: EA -* EA which are defined by (17.3)

(9iAco)j = cok

(co e EA, i, j e A, k = j — i mod A).

A finite measure \i e Ji{EA,SA) will be called A-periodic if 9A{n) = n for all i e A. The second class of transformations consists of generalized reflections which are defined as follows. Take any 1 ^ k ^ d and look at the reflection rk: A -* A in the plane {x = (x l 5 ..., xd) e Rd: xk = 1/2}. rk is given by fl-i, [i( where i = (il,...,id) (17.4)

if ^ = fc, otherwise,

e A. Clearly, rk is an involution which maps the set A+,k={ieA:l^ikSN}

onto the set A_>k = A\A + i k = {i'eA: - N < ik ^ 0}, and vice versa. rk induces a measurable involution of (£ A , SA) which will be denoted by the same letter: (17.5)

(rkco)j = (orkj

(co e EA,j e A).

Combining this mapping with an arbitrary measurable involution xk: E -> E we arrive at the generalized reflection (17.6)

rt:
(co e £ A )

which is a measurable involution of £ A . Using fk we shall now introduce the notion of reflection positivity. Let j / + j k be the space of all bounded measurable functions on £ A which depend only on the coordinates in A + k. For each fe J^+ik we consider the reflected function / * = fofk. Clearly, / * is a function of the coordinates inA_k. (17.7) Definition. A finite measure \i on (£ A , êA) is said to be rk-positive if M / / * ) ^ 0 for a l l / £ . < , * . Examples of rk-positive measures will be provided in Section 17.2. Here we ask for the use of this notion. By definition, a given measure /J. on (£ A , SA) is rk-positive if and only if the bilinear form

if,g)^Kfg*)

The chessboard estimate

369

on s/+ik is nonnegative definite. Of course, this property implies the CauchySchwarz inequality (17.8) \n(fg*)\2 £ n(ff*M99*) (f,ges*+,k). When combined with the property of A-periodicity, the last inequality will give us an improved inequality which is a fundamental tool for the derivation of the results of Part IV. At the heart of this improvement is the combinatorial lemma below which is known as the chessboard estimate. We consider the set A(N) = {l,...,2N}

x{0,l}

= {(n,e): 1 g n ^ 2iV,8 = 0 or 1}. The change of the second coordinate is an involution of A(N) which is denoted by a -• à. Each 1 ^ n ^ 2iV is identified with (n, 0) G A(N). Accordingly we write n instead of (n, 1). (17.9) Lemma. Let F: A(N)2N ->• [0, oo[ be any map which satisfies the following two conditions. (i) For all a l 5 . . . , a2N e A(N), F(au...,a2N)

=

F(a2,...,a2N,a1).

(ii) For all at, b( e A(N),

^F(a1,...,aN,äN,...,ä1)F{bl,...,bN,bN,...,bl). Then the inequality F(au...,a2N)

^ f]

n*t,äe,...,a(,ä()^

holds for all cii,..., a2N e A(N). In proving this lemma it will be helpful to visualize an element (al,..., a2N) e A(N)2N as a configuration of white and black chips on a 2N by 2N chessboard as follows. For each 1 ^ k ^ 2N, we put a chip at position nk in the /c'th column if ak — (nk,sk). We take a white or black chip according to whether sk = 0 or 1. In this picture, hypothesis (i) means that F remains invariant under cyclic horizontal shifts of all chips. In hypothesis (ii), a configuration of chips is compared with the two configurations that result from it by means of the following procedure: remove all chips on the right (resp. left) half of the board, take a colour reversed copy of the chips on the other half, and put a reflection image of this copy onto the empty half. Condition (ii) then implies that if F is positive or attains its maximum at the starting configuration then the same holds for each of the two resulting configurations. It is this

370

Reflection positivity

consequence of (ii) that will be exploited in the proof. In the conclusion of the lemma, the value of F at any starting configuration (a1,..., a2N) is compared with the values of F at purely horizontal configurations which fill one row of the board with alternating black and white chips. (In view of (i), it is no matter whether the leftmost chip is black or white.) Each such alternating row configuration appears with a multiplicity which is equal to the number of chips of(a1,...,a2N) in that row. Proof of (17.9). We introduce the set M = {aeA(N):F(a,ä,...,a,ä)

> 0}

and prove the claimed inequality in the two cases (a1,.. .,a2N) e A(N)2N\M2N and (a1,...,a2N) e M2N separately. 1) In the first case we need to prove that F(a1,..., a2N) = 0 when a( $ M for some (. So let ( a t , . . . , a2N) e A(N)2N be such that F(a1,..., a2N) > 0, and pick any a = (n,s) e {a1,...,a2N}. We will show that a e M. In view of hypothesis (i) we can assume that a = aN. For each 1 5Ï ( ^ N we look at the set RaJ={(b1,...,b2N)eA(N)2N:F(b1,...,b2N)>0, "N-I+\

=

VN-IT+2

=

"N-e+z = "• = 0jv+«f-i

=

"N+I

= a).

Thus each element of Ra e contains a strip of alternating black and white chips at the 2£ middle positions of the n'th row, the leftmost chip having colour e. Since ae M when Ra N # 0, we will show by induction on ( that each Ral is non-empty. First of all, we conclude from (ii) that (a l s . ..,aN,äN,... . â j e Ra,i- Hence Ral ^ 0. Next we assume that RaJ # 0 and set k = N A 2(. Then Rak # 0, as can be seen by the following procedure. We take any configuration of chips from Ra e and shift it to the left in such a way that the positions N — k + 1,..., N of the n'th row are occupied by a horizontal strip of chips of alternating colour. Using (ii), we obtain a new configuration which contains a horizontal strip of chips of alternating colour at the 2k middle positions of the n'th row and still belongs to the domain of positivity of F. Hence Rak =£ 0, and the induction step is complete. 2) For ax,..., a2N e M we define / 2JV

G(a1,..., a2N) = F(Ö! ,..., a2N) / \\ F(at, 5 „ . . . , ae, äf)1/2N. We need to show that G ^ 1. Since G{a, ä,..., a, a) = 1 for all ae M, this will be proved once we have shown that G attains its maximum, say c, at some vector of the form (a,ä,...,a,a). To this end we observe that G satisfies the same conditions (i) and (ii) as F, provided A(N) is replaced by M. (Note that ä e M when ae M, because of (i).) Therefore, if

The chessboard estimate

371

Re = {(a l 5 ...,a 2 N ) e M2N: G(au.. .,a2N) = c, a

N-s+i

=

äN-s+2 = a?v-/+3 = ' " = ctN+e)

then the same procedure as in Step 1) shows that Rx # 0 and RNA2lf ^0 when ^ # 0. Thus i?N # 0 and thereby c = 1. • Suppose now we are given d commuting measurable involutions x1, ..., xd of E. For each i = (i l 5 ..., id) e A we consider the iterated mapping (17.10)

T' = r\l o ... o x1/.

Clearly, xl = xJ when ik = jkmod2 defined by (17.6).

for all 1 ^ fc ^ d. For each k we let rk be

(17.11) Theorem. Let \xbe a finite measure on (EA, SA). Suppose \i is A-periodic and rk-positive for all 1 ^ k S à. Then for each choice (f)ieA of bounded or nonnegative measurable functions f on E, the inequality

\i'eA

jeA

holds. Just as Lemma (17.9), the preceding inequality is referred to as the chessboard estimate, this name being also suggested by the structure of the mappings xl (i e A) in dimension d = 2. Occasionally, the above inequality is also called generalized Cauchy-Schwarz inequality. In fact, the standard CauchySchwarz inequality for integrals corresponds to the particular case when d = 1, JV = l,x1 = i d , and /i is the image of any finite measure on (E, S) under the mapping x -> (x, x). Proof of (17.11). We can assume without loss that all / ; 's are bounded. In this case we proceed by induction on the dimension d. 1) Let d = 1. We define a family (fa)aeA(N) of functions on £ by putting f(n,o) = /n-jv an d/(„,i) = fn-N ° T i when 1 ^ « ^ 2N. We also define a function F:A(N)2N -• [0, oo [ by 2N

F(au...,a2N)

A* ( I l fat ° °e-N i=\

(By the very definition of fj-positivity, the absolute value sign can be dropped when (a 1,..., a2N) = (n, n,..., n, n), for example.) The claimed inequality then takes the form flF(n,n,...,n,n^2N.

F(l,...,2N)g «=i

We thus need only check that F satisfies the hypotheses (i) and (ii) of Lemma

372

Reflection positivity

(17.9). Condition (i) is an immediate consequence of the A-periodicity of JX. Hypothesis (ii) follows from the Cauchy-Schwarz inequality (17.8). Indeed, with the help of the functions f = fb„0Oi---fbl0 we can write

«AT»

F(a1,...,aN,bN,...,b1)2

g=färt0°i---fä10
J*+,i

= n(fg*)2

£ n(ff*)n(gg*) = F{bt ,...,bN,bN,...,

b1)F(a1,

...,aN,äN,...,ä1).

2) Now let d > 1. We write A = A 0 x A 1 , where A 0 = ] — iV, iV] and A1 = l - N . N ] " - 1 . Let (£0,
AM n.Wi) = H n ( n Aio.^0^.^)) \ieA

J /

VieA, VOEAO

^ n À n n/(w 1 »^ w,, °^ 1 » h eAi

Vi eA, ioeA0

/ / \l/|Ail

• /

Interchanging the rôles of A 0 and A1 and again applying the induction hypothesis we obtain the result, a Theorem (17.11) applies to functions fi o al of single spins a{. However, in Chapters 18 and 19 we shall need a similar estimate for functions of the joint behaviour of several spins. (This should not seem surprising because each interaction potential consists of such functions.) To obtain such an estimate we shall subject Theorem (17.11) to a coarse-graining. It is for this coarsegraining that the involutions xk are indispensable. Let (17.12)

C ^ A(l) = {0,1}"

be the unit cube in Zd. For each i e A we consider the elementary cube (17.13)

C(i) = C + i mod A

in the torus A. (We suppress the A-dependence of C(i) in our notation.) For example, if i = (JV,...,N) then C(i) consists of the 2d corner sites of A. For the sake of brevity we identify £ C(,) with Ec in the natural way. That is, we shall suppress the transformation 6f which establishes this identification. Accordingly, the projection
The chessboard estimate

373

reflection of co#, the first stage being a reflection of A and the second one a reflection of C. From this it is obvious that the rôle of the involutions TJL, ..., xd will be taken over by the reflections of Ec. So we shall now use the letters rl,..., rd to denote the reflections of Ec. (This corresponds to putting N = 1 in (17.5).) Of course, these reflections are commuting involutions of Ec. In accordance with (17.10) we also introduce the iterated reflections (17.14)

r' = r i ' o . . . o r j -

(i = (it,.. .,id) e Zd)

of£c Suppose next we are given a finite measure ß on (EA, SA). We then let ß* denote the image of ß under the mapping co -• w* from EA to (EC)A. We seek a condition on ß which implies that /i # is r^-positive, provided the transformation fk of (£ C ) A is defined by choosing xk = rk in (17.6). To this end we observe that the set

Lk = U

c

( 0n U

c

( 0 = {» e A: '* = 1 o r -N + 1}

consists of a pair of (hyper-)planes in A. If A is thought of as being bent to a torus in Ud+l, this pair of planes is the intersection of A and a hyperplane in Ud+l. The ffc-positivity of \i* should be related to the behaviour of \i under reflection in this hyperplane. So we let fk denote this reflection. That is, we set

(«.is)

VA

= \2-''m°iW

iU

= k:

(if otherwise when i = (i l 5 ..., id) e A and 1 S k S d, and we use the same symbol rk to denote the associated transformation of EA. We emphasize that fk is a reflection in a plane that meets the sites of A, whereas rk was a reflection in a plane between the sites of A. Thus fk preserves the position of some spins, namely all those in Lk, whilst rk does not. In analogy to (17.7) we will say that a finite measure ß on (£A, U. which only depend on the coordinates in Â +jk 4 {i e A: 1 S h S N or ik = -N + 1}. (17.16) Lemma. Suppose ß is fk-positive for some 1 ^ k ^ d, and let fk: (EC)A -> (£C)A be ^it;e« by (17.6) wiffr rfc = rk and Ec instead of E. Then ß* is rk-positive. Proof. Let / be any bounded measurable function on (£ C ) A which only depends on the coordinates in A + k. The function g = / o (oC{i))ieA then only depends on the coordinates in Â+>fc, and gofk = fo (rkofkC(i))ieA Thus ß*(ff*)

= fo (rkoC(rki))ieA

= ß(ggo fk) ^ 0. n

= / * o (oc (/) ), 6A .

374

Reflection positivity

(17.17) Corollary. Let p J ( £ V A ) be finite, and suppose JX is A-periodic and fk-positive for all 1 ^ k ^ d. Then for each choice (f)i eA of bounded or nonnegative measurable functions f on Ec we have .1/1M

0

r

a

AM n ^ ^ ) ^ n ^( u fj ° '° c(i) je A

\ieA

Proof Since n* is evidently A-periodic, this follows immediately from (17.11) and (17.16). •

17.2

Gibbs distributions with periodic boundary condition

In this section we shall provide examples of finite measures on (£ A , SA) which are both A-periodic and reflection positive. As we shall soon see, there is a trivial example of such a measure, namely the product measure kA of any finite k e Jiï(E, S). We can thus hope to find nontrivial examples by looking at measures with Gibbsian densities relative to kA. In order to ensure Aperiodicity we need to confine ourselves to shift-invariant potentials, and we also need to impose periodic boundary conditions. So we shall consider Gibbs distributions with periodic boundary condition, as were introduced in Example (4.20) (2). We shall first deal with a particular class of shift-invariant short-range potentials which will be called C-potentials. The associated Gibbs distributions with periodic boundary condition will be shown to be rk-positive. This fact will be essential for the results of the next two chapters, but will not be needed in Chapter 20. Then we shall establish the rk-positivity of the Gibbs distributions with periodic boundary condition relative to certain Heisenberg potentials. This result will not be used until Chapter 20. Throughout this section we let k e Ji(E, ê) be a fixed a priori measure. For simplicity we will assume that k is finite. For each potential <1> and A c A we shall freely think of Q>A as a function on EA, or as a function on EA which only depends on the coordinates in A. This is possible because of assumption (2.2)(i). Let C be given by (17.12). (17.18) Definition. Let us agree to call a potential <& with parameter set Zd and state space (E, S) a C-potential, and to write $ e <&@, if <& meets the conditions (i) to (iv) below. (i) <J>^ = 0 unless A = C + i for some i e Zd. (ii) cDc+. = (Dc o 0_j for all i e Zd. (iii) <1>C = <&c o rl for all i e C. (iv) Either ||O c || < oo, or there exists a sequence (K^^^ in ê with Ke\E as £ -> oo such that

Gibbs distributions with periodic boundary condition

375

for all t > 1

sup |Oc(a))| < oo WEKC

and inf Oc(co) ">• °°

as ( -> oo.

coiKc

(17.19) Comments. (1) For each <ï> e ^ e , assumption (iv) implies that all /J*'S are bounded. Thus $ is A-admissible. As we shall show in (18.12), condition (iv) also ensures that ^0(<ï>) ^ 0 (provided (£, ^) is standard Borel). (2) Up to equivalence, %>@ contains all next-nearest neighbour potentials which are invariant under reflections and translations and satisfy a version of assumption (iv). Suppose, for example, that d = 2 and can be written in the form "(p^Oi) \iA = {i}, (p2(<Ti,0)) if A = {i,j} and \i -j\ = ^ 2 , 0 otherwise, where the functions (Pi,cp2, 2 and q>z are symmetric. Then it is easily checked that the potential *F which is defined by (i), (ii), and T

c = T Z
Z

{i,j}<=C

Z

*»3(o"i»o>)

{W}<=Ç_

satisfies (iii) and is equivalent to <&. o Suppose now we are given a C-potential <ï>. We then let °y® denote the projection image on EA of the Gibbs distribution for $ in A with periodic boundary condition, and we write °p* for its density with respect to XK. According to Example (4.20) (2), °p% is given by (17.20)

> * = (°Z*)- 1 exp

I

ie A

$c(o

Here °Z* > 0 is a normalization constant, and C(i') is defined by (17.13). Of course, if i is such that C(i) ^ C + i then e%and

1 <; k ^ d, °y% is fk-positive.

Proof. The exponential in (17.20) can be written in the form h hofk with exp

c(i)<=A +ik

and h only depends on the coordinates in Â+ k. We thus only need to show

376

Reflection positivity

that 1 A is ^-positive, which is quite easy. Indeed, let L = Lk be defined as in the paragraph above (17.15), and let A = Ä +>k \L. Also, let / be any bounded measurable function which only depends on the coordinates in À+>k. Since / can be written as / = g{aL, erA), we conclude from Fubini's theorem that * A ( / M ) = I ^L(àœ)k\g(œ, The proof is thus complete,

-))2 ^ 0.

o

In the following, we intend to provide examples of periodic Gibbs distributions which are ^-positive, i.e., ^-positive when xk is the identity. We confine ourselves to the case when £ is a rotationally invariant subset of some Euclidean space W, and we only look at shift-invariant Heisenberg potentials. By definition, these are all pair potentials of the form (17.22) O A = J

otherwise.

0

Here J : TLd —> E is any even function with 7(0) = 0 and

(17.23) J2 |y(Â:) l < °°' keZd

and the dot denotes the usual inner product. For such a the projection °y£ onto EA of the Gibbs distribution in A with periodic boundary condition has the density (17.24) > * = C Z * ) - 1 exp [ I J2

y

AO"-;) * • o)]

i,./eA

relative to kA. In the above, (17.25) JA(k) = J ] 7(fc +

2Nl

(k

)

e z
).

and A. is assumed to be such that °Z* 4 /\ A (dû>) e x p U J ]

Mi-ntOftoj]

1,7'eA

is finite. We seek conditions on J which imply that °yf is r*-positive. The following lemma is a first step towards answering this question. (17.26) Lemma. Let fibe a finite measure on (£ A ,
Gibbs distributions with periodic boundary condition

377

(i) m is a finite measure on some measurable space (W, W); (ii) h and hw, w e W, are measurable complex functions on EA which only depend on the coordinates in A + k, hw(oS) is measurable jointly in w and <x>, and sup \hw\ is dominated by a measurable function; and wsW

(Hi) * indicates reflection by rk combined with complex conjugation. Then fi is rk-positive. Proof. We can assume that X is finite. By (ii), there exists a measurable function cp : EA -> [0, oo [ which only depends on the coordinates in A+Ji and satisfies \h\ ^ cp and \hw\ ^ q> for all w e W. For a given function fe j / + k and any number c > Owe set fc = fl(vic}. Expanding the exponential density of fi and using Fubini's theorem we then see that Kfef*)

= Z

7TJm(dwi)---jm(dw
where gCWl...Wl = fehhWi ...hW/. Clearly, the functions gCWl...We can be regarded as measurable functions on £ A+ k. Hence ÀA(a

a*

) = \XA+-k(a

)\2 > 0

Thus n(fcf*) ^ 0, and the lemma follows letting c tend to infinity, D Let K = {z e C: \z\ = 1} denote the unit circle in the complex plane. (17.27) Definition. Let us agree to call an even function J on Zd nonnegative definite relative to rY if there exists a finite measure a on ] — 1,1[ x X d _ 1 which represents J, in that J(i) = j a(dz l 5 ...,dz d )z['- l z^...z 1 / for all ie N x Z d_1 . For any other coordinate direction 1 < k S d the nonnegative definiteness relative to rk is defined analogously by an interchange of coordinates 1 and k. (17.28) Comments. (1) An even and absolutely summable function J on Zd is nonnegative definite relative to r1 if and only if Z

i.JEN x Z d ~ '

Z

J VO'i

+ h - !' »2 - J2> • • •. id - h) ^ °

for each choice of complex numbers zt such that {ie N x Z d _ 1 : z; ^ 0} is finite. This result which combines the Herglotz lemma (13.A9) and the solution of the Hamburger moment problem is a particular case of a theorem of Berg and Maserick (1984). (See also p. 96 of Berg, Christensen and Ressel (1984).) To apply this theorem it is sufficient to note that the operation (i,j) -»(fx + j \ - 1,i2 +j2,...,id

+ jd)

378

Reflection positivity

turns N x Z 1 " into a commutative semigroup with identity (1,0,...,0) and involution i -> (i1, — i2,..., — id). The mappings are nothing but the associated bounded semicharacters. (Since £ i j è l 1.7(^,0,...,0)| < oo, the semicharacters with |z x | = 1 do not contribute to J.) (2) Suppose J and J' are nonnegative definite for r1. Then so is the product J J'. For let a and a' be the measures that represent J resp. J'. Then J J' is represented by the image of a x a' under the mapping (z1,...

,zd,zl,...

,zd) -^\z^zx,..

,,zdzd).

(3) Any J which is nonnegative definite for r1 is necessarily ferromagnetic to some extent, in that J(i 1 ,0,...,0) ^ 0 whenever i t is odd. On the other hand, if the representing measure a is supported on ] —1,0[ x Kd~^ then J(i1,0,..., 0) < 0 when il ^ 0 is even. Also, if J is nonnegative definite for r1 then so is the function i -> — ( — l)' 1 J(i). Its representing measure is a reflection of a. o (17.29) Theorem. Suppose 3> is a Heisenberg potential of the form (17.22) with an even and absolutely summable function J which is nonnegative definite relative to some rk. Then °y* is rk-positive. Proof. We can assume without loss that X is supported on a bounded set. (This is because the general case then follows by an obvious limiting argument.) For notational convenience we will also assume that d = 2 and k = 1. (In all other cases the proof is similar.) For each L ^ 1 and (u, v) e Z 2 we put JA>L(u,!;) = (2Lr 1 X

Z

J(u + 2Nk,v + 2N(S

-1')\

and we let \xKL denote the finite measure which is obtained from the measure °Z* °y* when JA is replaced by JA L. Using hypothesis (17.23) we see that lim J A , L (0 = JA(i) for all i e Zd. Hence lim fiA L = °ZA 0y* in variational L—•oo

L~*co

distance. Consequently, it is sufficient to show that each fiA L is r r positive. To this end we fix any L ^ 1. In order to apply Lemma (17.26) we write the AA-density of [iA L in the form exp(h + h* + H), where h

= \

Z

Jh,L{i-j)Oi -CTj

'.J6A + ,i

and H

=

Z

4,L(*-./)
ieA.,i,J£At,i

We need to find a measure space (W,W,m) and complex measurable func-

Gibbs distributions with periodic boundary condition

379

tions hw (w e W) which only depend on the spins in A + j l and are such that H — j m(dw)hwh*. By definition of the inner product, H is the sum of n terms which correspond to the n coordinates of the a^s. We can look at each of these terms separately. Equivalently, we can assume that n = 1. In this case, H is equal to the double sum (2*>)

ZJ

2J

keZ

u,v,f

(J

(U-N,V)G(N

+

1-U',V')

u'.v'.r x J(u + u' - 1 + 2Nk, v-v' + 2N{t - /')), where the inner sum is taken over the range 1 :g u, u' ^ JV, —N
V Lu

u,v,e

n

0

n

(u-N,v)0(N

x2N)~l rxu+u'-2

+ l-u',v')Lx

I

z

x2N-u-u'

v+2NS -v'-

2NC

L

z~v-2Nt

zv'

+ 2Nr~\

The last expression has the desired form. Indeed, let W = {0,1} x ] — 1,1[ x K and W be the Borel er-algebra on W. Define a (finite) measure m on (W, W) by m(0,dx,dz) = m(l,dx,dz) = (2L) _1 (l - x 2N )~ 1 a(dx,dz), and consider the measurable functions "(0,x,z) — Z J

cr

(Af + l-u,i;)- x

Z

and I, _ V n xN-u "(l,x,z) — Z J ^JV+l-u,!))"*' u,v,(

Z

v + 2N( )

{x, z) e ] — 1,1[ x K. The functions hw (w e W) satisfy condition (ii) of Lemma (17.26). Moreover, since (u — N,v) = /^(iV + 1 — M,U) we have 7/ = jm(dw)/iM,/i*. The theorem thus follows from Lemma (17.26). D (17.30) Example. Ferromagnetic nearest-neighbour potentials. Suppose J is an even real function on Zd such that J(i) = 0 if ] i\ > 1 and J(i) ^ 0 if | i\ = 1, and

380

Reflection positivity

let $ be given by (17.22). Then °y* is ^-positive for all 1 :g k ^ d. Indeed, J is nonnegative definite for r1 with representing measure a = J(l,0,...,0)<5 0 x v" -1 . Here ô0 is Dirac measure at 0 and v is normalized Haar measure on K. Thus Theorem (17.29) shows that °y* is r1-positive. For k =£ 1 the rk-positivity follows by symmetry. We note that the nonnegativity of J is not only necessary for J to be nonnegative definite for rk (cf. Comment (17.28)(3)) but also necessary for °y* to be rk-positive. To see this we choose X = ôx + ô_1 for the a priori measure, and for fixed 1 ^ k ^ d we let u denote the /c'th unit vector in Zd. We put h

=

Z

WJJAÜ - J)

{i,j}<=A+,k

and

/ = e-\THl{„(=„u for all ie

A+,k}"

Then 0

ZZ°y%(ff*)=

Z

exp[(2AT)^1J(U)x3;]

x,y=±\

=

4sinh\_(2N)d-1J(u)l

Thus, if J(w) < 0 then 0y£(ff*)

< 0.

o

(17.31) Example. Next-nearest neighbour potentials. Let d^.2, and suppose J is given by 'a |i| = l, J(i) = -J fc if \i\2 = 2, 0 |i| 2 > 2. If a ^ 2(d — l)|b| then °y* is rk-positive for all k. For we can take cc(dz1,...,dzd)

= ^(dzj)

+ fc £ (zt + zt) v(dz2)...v(dz„), k= 2

(50 and v being as in the preceding example. Conversely, if a < 2(d — l)\b\ then °y* fails to be rk-positive. This can be seen in a similar way as in Example (17.30) by looking at the same function / and also at J

=

e

<7

iil{
In this connection it is interesting to note that no restriction on a and b is required for "y® to be rk-positive. This follows from Theorem (17.21). o (17.32) Example. Long range potentials. Let d = 1 or 2, and suppose $ has the form (17.22) with

Gibbs distributions with periodic boundary condition

j(i) = ß\i\-a

for

381

\i\ ^ 1,

where ß ^ 0 and a > d. Then °y* is rk-positive for 1 5Ïfc:g d To show this we only need to look at the case k = 1. By Theorem (17.29) it is sufficient to show that J admits a representing measure a on ] — 1,1[ x K d_1 , and to this end we may assume that ß = 1. 1) First we consider the case d = 1. From the formula 00

r(a) = J e - V " 1 ds o for the gamma function we obtain, by a change of variables, 00

k~a = r ( a ) - 1 J e ^ V " 1 ds o for each k ^ 1. Thus the measure a on ] — 1,1[ can taken to be the image of the gamma distribution r(a)" 1 e" s s a _ 1 ds on ]0, co[ under the mapping s -> e~s. 2) Now let d = 2. Because of Comment (17.28)(2) it is sufficient to look for a representing measure a in the case when 1 < a < 2. In this case the integral 00

c(a)= J dtt1-"/^ o

+ t2)

is finite, and the substitution s = (1 + t 2 (l + tf2/k2))112 yields the formula oo

(k2 + / T a / 2 - k1-" J p(ds)sk/(/2 + s2k2), i

where k e N, /feZ, and p(ds) = cia)'1^2 - l)" a/2 ds. From the case d = 1 above we know that the function (k, i) -» k1~a admits a representing measure o n ] — l , l [ x K (which is supported on [0,1[ x {1}). In view of Comment (17.28)(2) we thus only need to find a measure kernel s -» ns such that ns represents the function

{kJ)^sW2

+ s2k2),

s ^ 1. If k is held fixed and / is thought of as a real variable, this function is just the Cauchy density for the parameter sk (up to a positive factor). The Fourier transform of the Cauchy density is known to be exp [ —s/c|u|]. Thus OO

sk/(t2 + s2k2) = c J due^e-* 1 " 1 — oo

for some constant c > 0, and ns can be taken to be the image of the finite measure ce~ s|u| du under the mapping u -> (e"s,u|, e'u). (In the above, the symbol i stands for the imaginary unit.) o

Chapter 18 Low energy oceans and discrete symmetry breaking

This chapter is devoted to the proof of the existence of phase transitions for certain short-range potentials which exhibit a specific kind of ground state degeneracy. Specifically, we shall deal with the case when the parameter set S is a simple cubic lattice Zd of any dimension d^2, and we shall focus on C-potentials in the sense of Definition (17.18). A C-potential <ï> will be said to exhibit a ground state degeneracy of order N > 1 if the function <1>C attains its minimum at N pairwise distinct configurations on C. (Recall that C stands for the elementary cell of Zd.) These N configurations will be called the local ground states of <ï>. Assuming that this occurs, we shall also stipulate that the local ground states are incompatible in a certain sense but are related to each other by suitable symmetries of <ï>. (The first condition will imply that the relating ^-symmetries are necessarily discrete.) To get a feeling for the significance of these conditions it is worthwhile to look at a familiar example: the Ising potential at zero external field (cf. (6.8)). This potential has two local ground states, namely the constant configurations on C which are identically + 1 resp. — 1. These local ground states can be distinguished from each other at every single site but are interrelated via a symmetry, the spin-flip transformation. As should be clear from Section 6.2, it are precisely these features of the Ising potential which give rise to a breaking of the spin-flip symmetry at low temperatures in two (or more) dimensions. It is thus natural to ask whether these features are the driving force behind a more general mechanism of symmetry breaking which lies behind the phase transitions in the Ising model. The answer is: yes, there is such a mechanism. In order to describe it we start again from the Peierls argument which has been used in the Ising case. As we know from Chapter 6, the heart of the Peierls argument is the concept of a contour. The Peierls contours indicate the boundaries of the connected regions in which the local spin patterns coincide with a local ground state and thus require a minimal energy. Keeping this in mind, it is natural to look at the random subgraph of Zd which consists of all those spins whose -energy relative to the adjacent spins is minimal (or almost minimal, at least). For reasons similar to those which ensure that a Peierls contour is typically short, the "low energy graph" for $ should contain an infinite cluster which resembles an ocean with small (and, in particular, finite) islands, at least at low temperatures. This conjecture falls into the realm of what is known as percolation theory: the study of infinite clusters in random subgraphs of

Percolation of spin patterns

383

regular lattices. Using some techniques of percolation theory, and also using the chessboard estimate as a counterpart to the Peierls contour estimate, it will be possible to confirm the above conjecture as follows. If the spin distribution is governed by a limit of Gibbs distributions °y^ with periodic boundary conditions, and if ß is large enough, then each two-dimensional layer in Zd contains a "low energy ocean" as described above. Now the point is this: If <5 exhibits a ground state degeneracy as described at the beginning then the low energy ocean in any fixed layer is bound to show a pattern which corresponds to one of the N distinct local ground states. This pattern can be observed at infinity, and its N possible values are symmetry-related. Consequently, there exist N mutually disjoint but symmetry-related tail events of necessarily equal and thus positive probabilities. In view of the results of Chapter 7, this can only occur when the relating symmetries are broken. To summarize, there exists a mechanism of symmetry breaking which consists of two parts: the existence of a low energy ocean when ß is large, and the ground state degeneracy which implies that the pattern of the ocean is a fluctuating tail variable. This mechanism will give us the following general theorem. If <5 is a C-potential which exhibits a ground state degeneracy as above then, for any sufficiently large ß, the potential ß® shows a breaking of all symmetries which relate the local ground states to each other. This theorem will be stated and proved in Section 18.2. The first section will be devoted to the existence of low energy oceans. In Section 18.3 we shall present various classical examples of phase transition that fit into the theory just outlined.

18.1

Percolation of spin patterns

Throughout this chapter we put S = Zd, the dimension d being at least two. We also let (E, S) be an arbitrary state space, and we assume that (£, S) is endowed with a finite a priori measure L (The case of an infinite a priori measure can be reduced to the finite case using the remarks in (2.18).) As in (17.12) we let C denote the unit cube in Zd, and we let
m 0 4 2-inf d>c = sup {c e U: Àc(®c ^ c) = 0}

denote the essential infimum of <S)C relative to Xe. (In concrete cases,
384

Low energy oceans and discrete symmetry breaking

a local ground state of O if Oc(co) = m®. More generally, we shall consider the sets G £ (0) of all local e-ground states for O. These are defined by (18.2)

G£(
e ;> 0. By definition of the essential infimum, 2C(G£(G>)) > 0 for all e > 0. Of course, if ) coincides with the set of G0(d>) of all local ground states of d>. Needless to say, the significance of the sets G £ (0) comes from the fact that the Gibbs distributions for O give maximal weight to all configurations that minimize the energy. Next, we fix an arbitrary measurable set G e Sc of configurations on C. An important special case is G = G £ (0), but other choices of G will be considered as well. Generally speaking, G should be regarded as the set of all configurations on C that show some pattern which is expected to be typical for some Gibbs measure. G will be called r-symmetric if r'G = G for all i e C (or, equivalently, all i e S). Here r': Ec -* Ec is the iterated reflection which is defined by (17.14) and (17.5). In view of hypothesis (17.18)(iii), the sets G£(
V(G, co) = {i e S: (6^œ)c

e r'G}

be the set of all elementary cubes C + iinS for which coc+i exhibits the pattern that is described by the appropriate reflection of G. To understand the role of the reflections it might be worthwhile to look at an example. Suppose that d = 2, E = { — 1,1}, and G = {£}, where £) consists of a l l i e S which are such that a>j = ( — iyi+J2 for all j = (ji,j2) e C + i. This set is to be distinguished from V({r1Q,œ) /l+J 2 which consists of all i with aij = — (— l) ' for all j e C + i. See Figure 1 for an illustration.

- + - - + - + - + 1 1 1 1 1 + - - + - + - - - + 1

-

1

1

+ - + - + + - + 1

1

1

1

1

1

1

+ - + - + © 2

2 2

- -

2

2

2

2

+ - + +

- + - + + - + - + - 1 1 2 + - + + + - + - - + + 1

Figure 18.1

1

1

+ + + - + - -

2

+ - +

A configuration m of + 's and — 's. The origin is encircled. The plaquettes C + i with i e V({Çk},(o) are marked with a k. Here k = 1 or 2 and Ck e {-1, + 1}C is given by Cf = ( - \f+h+k (i e C).

Percolation of spin patterns

385

The set V(G, co) will be regarded as the vertex set of a subgraph of S. The set of edges is simply the set {{iJ}czV(G,co):\i~-j\

= l}

of all undirected nearest-neighbour bonds between the sites of V(G, co). Accordingly, we shall examine paths and clusters in V(G,co). By definition, a (self-avoiding) path in a set V c S is an injective mapping n -> iM from a finite or infinite interval / c Zinto F such that | j ( n + 1 ) — j ( n ) | = 1 whenever n,n + 1 e I. If/ has a minimum k resp. a maximum £ then the sites im and i(e) are called the starting point resp. the end point of the path. A cluster of V is any non-empty subset t, of F which exhibits the properties below: (i) £ is connected, i.e., for any two distinct sites i,j e t, there exists a path in t, with starting point i and end point j , and (ii) if i e t, and j e F are such that \i — j \ = 1 then j e £. In other words, a cluster of V is nothing but a connected component of V. We shall be interested only in infinite clusters of V(G, co). So we let (18.4)

£(G, co) = {i e V(G, co): There exists an infinite path in V(G, co) with starting point i}

denote the union of all infinite clusters of V(G, co). If £(G, co) ^ 0 then V(G, co) is said to percolate. We shall also need to deal with the restriction of V(G, co) to an infinite subset R of S. To this end we introduce the notation VR(G, co) = V(G, co) D R, and we let ÇR(G, co) denote the union of all infinite clusters of VR(G, co). ÇR(G,co) is defined by (18.4) with V(G, co) being replaced by VR(G, co). (One should note that £,R{G,co) 7^ Ç(G,co)r\R in general.) In particular, we shall be concerned with the case when R is a two-dimensional layer in S. For the sake of definiteness we shall look at the plane (18.5)

P={i

= (ilt...,id)eS:i3

= -- = id = 0}.

(18.6) Definition. A subset t, of P will be called an ocean in P if t, is connected and all ^-islands in P are finite. Here a ^-island is meant to be any subset V of P\£ which is maximal relative to the following property: For any two i, j e V there exists a finite sequence (iw,...,ii0) in V with i(1) = i, ii0 = j , (k+1) m and |i - i \ = 1 or x / 2 for all 1 < k < t. Clearly, an ocean in P is always infinite. Also, if a subset V of P contains an ocean then V contains a unique infinite cluster, and this cluster is an ocean. We put (18.7)

£P(G, co) if £,p{G, co) is an ocean,

. [0 otherwise. Thus ÇP(G,co) is the unique maximal ocean in VP(G,co) whenever such an ocean exists. In particular, we have #(G,U,)=,

386

Low energy oceans and discrete symmetry breaking

{Çp(G, •) # 0} = {VP{G, •) contains an ocean}. In this section we will seek to provide a condition on $ which ensures that ( H{ÇP(G, • ) # 0) > 0 for a suitable \i e S{%. In fact, we shall only look at Gibbs measures for $ which are limits of Gibbs distributions with periodic boundary condition. There are two reasons for this restriction. On the one hand, these limits have the advantage of inheriting all symmetries of $. On the other hand, we shall need to make use of the chessboard estimate. So we let %() denote the set of these limits. That is, %() is the set of all cluster points (in the j§? -topology) of any sequence of the form (°y*(JV) x ô(0i)Nèl. Here A{N) is the cube (17.1), °y%m is given by (17.20), and œN e ESX-MN) is arbitrary. Recall that the set %(Q>) has already been considered in Example (5.20)(3). In particular, we know that %() a ^@(). In (18.12) below we shall prove that %{) whenever t(G, $) is sufficiently small. We put (18.8)

t(G,
c

ô>0

1

Here G e <S , Q>e%, l = À(Ey À, and I- inf
C

infimum of <5C on the set E \G relative to Xe. Here are some elementary properties of t(G, $). (18.9) Remarks. (1) For each $ e ^ e , the function t(-,(D) on Sc is decreasing. (2) For each $ e # e we have inf e7lc(G^(
inf

exp [2-sup
where A-sup <5)c = inf {c e U: XC(M D {C ^ c}) = 0} is the essential supM

remum of C on M. (3) For each G e Sc, the function t(G, •) is upper semicontinuous, in that f (G, *F) < t(G, 0, and *F e % is such that ||*FC — O c || is sufficiently small. (4) For all e > 0 and $ e # e we have t(G£(
asß^oo.

Proof. (1) and (2) are obvious. (3) is an immediate consequence of (2) and definition (18.8). To prove (4) we set Ö = e/2 and observe that t(G£(<&), jffO) <; e-ßEeß3ßc(Ga{^)).

a

Percolation of spin patterns

387

The significance of the quantity t(G, O) comes from the following key estimate which might be regarded as a general version of the Peierls contour estimate. This estimate is the only result of this chapter which makes use of reflection positivity. (18.10) Lemma. Suppose G e §c is r--symmetric, and let O e ^&, ß e %($>), and D ^ if be given. Then the inequality n(DnV(G,-)

=

0)^t(G,®)m

holds. Proof. It is sufficient to prove that

°y*(DnV(G,-) =

0)St(G,®r

whenever A = A(N) is so large a cube that A ZD \J C + i. In this case we can isD write l{DriK(G,-)=0} = 11 ft ° ac(i)> ieA

where C{i) is defined by (17.13) and fi = f = l£C^G when i e D and ft= 1 otherwise. Applying Theorem (17.21) and the chessboard estimate (17.17), we thus obtain that / \ l/IAI °tf(D H V(G, • ) = 0) ^ f i °7A M l fj ° *<*> = tf'• jeA

\i'eA

/

Here MM \ie A

To estimate tA we can assume without loss that 1(E) = 1. (Otherwise we replace X by I = À(E)"1 À.) Then

a

z;^ = ^ ( n ^ / ^ c , o ) ^ exp

|A| A-inf Oc C E \G

nil/**™ ieA

Letting ]"]' denote the product over all i = (^,..., id) e A with ik = 0 mod 2 i

for all 1 ^ k ^ d, we also have

^ ( n. /° ^(o) ^ A (rr /° **>) = ^(/) IAI/IC| . On the other hand, for each (5 > 0 we can write, letting g =

\Gim,

388

Low energy oceans and discrete symmetry breaking

^ e x p [ - | A I K . + ö)-]XA(ug°

acA

To estimate the last integral we can again apply Corollary (17.17) (with p. = AA). This gives \ 1/|A|

/

1C(G,((D)) = XA(g o <jc) ^ AA m

9 ° *c
Putting together all preceding estimates we see that tA ^ t(G, 3>). The proof is thus complete. • We insert a comment which will be needed in the next chapter. (18.11) Comment. The conclusion of Lemma (18.10) remains true when t(G, $) is replaced by the quantity t{G,<S>) = e x p [ - l - i n f 0> c ]A c (£ c \G) 1/|C| EC\G

x infexp[l-sup(D c ]/A c (i?) 1/|C| , R

R

where the inf is taken over all rectangles R = f\ &i with Rt e S and A(K;) > 0. ieC

Indeed, in estimating °Z* from below we can take advantage of the identity

^A(nA u ° ^c(o) = ^ A (n' i« ° *««>) = m iAi/ic| . (In view of Remark (18.9)(2), the main difference between t(G,<&) and f(G,3>) is in the power 1/| C\ of the last factor of t(G, be a C-potential. Then %(d>) # 0 and thus %{<$) # 0. Proof. We intend to apply Corollary (4.13) in the special case when D = N and, for each N e D, AN = A(N) and C is bounded on each K$ and inf <J>c(eo) -> oo

as

( -> oo.

Percolation of spin patterns

389

The last condition implies that t{Kce, periodic, we see that hypothesis (ii) of Corollary (4.13) is satisfied. In view of Comment (4.14) (2), we can therefore apply this corollary to conclude that %(Q>) ^ 0. • In the following, we shall use the estimate (18.10) to establish a lower bound of the probability that the origin belongs to an ocean in VP(G, •). To this end we first look at the event that the origin belongs to an infinite cluster of V(G, • ) in any quadrant of P. We put (18.13)

z(t) = 1 A X ^(5075

(t ^ 0).

Clearly, z(t) - > 0 a s t - » 0 . (18.14) Lemma. Let Q be any of the four quadrants with vertex 0 in the plane P. Suppose Geêc,{ie 0>(Q, &\ andt^O are such that n(D n V(G, • ) = 0) ^ tm for all DeSf. Then

M0ee f l (G,-))^l-z(t). Proof If 0 £ ÇQ(G, •) then either 0 £ V(G, •), or 0 belongs to a finite cluster of VQ(G, •). In either case, there exists a finite set D = {u{1\...,uin} in ß such that (i) u^ = 0 and u = 0, (ii) \um - u (k+1) | = 1 or Jl for all 1 ^ fc < /, (iii) \um - u{k+2)\ > J2 for all 1 ^ k < t - 1, and (iv) D D F(G, •) = 0. (Thus D is the vertex set of a path in Q which connects the two half-axes at the border of Q and is allowed to make next-nearest neighbour jumps.) For in the first case we can simply set D = {0}, and in the second case we let D be a suitable subset of the "outer boundary" of the finite cluster of VQ(G, •) that contains the origin; cf. the proof of Lemma (6.14). Hence { 0 ^ Q ( G , 0 } c U { ö n F ( G , - ) = Ö}, D

where the union is taken over all finite sets D c Q with properties (i) to (iii) above. Therefore we only need to count all such D with a given cardinality | JO | =
390

Low energy oceans and discrete symmetry breaking

for each quadrant Q c P, r-symmetric G £ ^ c , $ e ^ 8 , and p e %(®). We shall now combine this estimate with the shift-invariance of p to obtain a lower bound for p(0 e ££(G, •)). As a tool we shall need the well-known recurrence theorem of Poincaré which reads as follows. (18.15) Lemma. Let (Q.,tF,p) be an arbitrary probability space and 6 a measurable p-preserving transformation of Q. Also, let A e J^ and Aœ(6) = {(û e Q: 6"œ e A for infinitely many n ^ 0}. Then p(A\AJ6))

= 0.

Proof. For each œ e Q we let T(CO)

= sup {n ^ 0: 0"œ e A}

whenever there is some n with 6"œ e A, and T(CO) = — 1 otherwise. Then the inclusion A\AJ0)

c {0 g T < ex)} = U (^ = »}

holds. For each n ^ O w e have d~x {T = n) = {z = n + 1} and thus /X(T = n) = /i(r = n + 1). Since the sets {T = n) are pairwise disjoint and their union has at most probability one, we conclude that p(x = n) = 0 for all n ^ 0, and the lemma follows, D Let us agree to call a random field p quasi-Gibbsian if it enjoys the following property: There exists some X e &{E, S) such that for each A e 2T with p(A) > 0 the measure p(-\A) is equivalent to Xs on J \ for all A e if. It follows from Theorem (7.7)(b) and Remarks (1.28)(2) and (3) that each Gibbs measure relative to an arbitrary potential is quasi-Gibbsian. (18.16) Lemma. Let G e Sc be r-symmetric, and suppose /ie^@(Q, J^) is quasi-Gibbsian and satisfies p(0 e ÇQ(G, •)) ^ 1 — z for some z < 1/4 and each quadrant Q in P with vertex 0. Then /i(0e^(G,-))^l-4z. Proof. Let X e 2P(E, S) be such that p is quasi-Gibbsian with reference measure X. If Xe(G) = 1 then p(i e V(G, •)) = 1 for all i e S and thus p(V(G, •) = S) = 1. So we can assume that XC(G) < 1. To simplify our notation we shall further assume that d = 2. Hence P = S. We write Qx,..., Q 4 for the four quadrants with vertex 0 and consider the event * = {oen^(G,-)l=n{0^ß„(G,-)} that the origin is the point of intersection of an infinite "cross" in V(G, •)•

Percolation of spin patterns

391

By the subadditivity of ß, ß(X) ^ 1 — 4z. We thus need only to show that X is contained in {££(G, •) ¥= 0} ^-almost surely. This will be done in three steps. 1) For each k < 0 <
i1 ^ £, i2 = 0},

and we consider the events B(k,/) = {D(k,/)nV(G,-) 5W = # / ) J , O , - D ) .

= 0}, and

B=

f] k<0<(

BkJ.

(Here we use the notation of Lemma (18.15).) The significance of B is the following: The event B guarantees the existence of barriers which prevent V(G, •) from containing an infinite cluster which is limited to a strip of the form {i G 5:fc^ ij ^ t v |fc| and A = A(N) then

Using the chessboard estimate (17.17) (or a direct Fubini-type argument) we conclude that 1C(EC\G) = 0, in contradiction to the assumption at the beginning of this proof. Hence ß(B) = 1. 2) In this step we will show that, ju-almost surely on X, V(G, •) contains a doubly infinite path in the upper half-plane which meets infinitely many points of each of the horizontal half-axes. For definiteness, we put Q1 = {ieS:i1~£0,i2^ty

and

Ô2 = {' e S: ^ ^ 0,i 2 ^ 0}.

We look at the events 4 = {0^Cl(G>-)nyG)-)} and -4«, =-4œ(o(i,o))n/4œ(o(_1>0)). Clearly, X cr A By Poincaré's theorem (18.15), A cz Aœ ju-almost surely. Combining this with Step 1), we conclude that X a AmC\B ju-almost surely. Now let F denote the event that VQiUQi(G, •) contains an infinite cluster which meets each of the half-axes {i2 = 0,i1 ^ 0} and {/2 = 0,il ^ 0 } infinitely often. The reader should check that F is measurable. We claim that Am H B cr F. Indeed, if w e Aœ then there exists a strictly increasing sequence (fc„)„sZ in Z with the property below: For each n e l there exist two infinite paths n* and n~ in VQ UQ(G, co) with starting point (fc„,0).

392

Low energy oceans and discrete symmetry breaking

n~ runs on the left side of the vertical axis [i1 = k„, i2 e Z}, and n^ extends on the right side. If, in addition, coe B then n* is bound to intersect n~+l, for all ne Z. Consequently, for each n e Z there exists a path in VQlUQ (G, co) which runs from (/c„,0) to (/c„+1,0). Sticking these paths together we see that coe F. This completes the proof of the inclusion Ax D B c F. Combining this result with the previous inclusions we finally conclude that I c F /i-almost surely. 3) To complete the proof of the lemma we consider the event Fx = Fx(9i0i _1() as well as the events F2, F 3 , F 4 which are obtained from F^ by a rotation of S by the angles n/2, n, and 3n/2, respectively. By Poincaré's lemma, F a F1 /i-almost surely. Hence, by Step 2), I c F j /i-almost surely. Interchanging the roles of the four quadrants we obtain by the same argument that I c F „ /i-almost surely for n = 2, 3, 4. Hence X <= F1 D • • • D F4 /i-almost surely. However, if œ e Fx D • • • fl FA then V(G, co) contains an infinite network of doubly infinite paths which run along some horizontal or vertical line, in that they keep to the "outer" side of that line and return to it infinitely often; see Figure 2. Clearly this network is an ocean. Thus Fl D • • • D FA a {££(G, • ) j= 0}, and the proof is complete, G

Figure 18.2 The event Fx 0... H F4; cf. the proof of (18.16). Combining the preceding lemmas we arrive at the following theorem. (18.17) Theorem. Let G e ) the inequality

/i(0e^(G,-))^l-4z(t(G,0)))

Percolation of spin patterns

393

holds. In particular, if (E, S) is a standard Borel space and z(t(G, <ï>)) < 1/4 then there exists some n e (S(^>) which is invariant under the shift-group (Ö,); e P and the reflections rx, ...,rd and satisfies

M#(G,-)*0)=1. Proof. The first assertion is immediate from Lemmas (18.10), (18.14), and (18.16). The second statement then follows from Proposition (18.12) by setting H = v(- \&(G, •) ± 0) for some v e %( lim

inf

0,

/i(0e^(G£(
In particular, if (E, S) is standard Borel and ß is large enough then there exists some /j. e ^(/?<ï>) which inherits the symmetries (ö ; ) ;eP and rx, ..., rd of <& and satisfies M(Gt(0), •) * 0) = 1. The above corollary is most appealing when d = 2 and thus P = S. It can then be rephrased as follows. In the low temperature limit ß -* oo, the set %(ßQ>) approaches the set {
m
for all ieS} = K°(G0(
of all absolute ground states. More specifically, if ß increases then with increasing probability the origin belongs to an ocean of spins which try to form an absolute ground state. The following example might serve as an illustration of this kind of approach to the absolute ground states. In particular, this example will show that a low energy ocean need not exist at high temperatures. (18.19) Example. The classical Heisenberg model. Let d = 2, n ^ 2, E be the unit sphere in W, and X the surface measure on E. We define a potential CD e <^e by putting {i,j}^C:\i-j\

=l

(In view of Comment (17.19)(2), <& is equivalent to a multiple of the potential A

(

0

otherwise

394

Low energy oceans and discrete symmetry breaking

which is usually called the Heisenberg potential.) Clearly, m(<S>) = — 4, and the local ground states are the constant configurations on C. For each e > 0 we have Ge(
{(oe Ec: co; •
= 1}.

Corollary (18.18) thus shows that for sufficiently large ß there exists some H e %(ß<£) such that with probability 1 there exists an ocean with the property that any two adjacent spins in this ocean point in almost the same direction. Of course, this property does not imply any relation between the orientations of spins which are far apart from each other. This fact contrasts with the behaviour of those low energy oceans which will be considered in the next sections. Next, we will show that n(^(G8, •) # 0) = 0 for all p e #(j80) when both ß and e are small. To this end we fix some integer £ ^ 1. For each \i e ^(ßO) and a e S we can write flic)" O j l l c + l ) ^

1

-

e for

all

0

^

k

<

£),

where the sum runs over all self-avoiding paths D = {i <0) ,..., i(n} with i<0) = a. There are at most 4- l>e'v such paths. Moreover, for all i,j e S with \i —j\ = l and ^-almost all œ we obtain by a straightforward estimate n(oroj £ 1 - e | ^ ) ) M = yffioi-coj ^ 1 - e\a>) ^e16ßA((xi,...,x„)eE:x1

^ 1 - e)/A(£)

An iterated application of this inequality shows that

Li(aeaGe,-))^4-y-h(ß,sY for all *f ^ 1 and a e S. Consequently, if ß and e are so small that 3r(ß, e) < 1 then n(a e Ç(Ge, •)) = 0 for all a e S. By the subadditivity of ß, this implies t h a t ^ ( G £ , - ) # 0 ) = O. o

18.2

Discrete symmetry breaking at low temperatures

In this section we will show how the existence of low energy oceans, together with a degeneracy of the local ground states, can give rise to a phase transition. As we shall see, for this to occur it is first essential that the degeneracy of local ground states induces a long range order on the ocean, in that knowledge of the actual spin pattern on any part of the ocean provides some information on the pattern of the ocean at infinity. The low energy ocean then establishes

Discrete symmetry breaking at low temperatures

395

a link between the microscopic and the macroscopic behaviour of the spins. In the presence of symmetries, this link will imply a phase transition. We begin with specifying a class of transformations which we will consider below. Roughly speaking, this class consists of all transformations in T which are completely determined by their restriction to Ec. Specifically, we shall deal with the following two classes of transformations of Ec. (i) Spatial transformations. These are the reflections r1, ..., rd which have been defined in (17.5), as well as the P-preserving rotation r0: Ec -> Ec which is given by the formula (18.20)

(r0cü)j = cü(i2,1_i1>i2,...jiil)

(i =

(il,...,ii)eC,coeEc).

(ii) Pure spin transformations. These are all transformations T: EC -> Ec of the form (18.21)

zœ = {ziœi)ieC

(œ e Ec),

where the T;'S are A-preserving invertible transformations of (£, $). Each of the preceding transformations can be extended in a natural way to a measurable invertible transformation of (Q, #") which will be denoted by the same letter. This is obvious for the transformations of type (i). In case (ii) we put T(O = (Zj(Dj)jsS (œ e Q), where T,- = xi whenever; e S and i e C are such that jk = ik mod 2 for all 1 ^ k :§ d. We let Tc x denote the transformation group on Ec resp. Q which is generated by the transformations in (i) and (ii) above. Also, for each O e <^0 we let / c A() denote the subgroup of Tc k which consists of all symmetries of Q> in Tc A. In view of hypothesis (17.18) (iii), / C t i (0) certainly contains the iterated reflections r\ ie C. We also recall the following fact which was proved in Example (5.20) (3) and will soon turn out to be important. (18.22) Remark. If .(®) then each /x e %(<&) is x-invariant. Next, we introduce the fundamental notion of a stable ground state degeneracy. As before, we let P denote the plane (18.5). A subset F of C will be called a P-face of C if F is equal to one of the four sets {i = (J\ ,..., id) e C: ik = a}, ke{\,2},ae{0,i}(18.23) Definition, (a) A potential C> e c€@ will be said to exhibit a P-stable ground state degeneracy of order JV > 1 if there exist JV distinct sets Gi, ..., GN e Sc with the following two properties: (i) GE(Q>) cz Gj U • • • U GN when e > 0 is small enough. (ii) For each P-face F of C, the projections aF{GY),...,
396

Low energy oceans and discrete symmetry breaking

(b) {G 1 ,...,G JV } will be called (^-symmetric if for all 1 ^ m, n ^ N there exists some x e /CjA() with xGm = G„. What is the significance of a stable ground state degeneracy? To answer this we assume that the conditions of Definition (18.23) (a) hold, and we let œ e Q. be any configuration. Suppose i,j e VP(Ge(®), co) are adjacent (i.e., |z —j\ = 1). Since rl(6_ioj)c, rj(6_jCo)c e Ge(), we can find two indices 1 ^m,n ^ N such that r'(ö^;co)c e Gm and rj(6_jCo)c e G„. However, the configurations r'(ö_;co)c and rj(6_jCo)c coincide on some P-face of C. The stability condition (18.23)(a)(ii) thus implies that m = n. In other words, there exists some 1 ^ n ^ N with i,j e VP(Gn, co). This in turn implies that each connected subset of VP(GE(<&), co) is a subset of VP(Gn,co) for some n. Consequently, there exists some n with £°(G£(<5),co)c£°(G„,co). Hence

(18.24) K°(Gs(0), - ) ^ } c U K?(G„, ') # 0}. n= l

One should note that the sets on the right side of (18.24) are pairwise disjoint. Indeed, let m ^ n be given. Then ÇP(Gm, co) D ÇP(G„, co) = 0 for all co e Q because Gm and G„ are disjoint. But the definition of an ocean implies that any two oceans have a non-void intersection. Hence ÇP(Gm, co) = 0 or ÇP(G„, co) = 0 for all co, and thus

K°(G m ,-)^0}nK°(G„,-)/0}=0. The inclusion (18.24) provides a key to the following theorem which is the main result of this chapter. (18.25) Theorem. Let (E, S) be a standard Borel space and <5 any C-potential. Suppose <5 admits a (^-symmetric P-stable partition {G1,...,GN} of its local ground states. If ß is sufficiently large, there exist N distinct Gibbs measures / ^ , . . . , / % e ^(ßQ>) which enjoy the following properties. (i) For all 1 ^ n ^ N, n„(Ç0P(Gn, •) ^ 0) = 1. Moreover, if ß-^ co then /x„(0e^(G„,-))-l. (ii) For each 1 ^ n ^ N, the measure p,„ is preserved by the shift-group @„ = {6t: ie P,rlG„ = G„} as well as all transformations x e Ic A(<5) which satisfy xG„ = G„. (Hi) If 1 ^ m, n ^ N and x e /CiA() are such that xGm = G„ then x{p,m) = p,„. If, in particular, x = r1 or r2 then, in addition, 6u(p,m) = p,n, where u is the unit vector in direction 1 resp. 2. The potential ß® thus exhibits a breaking of all symmetries x e / C A ($) which N

map some Gm onto a different G„,n / m. In particular, if © = f] ©„ then \ex%(ß®)\^N. Proof Let e > 0 be so small that Ge(<5) c Gt U • • • U GN and ß so large that

Discrete symmetry breaking at low temperatures

397

5{ß) = 4z(t(Gt(0), ßO)) < 1. (This is possible because of Remark (18.9)(4).) By Theorem (18.17), there exists some ß e %(ßd>) with /x(£°(GE(0>), •) # 0) ^ 1 - ö(ß). We put An = {^(G„, •) ^ 0}, i ^ „ ^ jv. The inclusion (18.24) then ensures that I p{An) > 0. n= l

Since {G1,...,GN} is O-symmetric, the events Av,..., AN are related to each other by suitable transformations in Ic,i{^>)- In view of Remark (18.22), this implies that ^{A^ = ••• = n(AN). We thus conclude that n{A„) > 0 for all n. The rest of the proof is routine. We introduce the conditional probabilities n„ = n(- \A„\ 1 ^n^N. Since An e ST, Theorem (7.7) (b) yields that Hn e ^(/?). As Av,..., AN are pairwise disjoint, the measures / i l 5 . . . , fiN are mutually singular and thereby distinct. Also, since A„ and ß are invariant under the shifts in ©„ as well as all G„-preserving transformations in IClx{^>\ fin is invariant under the same class of transformations. This proves assertion (ii), and the same kind of argument establishes (iii). To complete the proof of (i) we put G = Gt U • • • U GN. Using the hypothesis of -symmetry and the Ic A()-invariance of n, we obtain H„(0 e É?(G„, •)) = A*(0 6 ^(G„, -))ln(An) = MOe^(G,-))/M^(G,-)^0) ^M0^?(G£(! as ß^> oo. The last statement of the theorem follows from the remark in the paragraph below Corollary (14.18). a (18.26) Comments. (1) Suppose 0 which are such that the conclusion of the theorem holds for all ß > ß'. ßc{^>) is called the (upper) critical inverse temperature. The above proof then provides a device for establishing an upper bound of ßc(),/?(D)) < 1 for all ß ^ ß0. Then ßc(d>) ^ ß0. (2) Suppose d = 2 and G„ contains a unique local ground state £". Let co" eQ.be the periodic continuation of £". Assertion (i) of the theorem above can then be rephrased by saying that ßn is a random perturbation of co". More generally, if G„ contains several local ground states then \in can be regarded as a random perturbation of the ensemble of all absolute ground states whose local pattern fits with G„. In dimensions à > 2, the theorem is less satisfactory because of the special role of the plane P. As a compensation, the condition of P-stability then still involves only four of the 2d faces of C. This can be advantageous, as will be seen in Subsection 18.3.10 below, o

398

Low energy oceans and discrete symmetry breaking

18.3

Examples

In this section we shall visit a zoo of various classical models which are known for their symmetry breaking at low temperatures. In all these models it is easy to find a stable symmetric partition of the local ground states, so that the symmetry breaking is an immediate consequence of Theorem (18.25). Most of the models are usually defined in terms of a nearest-neighbour or next-nearest neighbour potential. Therefore, in order to apply Theorem (18.25) we need to pass to an equivalent C-potential. This is possible because of Comment (17.19) (2). Except in the last example which needs three dimensions, we shall always assume for simplicity that d = 2. (The extension to higher dimensions provides no difficulties.) A configuration œ e Ec can then be described conveniently by writing

\u

vj

when u = co(0i0), v = <w(1>0), w = co (0jl) , z = co(1>1). 18.3.1

The Ising ferromagnet

Let E = {— 1,1}, X be counting measure, and O e # e be the unique C-potential with (18.27)

X {i,j}^C:\i-J]

=l

Up to a factor of 2, ) consists of the local ground states of — co which belongs to / c , A(®)- Theorem (18.25) thus gives the following result (which is a slight refinement of Theorem (6.9)): For sufficiently large ß there exist two mutually singular Gibbs measures H+, n- e ^e(jßO) which are invariant under r0, r l5 r2 and satisfy T(^+) = /U_, AMÉ°(G+, •) ^ 0) = 1, and ^+(0 e £°(G+, •)) -> 1 as ß -• oo. Thus | e x ^ e ( ^ ) l ^ 2 for these ß. 18.3.2

The Ising antiferromagnet in an external field

Let E and X be as above. We consider the C-potential <£> which is given by

(18.28)

X U,j}^C:\i-j\ = l

wj-h

5>iieC

Examples

399

Thus the first term differs from the ferromagnetic case (18.27) only in its sign. The real parameter h represents the action of an external magnetic field. We assume that \h\ < 2. ) = Gl U G2. The partition {GX,G2} clearly satisfies the stability condition (18.23)(a)(ii), and Gx and G2 are related to each other by each of the symmetries r0, rx, r2e IC,A(®Y We thus conclude from Theorem (18.25) that for sufficiently large ß there are two distinct Gibbs measures /J,X, \i2 e ^(/J1 = ((— l)' 1+l2+1 ); e s resp. co2 = ((— l)' 1+ ' 2 ); eS , are invariant under all translations 8t with ix + i2 = 0mod2, and are related to each other by the ^-symmetries r0, rx, r2, ö(0,i), and0 (1>o) . 18.3.3 The Gertsik-Dobrushin model This example includes the preceding ones as particular cases. Let again E = {— 1,1} and X be counting measure. Given any three real parameters J l 5 J2, and h, we define a C-potential by (18.29)

Oc=-J1

X

°i°j-Ji

Z

Wj-hZ ieC

Thus h is an external field, and Jx and J2 are the nearest-neighbour resp. next-nearest neighbour coupling constants. (Positive values of Jx or J2 correspond to a ferromagnetic coupling.) Ic,x(®) always contains the rotation r0 and the reflections rx and r2. If h = 0 then JTC.AO^) also contains the spin-flip, and if h = Jx = 0 then the sublattice spin-flip m -> ((— l) il+i2 cü ; ) ieS is a further symmetry of
Figure 18.3 The phase diagram of the Gertsik-Dobrushin model. The "etc." stands for the rotation images of the local ground state shown.

400

Low energy oceans and discrete symmetry breaking

Theorem (18.25) applies in each of the following four cases. (a) J t > 0, J2 + J1 > 0, h = 0. The set of local ground states then consists of the configurations ( ++) and (II), and we obtain the same result as in the case of the Ising ferromagnet. (b) J1 < 0,J2> J1,\h\ < — 2J t + (J2 A 0). In this case there are again two ground states, namely ( ++) and (+*). Theorem (18.25) thus implies the same result as in Example 18.3.2 (which corresponds to the particular choice Ji= - 1 , J 2 = 0). (c) J t = 0, J2 > 0, h = 0. These parameter values constitute a half-line in 1R3 which is the common border of the regions (a) and (b). The Gibbs specifications yß) contains four mutually singular Gibbs measures which are shift-invariant in one direction and shift-periodic with period 2 in the other direction and are transformed into each other by suitable powers of the rotation r0. In all other cases, Theorem (18.25) does not apply. This is because the set of local ground states then either (i) is a singleton, or (ii) contains configurations which violate the stability condition, or (iii) splits into configurations which are not related by symmetries. Case (ii) occurs at the common border of cases (a) and (d) or (b) and (d) and also when there exists a local ground state with an odd number of + 's and — 's. Case (iii) occurs when Jx < 0 < J2 and \h\ = 2\Jx\. This case will be reconsidered in (19.9). As an exercise, the reader might investigate what happens when a four-spin term J3 \\ ai is added to the potential (18.29). ieC

18.3.4

Hard-core lattice gas models

Let E = {0,1} and X be counting measure. We consider the function (18.30)

f — Y, <*i if ai°j = 0 f° r a ll Uj e C with | i — j \ = 1,
and we set O c + i =
Examples

401

+ 00, <ï> is not a potential in the strict sense of Definition (2.2) to which we have limited ourselves so far for simplicity. However, since <ï> has finite range the Hamiltonians H% are well-defined functions with values in ]—00,00]. Using the convention exp( —00) = 0 in (2.4), we can thus define the Gibbsian specifications yß<* for ß<&, ß > 0. An obvious limiting procedure shows that the Gibbs distributions °y{®N) with periodic boundary conditions are still defined and rk-periodic for 1 ^ k :g d. So the chessboard estimate and thus Lemma (18.10) are still applicable. In fact, there is only one point in the proof of Theorem (18.25) which needs some care in the present case: Since yßi" fails to be positive, a Gibbs measure for ß<£> is not necessarily quasi-Gibbsian in the sense of Lemma (18.16). We thus need to use an ad hoc argument in Step 1 of the proof of that lemma in order to show that v(B(k,t?)) > Owhen v e $(ßQ)) and G = G0(<ï>) is the set of local ground states. But this is an immediate consequence of the fact that B(k,t?) => {aA = 0} when A is large enough, and y{*(oA = 0| •) > 0 for all A. Hence Lemma (18.16) and thus Theorem (18.25) still hold for the present <X>. In order to apply Theorem (18.25) we note that G0(<X>) consists of the two configurations (%) and (0°). The situation thus resembles that of the Ising antiferromagnet, and Theorem (18.25) implies the same kind of result as in Example 18.3.2. A hard-core lattice gas with two types of particles, + 1 and — 1, is described by the following model of Widom and Rowlinson. Let E = { — 1,0,1} and X be counting measure. We define a hard-core interaction <X> between particles of different type by letting (18.31)

f — YJ af d> c = -l i*c [ 00

ifffjffj7e — 1 for all i,j e C with \i —j\ = 1, otherwise.

In view of the remarks above, Theorem (18.25) can also be adapted to this <ï>. The reader may check quite easily that the behaviour of the Widom-Rowlinson model resembles that of the Ising ferromagnet. 18.3.5

Potts' model

This model is a generalization of the Ising ferromagnet to an arbitrary finite number of states. Let N ^ 2 be an integer, E = {1,...,JV}, and X counting measure. Define a C-potential $> by (18.32)

Oc=

X

rtWi-OjM

{>,j}<=C-.|i-j| = l

where 1R is any function which satisfies cp(0) for all n > 0. (The classical choice is
is invariant under

402

Low energy oceans and discrete symmetry breaking

all transformations of the form co ->• (7tcu;);eS, where n is a cyclic permutation of E. These transformations map the local ground states (™) (n e E) of
The anisotropic Heisenberg ferromagnet

Let n ^ 2, E be the unit sphere in R", and X be surface measure. (Alternatively, we might let E be the unit ball in R" and X the Lebesgue measure on E.) For a given parameter 0 ^ a < 1 of anisotropy we let $ e
Oc=-

X

L°;,I +

{i,j}czC:\i-j\ =

a

Ë

Ö";

( =2

where c, y stands for the /'th coordinate of C;. Of course, the term in square brackets should be regarded as a deformation of the inner product
e E: + x t ^ 1 — <5}

as well as the sets G+ = U+ and G_ = £/£. A straightforward computation shows that G£(0)) c G + U G _ whenever 0 < e < (1 - <x)<5. Hence {G+, G_} is a stable partition of the local ground states. The reflection of the first coordinate of all spins is a symmetry of <5 which maps G+ onto G_, and vice versa. Theorem (18.25) thus shows that |ex @@(ß<S))\ ^ 2 when ß is large enough. 18.3.7

Anisotropic Heisenberg antiferromagnets in external fields

Let E and X be as before and define $ e ^ 0 by

(18.34) (D c =

X {«,J}CC:|(-J| =

a

u\ai.\

+

a

u Ë o'.i,/ -y^j ;y - * E ^ i .

^= 2

ieC

i»

where 0 ^ a < 1 and \h\ < 2. (h describes the action of an external field in the 1-direction.) The behaviour of this model is similar to that of the Ising antiferromagnet. Indeed, let u = (1,0,..., 0) e E denote the unit vector in direction 1. It is easily seen that (_£ ~uu) and (~uu _£) are the only local ground states of O. Let U+ and (/_ be as in the last example, and let Gx = Y[ ^(-i)'i+i2+1 a n d ieC

G2 = r0 Gx be the associated neighbourhoods of the local ground states. By continuity, GC(<1>) c Gt U G2 when e > 0 is small enough. Theorem (18.25) thus ensures that |ex^(ß)| ^ 2 when ß is large enough. It is natural to ask what happens when the external field is transverse to the 1-direction which is favoured by the anisotropy. To deal with this case

Examples

403

we assume for simplicity that n = 2. So E is the unit circle, and we have a plane rotor model. We look at the C-potential (18.35)

Oc=

X

[°i.i°j,i +

aCT

«,2Ö),2] -h

{;,;}<=C:|>-j| = l

X OÏ,2 ieC

where |/i| < 2(<x + 1) and 0 ^ a < 1. If /i # 0 we can also allow for the isotropic case a = 1. To find the local ground states of Q> it suffices to determine the minima of the function
ccx2y2 - h(x2 + y2)/2

on E x E. A straightforward application of Lagrange's multiplier method shows that cp takes its minimum at the two pairs (v, w) and (w, v), where v,w e E are such that vx = — wt and v2 = w2 = h/(2 + 2a). Hence $ has again two local ground states, namely (££) and (££). Consequently, the same arguments as before show that $ admits a stable «^-symmetric partition of local ground states, and Theorem (18.25) implies that |ex^(/?C>)| ^ 2 when ß is large. 18.3.8

Shlosman's plane rotor model

This model was already considered in (9.22). It differs from the preceding examples in having uncountably many local ground states. Let E = [0,27r[ and X be Lebesgue measure on E. We think of £ as the unit circle. Accordingly, the addition of any two elements of E will be understood modulo 2n. We consider the C-potential (18.36)

Oc=-

X {i,j}<=C

|i-J|=V2

COSfa -
X

COS2((Tf-(T,.),

[i.j}<=C

|»-j| = l

where J > 0 is arbitrary. Besides r0,r1, and r2, /CiA() also contains the spin rotations x': a> -> (co; + t) i e S (co £ Q, t £ E) as well as the spin reversal transformation T on the even sublattice. T is defined by TCO = (cot + nt)ieS, where co £ Q and 7r; = n when i\ + i2 is even, and 7r; = 0 otherwise. A configuration on C is a local ground state of Q> if and only if any two adjacent spins are orthogonal and any two next-nearest neighbour spins coincide. The local ground states thus fall into two classes. The first class contains all co e Ec with CO; = x + nJ2 for some x e E and all i e C, whilst the second class consists of all co e Ec with cot = x — nt/2 for some x e E and all i e C. These classes are (Tr)Ie£-invariant and are related to each other by the symmetries T, r0, r1, and r2. For each face F of C, the distance between the oy-images of these classes is positive. We thus can find two (T()[e£-invariant open neighbourhoods Gj and G2 of these classes which are related to each other by the symmetries T, r0,r1, r2 and form a stable partition of the local ground states. We thus arrive at the following conclusion: If ß is sufficiently large then ^(/?
404

Low energy oceans and discrete symmetry breaking

is even} and satisfy /i2 = T ( / ^ ) = r^n,) = r ^ ) = r2(nx) = 0 ( l i O ) ( ^ ) = Ö(O,I)(J U I)- (The (T') teE -invariance of /j,1 and JX2 should not be surprising. For we already know from Example (9.22) (2) that each ß e ^(/?) inherits this continuous symmetry of .) A variation of Shlosman's theme is provided by the potential (18.37)

Oc = -

£

cos fa - Oj) + J

{i.J}^C \i~J\ = s/2

£

cos 2 fa - cr,) sin 2 fa - a,),

{i,j}^C |i-j| = l

where again J > 0. In addition to the symmetries of Shlosman's potential (18.36), this has the further symmetry f: co -> {cot + nJ2)us. The local ground states of fall into four classes. The first class consists of all constant configurations, and the other classes are the images of the first under the powers of f. thus exhibits a stable ground state degeneracy of order 4, and Theorem (18.25) gives the following result. If ß is large enough then ^(/?) contains four pairwise mutually singular measures n1,..., ju4. These are linked to each other by the relation ßk = fk(jU4) {k = 1,2,3), and ju4 is invariant under the shift-group 0 as well as {x')t£E, r0, rx, r2. In particular, the discrete symmetries f, r0, rx, r2, 0 (liO ) ar>d ö (0 ,i) °f ^ a r e broken, whereas Theorem (9.20) ensures that the continuous symmetry (T f ) (e£ remains unbroken.

18.3.9

Unbounded spin systems with double-well potentials

Here we put E = M and let X be Lebesgue measure. We consider any C-potential (£> of the form (18.38)

X {i,7}cC:|i-j| = l

ufa - aj) + £

r fa).

ieC

Here u, Ü: £ -» U are supposed to be even and, for simplicity, continuous. As for the nearest-neighbour interaction u, we assume that u is attractive in that u attains its minimum at 0 only. A standard choice is u(x) = x2. The selfpotential v is required to satisfy the two conditions below: (i) As |x| —> oo, v(x) tends to oo in such a way that X{e,~ßv) < oo for all ß > 0, and (ii) v(0) > min v. Hypothesis (i) implies in particular that ßQ> is A-admissible for all ß > 0. Condition (ii) ensures that v forms (at least) two potential wells. Typical examples are the functions x ->• a x 4 — bx2 (a,b > 0) and x —» ||x] — a\ v b (0 ^ b < a). The local ground states of consist of all configurations (**) with v(x) = min v. These configurations fall into two classes according to whether x > 0 or x < 0, and these classes are related to each other by the spin-reflection T: co -> -co which obviously belongs to Ic,x(®)- ® thus exhibits a i-symmetric stable ground state degeneracy of order two. Consequently, all hypotheses of Theorem (18.25) are satisfied, except for the assumption that the a priori measure should be finite.

Examples

405

The last problem can be resolved as follows. First of all, we let {G+,G_} be a stable z-related partition of the local ground states of O. Then we let £ > 0 be so small that G2e((5)) c G + U G _ . Next, we let ß ^ 2 be so large that 4z(t(Ge(Q)),ß) c Gô(<&) for all 5 > 0, we conclude from (18.8) that 4z(t(G£(Ö),ßÖ)) < 1. As G£(Ô) c G2£(0>) c G+ U G_, this implies that the conclusion of Theorem (18.25) holds for the partition {G+, G_}, the a priori measure A, the potential and X coincides with that for ßQ> and X. Hence |ex ^@(ß^>)\ ^ 2 when ß is large enough. 18.3.10

Slawny's D-model

In this last example we assume that S = Z 3 . The state space E is the two-point set {— 1,1}, endowed with counting measure. We consider the C-potential

(18.39) < 5 C = - X F^C

FK ieF

The sum extends over the six faces F of C. First of all, we note that ^(eo) is minimal if and only if on each face F of C there exists an even number of sites i e F with œt = 1. The latter property implies that if eo(u „ 0) = eo (u vl ) for some choice of u, v e {0,1} then the same identity holds for all u, v e {0,1}. The set G = G0( for all w, t; e {0,1}} and G2 = {coe£ c :co (Uii;i0) # co{u ->•((— l)'3co£);=0i i2 ,-3)eC which belongs to 7Ci/l(). We thus conclude from Theorem (18.25) that |ex %(ß has an uncountable number of (discrete) symmetries, including the transformations t,:(li3<»i)i=(i1,i1,i,)ES

(coeQ,?je{-l,l}z).

To exploit these symmetries we proceed as follows. If ß is large enough then Corollary (18.18) provides us with some v0 e %(ßQ>) such that v0(£°(G, •) ,6 0) > 0. (Recall that G = G0(
406

Low energy oceans and discrete symmetry breaking

A = {£p(G,0(O>o>_k)-) ^ 0 for infinitely many k} has positive v0-probability. Next we take advantage of the fact that the ergodic decomposition theorem (14.17) remains true when the full shift-group 0 is replaced by 0 . The decomposition of v0 thus provides us with some n e ex^@(/? 0. \i is trivial on the cr-algebra of all Ôinvariant events, and this cr-algebra contains the events {^p(G„,ö(0 0 _t)-) ^ 0}, ne {1,2}, k e Z. Consequently, there exists an infinite set M c Z and a mapping v: M -> {1,2} such that ^(£P(G, 0(O 0 _k)-) ^ 0) = 0 when k<£ M and Ä ( G 0 ( k ) , ö(o.o,-k)-) ^ 0) = 1 when k e M. ' Now let w: M -> {1,2} be an arbitrary function. We claim that there exists some ßw e ex ^(ß®) with (18.40)

M^(GwW,Vo,-k)-)^0) = l

for all

k e M.

Indeed, it is possible to construct a sequence n e { — 1,1 } z in such a way that nknk+1 = — 1 if and only if k e M and w(k) ^ u(fc). It is then clear that Hw — tq(n) satisfies (18.40). ßw belongs to ex ^©(/JO) because xn is a symmetry of <5 which commutes with Ô; cf. (5.10) and (14.11). In view of (18.40), any two distinct /iw's are mutually singular. Since M is infinite, {1,2} M has the cardinality of the continuum. We thus conclude that

\cx%(ßO)\^\U\. when ß is large. With some additional effort we can even show that Slawny's model admits an uncountable number of 0-ergodic Gibbs measures when ß is large enough. (18.41) Proposition. Consider Slawny's a-model. If ß is sufficiently large then |ex^0(i? 1/2. Putting g = liminf (2n + l)" 1 £

1{O6«?
Vo,*)>

we conclude from the (one-dimensional) ergodic theorem that v0(g) > 1/2. Decomposing v0 into Ô-ergodic components, we find some Vj eex^iß®) with v1(g) > 1/2. By means of a suitable transformation T,, VJ can be transformed into a measure v2 = T ^ V J e ex%(ß 1/2 and V 2 (£P(G 2 , ö(o,o,t)0 ^ 0) = 0 for all IceZ; cf. the proof of (18.40). In particular, we have liminf (2n + l)" 1 X v2((0,0,/c) e F(G 1; •)) ^ v2(g) > 1/2. Next, we let v3 be any cluster point of the sequence (In + 1)_1 £ |k|S«

0(O 0 k)(v2),

Examples

407

n ^ 1. (Such a cluster point exists because of Example (4.11) (2).) Then v3 e %(ßO) and v3(0 e V{GU •)) ^ v2(g) > 1/2. Theorem (14.17) thus ensures the existence of a measure ß e ex^@{ßQ>) which satisfies MO e V(Glt •)) * ct > 1/2 > c2 A ^(o e J/(G2, •))• 2) We now use the above measure fi to construct a one-parameter family (/V>oâpâi/2 in ex%{ßO). For each 0 ^ p ^ 1/2 we let <xp = (pôl + (1 - p)^_J z denote the Bernoulli-measure on { — 1,1}Z with probability p of success, and we put

The measures ßp are pairwise distinct. For it is immediately checked that fip(0 e V(Gu •)) = (P2 + (1 - p)2)Cl + 2p(l - p)c 2 . Since xn is a symmetry of O which commutes with Ô, it is clear that \ip e ^(ß) = 1 aP(d»?)0(o,o,i) ° xr,(ß) = 1 a p ( d/ ?) T ^ ° 0(o.o.i)(A^) = 1 <xp(dri)Tgn(n) = /i p . Hence /xp e 99(ßQ). 3) It remains to show that ßp is ergodic. Let A e J be given. We will show that fip(A) = 0 or 1. The function tö-> J <xp(d»j) l ^ t ö ) is shift-invariant and thus constant /^-almost surely. Hence 2

ßp(A)

=

^(dco)^ap(dr,)lA(^co)V.

By a straightforward extension of Proposition (14.9), there exists a set B e f] ^Z2X[_„,„] with /iJAAB) = 0, and thus lA{i„œ) = lB(x to) for /1 x a «èi

almost all (to,?/). For each to, the function Y\ -> lB(x^tu) is measurable with respect to the tail er-field in { — 1,1}Z. Kolmogorov's 0-1 law therefore implies that this function is constant a p -almost surely. Hence ^p(A)2 =

^(dœ)^ap(dr,)lB(Tnœ)r

= j/x(dto)jap(dfy)lB(T,to) = np(B) = np(A), and the proof is complete,

a

Chapter 19 Phase transitions without symmetry breaking

This chapter is a continuation of the last chapter. For the sake of convenience we shall confine ourselves to the case d = 2. Thus S = Z 2 , the square lattice. As in the last chapter, we shall consider C-potentials -symmetric. In this case, a multiple ß® does not necessarily admit N distinct Gibbs measures which correspond to the sets Gy,..., GN, even when ß is arbitrarily large. However, for large ß it will be possible to find a small perturbation Ô of exhibits a phase transition of the expected type. This result can be regarded as an extension of Theorem (18.25) to the non-symmetric case. The precise statement and some examples will be given in Section 19.1. The proof will follow in Section 19.2. Roughly speaking, the rôle of the symmetry argument in the proof of Theorem (18.25) will be played by a topological lemma which is a consequence of the well-known Sperner's lemma. For the convenience of the reader we shall provide a proof of this lemma in Appendix 19.A. As a matter of fact, the methods of Section 19.2 are not limited to the above-mentioned case of a phase transition at low temperatures which arises from a degeneracy of the local ground states. These methods are also suitable for establishing the existence of a different kind of phase transition which only occurs at a specific value of temperature. Its origin is based in a conflict of energy and "entropy" which occurs when we offset the energetic advantage of the local ground states by imposing the condition that their a priori weight is very small. At a suitable value of temperature there is then a balance of power which leads to the existence of distinct Gibbs measures whose properties are respectively determined by the predominance of energy or entropy. This kind of phase transition will be the subject of Section 19.3. On the one hand, we shall deal with the competition of two or more potential wells which differ in depth and width and are separated by potential barriers. On the other hand, we shall investigate the Potts model with a large state space.

19.1

Potentials with degenerated ground states, and perturbations thereof

As in the last chapter, we let (£, ê) be an arbitrary measurable space which is equipped with a finite a priori measure X. The parameter set is S = Z 2 . We

Potentials with degenerated ground states, and perturbations thereof

409

consider a fixed C-potential <X>, and we assume that <X> exhibits a stable ground state degeneracy of any order N > 1; see Definition (18.23)(a). The associated stable partition of the local ground states will be denoted by {G1,..., GN}. We JV

let G = (J G„. As we shall see in (19.7), there is no loss in assuming that n= l

each G„ is r-symmetric. We will also assume that (19.1)

;i-infO c = m^

(1 ^ n S N).

By definition of the essential infimum, this means that AC(G„ fl G£(<X>)) > 0 for all £ > 0 and 1 ^ n ^ JV. In other words, none of the sets G„ is at an energetic disadvantage. Clearly, this condition only ensures that {G1,...,GN} is no larger than is necessary to cover the local ground states. It is trivially satisfied when {G1,...,GN} is ^-symmetric. Keeping Theorem (18.25) in mind, we might expect that ß<& exhibits a phase transition when ß is large enough. Indeed, the fundamental inclusion (18.24) is still valid, and Theorem (18.17) still ensures that the set on the left of (18.24) has positive probability relative to any /x e %(ß®) when ß is large. We would like to conclude that each set on the right of (18.24) has positive /x-probability. For, this would readily imply that | ex %(/?) | ^ N. Although the above conclusion was trivial in the ^-symmetric case, there is no reason for it to hold in general. In fact, there exists a counterexample; see (19.8) below. Intuitively speaking, the above conclusion requires a certain amount of balance between the sets Gx,..., GN. Such a balance can be achieved by a suitable perturbation of <X>. We now describe what "suitable" means. Let O 1 , ..., d»*-1 e % be bounded, in that t|0£H < oo for all 1 ^ n < N. These potentials define an (N — l)-dimensional family of perturbations via (19.2)

(fc = (fc 1 ,...,fc w _ 1 )elR w - 1 ).

O* = 0> + "£ VD„ n= l

Relative to <X>\ some of the sets Gx,..., GN will sustain an energetic disadvantage. The distribution of disadvantages can be measured by the quantity (19.3)

?W =

(#n{h))imN,

where pn(h) = A-infO£- A-infd>£. G„

G

g. is a continuous mapping from IR^"1 to the boundary dUN+ = \z = (Zl,...,zN)eUN:

min z„ = o i

of the positive octant in UN. Assumption (19.1) implies that ^(0) = 0.

410

Phase transitions without symmetry breaking

(19.4) Definition. The potentials O 1 , ..., Œ^"1 will be said to resolve the partition {G l 5 ..., GN} of the local ground states of <5 if the following condition holds: There exists a neighbourhood U of the origin in UN^ such that (i) g- is a homeomorphism from U onto a neighbourhood of the origin in dUl,and (ii) inf Ac(Ga(0*)) > 0 for all <5 > 0. fret/

(19.5) Comments. (1) There is a simple device for constructing bounded C-potentials <X>\ ..., Q>N1 which resolve a given partition {G1,..., GN}: One can take any bounded c = c„ on G\G„ and <££ = c„ + <5„ on G„, where c„ e R and ô„ =£ 0. Then we can write, putting hN = 0 and taking any cN and c^, A-inf
G„

k^n N-l

= m* + h„ô„ + £ hkck

( l g n g TV).

k= l

Hence Ah) = \

h

A - min ftk<5k )

for all ft e (RN_1. This shows that ^ is a homeomorphism from R N_1 onto 31R+. The inverse of 9 is given by f\z) = ((z„ - z ^ " 1 ) ^ ^ ^ zedUN+. To verify condition (19.4)(ii) we let e > 0 be so small that GE( and thus, by (19.1), A-inf
G

the other hand, for each ft e U we can find some 1 ^ n ^ iV with ^„(ft) = 0. Since $£ — <5C is constant on G„, we conclude that c(co) = /l-inf 0. We thus end up with the inequality inf ÀC(GÔ(®")) ^ min AC(GÔ(
Potentials with degenerated ground states, and perturbations thereof

411

ing measure. More generally, Condition (19.4)(ii) holds whenever E can be endowed with a metric d which generates S and is such that Gj, ..., GN are compact (in the product topology on Ec generated by d), <ï>, O 1 , . . . , O^ - 1 are continuous at each point of G, and inf A(d(co,-, •) ^ 5) > 0 for all ô > 0 and WGG

i e C. This can be proved in a similar manner as in the preceding comment, o The significance of Definition (19.4) becomes apparent from the following theorem which is the main result of this section. We shall use the notation Ic ^ ( O j O 1 , . . . , ^ - 1 ) for the group of all common symmetries of <ï>, O 1 , ..., e <&@. Suppose <ï> admits a stable partition {Gl,..., GN} of its local ground states into r-symmetric sets G„ e ëc which satisfy (19.1). Also, let O 1 , ..., O*" 1 be any bounded Cpotentials that resolve {Gl}...,GN}. If ß is sufficiently large, there exist a parameter h(ß) e UN~l as well as N pairwise mutually singular Gibbs measures Hl,...,HNe ^&(ß^h{ß)) such that the following assertions hold. (i) h(ß) -> 0 as ß^ oo. (H) For each l^n^N, ßM°{Gn, •) * 0) = 1 and /z„(0 e £°(G„, •)) - 1 as ß^oo. {Hi) T(H„) = n„ for all Te ICtX(0,<S>1,...,(!>N-1)and 1 ^ n ^NwithxGn = G„. In particular, \ex%(ß®h(ß))\

^N.

We postpone the proof of this theorem until the next section. Here we will first state a corollary and then look at several examples. The corollary deals with the case when some (but not all) sets in a stable partition of local ground states are symmetry-related. In particular, this includes the case when some sets of this partition, instead of being r-symmetric, are reflection images of each other. In the following, the adjective "stable" will refer to property (18.23)(a)(ii). (19.7) Corollary. Suppose the hypotheses of Theorem (19.6) hold. Suppose further that for each 1 ^ n 5Ï N there exist an integer k„^l and a stable (possibly trivial) partition {G„A,..., G„tkJ of G„ into sets Gn( e Sc which can be transformed into each other by suitable transformations x e Ic A ( 0 , 0 1 , . . . , <&N1) with xG„ = G„. / / ß is large enough, the measures \xn in the conclusion of Theorem (19.6) can be decomposed into mutually singular Gibbs measures ßnA, ..., ß„tkn e r&(ß
412

Phase transitions without symmetry breaking

/C>A(
\^€^€'^k„

1 S

|ex^(/?a> hw )|^ £ Kn= l

Proof. Let ß be so large that the conclusion of Theorem (19.6) holds. Consider a fixed 1 ^ n ^ N. In analogy to (18.24) we have

{?(Gn,-)*0}=

U K°(G„.„-)^0}.

Properties (19.6)(ii) and (iii) thus imply that /xn(£°(Gny, •) ^ 0) = l/fc„ for all 1 S f S kn- We can therefore define the conditional probabilities By Theorem (7.7)(b), the /x„ / s belong to &(ßQ>hiß)). They evidently have the required properties; cf. the proof of Theorem (18.25). D We will now look at some specific models. (19.8) Example. The Blume - Capel model. Let E = {— 1,0,1} and X be counting measure. Consider the C-potential $ given by {i,j}<=C:\i-j\

=l

The local ground states of $ are the three constant configurations (++),("), and (oo)- (For the notation just used we refer to the beginning of Section 18.3).) The first two of these local ground states are related to each other by a symmetry of , namely the spin flip. However, there exist no ^-symmetries which map these two configurations to the third one. So we put

and G2 = G 2il U G2 2 . The partition {G1,G2tl,G22} partition {G1,G2} is resolved by the potential

is stable, and the coarser

*è = I of; ieC

cf. Comment (19.5)(1). Consequently, if ß is large enough then there exists some h(ß) e M with h(ß) -> 0 as ß -> oo such that \ex%(ß® + ßh(ß)®l)\ ^ 3 . As a matter of fact, Bricmont and Slawny (1986) have shown that a perturbation by O 1 is necessary to obtain a phase transition: For arbitrary large ß,

Potentials with degenerated ground states, and perturbations thereof

413

the unperturbed potential /M> only admits a unique shift-invariant Gibbs measure. This Gibbs measure is related to the ground state (go). O (19.9) Examples. The stabilized critical I sing antiferromagnet, and a lattice gas model. Here we will look at two models of similar behaviour. The first of these is a particular case of the Gertsik-Dobrushin model which has been considered in Subsection 18.3.3. We put E = { — 1,1}. Of course, A is counting measure. We fix a number J > 0 and let <5) e %>& be defined by a,. (i,j)cC \i-j\ = l

|i,j)cC \i-J\ = y/2

ieC

(In the case J = 0, the above Q) would simply be the antiferromagnetic Ising potential with external field h = 2. As we have seen in Subsection 18.3.2, this value of h lies on the borderline of the antiferromagnetic behaviour. The associated local ground states include the configurations (++), ( ++), (+*) and thus do not admit a stable partition. This instability is removed by the nextnearest neighbour interaction J.) In accordance with the phase diagram of the Gertsik-Dobrushin model (Subsection 18.3.3, Figure 3b),
*-{(:;K-{(:;)}•*•- {(;* The sets Gx and G2 = G2 , UG 2%2 are r-symmetric, and Comment (19.5)(1) shows that the partition {G l5 G 2 } is resolved by the potential ieC

1

(O is an external field.) Corollary (19.7) thus gives the following result. If ß is sufficiently large, there exists a real number h{ß) with lim h(ß) = 0 such that|ex^(/3 ^2,2 of the absolute ground states a>1 = 1, a» 2 ' 1 = ( ( - l ) ' 1 + ' 2 ~ 1 ) j e S , and a>2,2 = -co2,1, respectively. fiY is preserved by 0 , r0,rl,r2, whilst /z 2 ,i a n d ^2,2 a r e invariant under {9i:i1 + i2 even} and are related to each other by the one-step translations 0 ( O i l ) and öd.o) a s w e U a s the rotation r0 and the reflections r1,r2. The same kind of behaviour can be found in the following model of a lattice gas. We put E = {0,1}. Again, X is counting measure. Consider the C-potential Q> defined by
E

f f

r 2

X

J-Pjï c

GiGj if CjOj = 0 for all i,j e C

with |i-.71 = 1, otherwise.

414

Phase transitions without symmetry breaking

Here 0 < c < oo. ( should be regarded as an approximation to the hard core lattice gas model which is obtained by putting c = oo.) The local ground states of are (o°), (?o)> a n d (o?)- These have the same structure as the local ground states in the foregoing model. Corollary (19.7) thus implies the same type of result as in the first model above. In fact, it is not difficult to see that Corollary (19.7) can be extended so as to include the hard core case c = oo; cf. the remarks in Subsection 18.3.4. o The next and final example will be formulated in an abstract setting which includes Example (19.8) as well as some other particular cases of interest. (19.10) Example. Wells of the self-potential, and nearest-neighbour attraction. Let £ be a complete separable metric space with Borel cr-algebra S and X e J?{E,S) be finite and everywhere dense. Also, let u: E x E -> M be a symmetric continuous function which satisfies the attractivity condition u(x, y) > u(x, x) = u(y, y)

(x,yeE,x¥=

y).

Finally, let v. E -> U be a continuous function which attains its minimum at N 5: 2 distinct points x l 5 . . . , xN e E. We consider the C-potential O with
X {i,j}^C:\i-j\

M(oi,o))+ I f(o-j). =l

ieC

To have something definite in mind the reader can think of the case E = {1,..., N}, v = 0, and u arbitrary; this is a general version of the N-state Potts model. Alternatively, he can think of the case when E = Uk for some k, X is a Gaussian distribution, u(x,y) = \x — y\2 for all x, y e E, and v(x) = N

Y\ \x — x„\2 for all x and suitable x l 5 ..., xN e E; this case is a model of an n= l

anharmonic oscillator. Since x1,..., xN are distinct, we can find open sets C/ 1; ..., UN in E which are pairwise disjoint and respectively contain x l 5 ..., xN. We put G„ = C/„c, 1 S n ^ N. The family {G 1 ; ..., GN} then clearly is a stable partition of the local ground states . By Comment (19.5) (1), this partition is resolved by the potentials «)

(1 ^ « < AO.

ieC

Here v„: E -> U is any bounded function such that vn(x) = 1 when x e Un and u„(x) = 0 when x e Um for some m # n. Theorem (19.6) (as well as Theorem (14.17)) thus imply that for sufficiently large ß there exists a parameter h(ß) e UN~* of arbitrarily small norm as well as N pairwise distinct Gibbs measures / i 1 ; . . . , fiN e ex ^@(ß<5>h(ß)) such that H„({i e S: ate U„} is an ocean) = 1 and fi„(<J0 e U„) -> 1 as ß -> oo, for all n. (In the presence of partial symmetries we can obtain some further information on h(ß); see Example (19.8).) o

Exploiting Sperner's lemma

19.2

415

Exploiting Sperner's lemma

This section is devoted to the proof of Theorem (19.6). In fact, we shall derive a more general result which also applies to phase transitions different from those in Theorem (19.6). These will be studied in Section 19.3. Throughout this section we fix an integer JV > 1 as well as JV distinct r-symmetric sets G l 5 . . . , GN e Sc which satisfy the stability condition (18.23) (a)(ii). We set G = G1 U... U GN. (In the proof of Theorem (19.6), {Gu..., GN} will be a partition of the local ground states of the potential O under consideration.) We pose the following problem: Suppose we know that t

sup

M£°(G„,-)^0)>O

for all O in some subclass of (€@. How can we then ensure the existence of a O which makes each term in the above sum positive? Once we have solved this problem, we can substitute its solution for the symmetry argument in the proof of Theorem (18.25), and this will suffice to establish the existence of a phase transition. The covering lemma below provides a key to the problem which we just stated. This lemma is an immediate consequence of a classical topological result, the lemma of Sperner. We shall supply a proof of both lemmas in Appendix 19.A. Let (19.11)

0>N = | a = (a(n)) 1SnSJV e [0,1]": J ] a(n) = 11

denote the simplex of all probability vectors of length JV. &N will be endowed with its natural topology. (19.12) Lemma. Let {K1,...,KN} be a covering of &N with closed sets Kn satisfying K„f1{ae 0>N: a(n) = 0} = 0 for all 1 ^ n S N. Then n= l

Proof See (19.A4).

•

How can we apply this lemma to the problem which we have set? First of all, we need to find a family (vP°I)0(e;?.JV of potentials satisfying sup

^(É°(G, •) * 0) = 1

( « e ^

We then might attempt to apply the lemma to the sets K„ = Le9N:

sup

^°(Gn,

•) * 0) = 11

(1 ^ n ^ JV).

416

Phase transitions without symmetry breaking

Indeed, these sets form a covering of 0>N, and each a in their common intersection would give us a potential y with the desired properties. Unfortunately, the sets K„ fail to be closed in general, even when ¥ " depends continuously on a. This is because the sets {£°(G„, •) ^ 0} are non-local. To circumvent this difficulty we shall resort to the estimates of Section 18.1. These will enable us to localize the above ansatz for K„ in a suitable way. We will break the main idea into two steps, namely the two propositions below. As before, a topological statement concerning random fields will always refer to the =£? -topology. (19.13) Proposition. Let (1"*)a6^s be a family of C-potentials with index set SPx- Suppose for each a e f „ there exists a non-empty set T(a) a <ßB(x¥a) which enjoys the three properties below. (i) The set Y = {(a,//): a e SP^,\i e T(a)} is countably compact, i.e., every sequence in F has a cluster point in F. (ii) There exists a number 0 ^ c < 1/JV such that ju(0 e £°(G, •)) ^ 1 — c for all fie [j r(a). (Hi) If ae ëPN and 1 ^ n ^ N are such that oc(n) = 0 then H(0 e V(G„, -)|0 e V(G, •)) < l/N for all \i e T(a). Under these conditions, there exists an index a* e â?N and N Gibbs measures Hlt...,HNe %(V**) such that n„(Ç°(G„, •) * 0) = 1 and

/i„(0^°(G„,-))^l-JVc for all 1 ^ n ^ N. Moreover, if 1 ^ n ^ N and T e Tc x are such that rG„ = G„ and T(V) = v for all v e [j T(a) then T(/I„) = /i„. / / (E, S) is standard Borel then a

\ex%Ç¥'')\

^N.

Proof. 1) By hypothesis (ii), /i(0 e V{G, •)) ^ 1 - c for all \i e [j I » . For each such \i we can thus introduce the conditional probabilities " qn{n) = MO e V{Gn, -)|0 e V(G, •))

(1 ^ n g iV).

We consider the sets K„ = {ae 0»w: q„(/i) ^ 1/JV for some n e I » }

(1 ^ n ^ JV).

These sets satisfy the hypotheses of Lemma (19.12). Indeed, as T(a) # 0 for all a and ]T qn(/i) = 1 for all /i e T(a) it follows immediately that [j Kn = 0>N. n=l

n=l

Hypothesis (iii) just means that K„fl{ae 0>N: a(n) = 0} = 0 for all n. Finally, hypothesis (i) ensures that each K„ is closed. For let (a fc ) tâl be any sequence in Kn which converges to some a. Then there exists a sequence (fJ.k)kil such

Exploiting Sperner's lemma

417

that nk e T(afc) and q„(nk) ^ 1/N for all k. By (i), the sequence (/J-k)kil has a cluster point /leT(a). Since q„(-) is defined in terms of local events and ^fc(0 e V(G, •)) ^ 1 - c for all k, qn(n) ^ 1/N. Hence a e K„. As a result, we N

can conclude from Lemma (19.12) that (~) K„ # 0. iV

n= l

2) We pick any a* e f] Kn. For each 1 ^ n ^ TV, there is then a measure n= l

v„ e T(a*) with q„(v„) ^ 1/JV. In particular, v„(0 e V(G\G„, •) ^ v„(0 e V(G\G„, -)|0 e F(G, •)) ^ 1 - ^ _ 1 We put a„ = v„(0 e £°(G„, •)). The stability of {G 1 ; ..., GN} and hypothesis (ii) imply that an =

vn(OeÇ°(G,-),0$V(G\Gn,-))

^ (1 - c) - (1 - iV"1) = N" 1 - c> 0. Hence v„(<^0(Gn, •) # 0) > 0, and the conditional probability ^ = v n (-K°(G„,-)#0) is well-defined. As v„ and {<J°(Gn, •) # 0} are shift-invariant, ^„ e ^©(T**); cf. Theorems (14.5)(b) and (14.15)(b). Moreover, using the stability of {G l5 ..., GN} we obtain the estimate vn(£°(Gn, •) # 0) - an = v„(0 £ ê(G, - ^ ( G , , , •) # 0) ^ c and thus Lin(Oee(Gn,-))^aJ(an

+

c)^l-Nc.

The remaining statements follow in the same way as in the proof of Theorem (18.25). D Using the estimates of Section 18.1, we shall now provide sufficient conditions for the hypotheses of Proposition (19.13) to hold. For definiteness, these conditions will be stated in terms of the function t( •, • ) which we have defined in (18.8). However, the proposition below remains true when t(-, •) is replaced by the function t(-, •) of Comment (18.11) or any other function which can be substituted for t{-, •) in Lemma (18.10) and exhibits the semicontinuity property of Remark (18.9)(3). Recall from (18.13) that z(t) = 0(t) as t -» 0. (19.14) Proposition. Let (E, S) be a standard Borel space and ( ¥ ' ) , 6 ä ) i a family of C-potentials which satisfies the conditions (I) to (III) below. {I) The mapping a -> *Fa from &N to ^@ is continuous, in that H^ê" ~ *^cl! "-*• 0 whenever ock -> oc. (II) t = sup tiG,^) is so small that 4Nz{t) < 1. (///) For all(xe0>Nand\Sn^N

with a{n) = 0, t(Ec\G„, *P) < (1 - t)/N.

418

Phase transitions without symmetry breaking

Then it is possible to define a correspondence cc -> T(a) which satisfies the hypotheses (i) to (Hi) of Proposition (19.13) and is such that t(/i) = /i for all fi e [jr(a) and T G f)lCA(^). a

i

Proof 1) In view of condition (I), the sets {a'e^:||¥£-¥«c||
(ae&N)

are both open and closed and thus equal to é?N. If ||¥£|| = oo for some (and thus all) a then we let (Kf)/!^1 be a sequence in S which satisfies property (17.18)(iv) for some (and thus all) T". In the alternative case we simply put Ke = E for all /. In either case we conclude from (18.8) that lim t(Kf, ¥*) = 0 for all a e 9N. ('-»OO

2) For each a e SfN we let T(a) be the set of all jU e ^©(T01) which are such that (i) (ii) (iii) (iv)

n(D 0 V(G, • ) = 0) ^ t|D| for all De 6?, n(0 e V(Gn, •)) ^ t(£ c \G„, T a ) for all 1 g n g JV, /J(0 i V(Kf, •)) ^ t ( J v , ¥ " ) for all ( ^ 1, and T(/x) = ^for all r e f V c , * ( ¥ " ) . a

We will show that the mapping a -> T(a) satisfies the hypotheses (i) to (iii) of Proposition (19.13). First of all, we note that each T(a) is non-empty. In fact, T(a) => %0F"), the limiting set of all Gibbs distributions with periodic boundary conditions; recall the paragraph below (18.7). This inclusion follows from Lemma (18.10) and Remark (18.22). By Proposition (18.12), 90Ç¥") # 0. Hence T(a) * 0. Next, we observe that condition (19.13) (iii) is an immediate consequence of hypothesis (III). Also, hypothesis (II) and Lemmas (18.14) and (18.16) readily imply condition (19.13)(ii) (with c = 4z(t)). So it only remains to check condition (19.13)(i). Let ((<xk,nk))kèl be any sequence with a.k e &>N and fik e r ( a j . Since 3PN is compact, we can assume that (ock) converges to some a e 0 as k -> oo. Using Remark (18.9)(3) and Step 1) of this proof we conclude that lim sup iik(0 $ V{KC(, •)) ^ lim sup t(Kce, ¥" k ) fc-*oo fc-»oo

^ f(K£, ¥ a ) - • 0

as/->oo.

Comment (4.14)(2) and Corollary (4.13) thus show that the sequence (/ij k à i has a cluster point /i (in the if-topology). Looking at Proposition (4.19) and Theorem (4.17) we then see that \i e ^(*P). In fact, \i e T(a). Indeed, properties (i) and (iv) in the definition of T(a) and the shift-invariance of /i are obvious, whilst properties (ii) and (iii) follow from the upper semicontinuity of the

Exploiting Sperner's lemma

419

functions a -> f(£c\G„,xI'0') and a -> *(*:£, *F"); recall (18.9)(3). The proof is thus complete, n We shall now use the hypotheses of Theorem (19.6) to construct a family ^a)ae»N that meets the conditions (I) to (III) of Proposition (19.14). (19.15) Proof of Theorem (19.6). 1) Let U be an arbitrary open neighbourhood of the origin in UN~1. We need to show that for sufficiently large ß there exists a parameter h(ß) e U such that y@{ß<&hm) contains N distinct Gibbs measures that satisfy properties (19.6)(ii) and (iii). Let {Gx,..., GN} be as in the hypothesis of the theorem, and let e > 0 be so small that G£(* - O c || < a/4. heU

(This is because the perturbing functions £> • • • > ^ c * a r e bounded.) We can also assume without loss that U is small enough to satisfy conditions (i) and (ii) of Definition (19.4). The ^-image #(U) of U is then an open neighbourhood of the origin in dU+. Consequently, there exists a number ô > 0 such that ô < e/2 and

duN+n[o,2ôT <=?(V). In particular, the inverse mapping ^ _ 1 is defined and continuous on dU+ D [0,2ô"]N- We can thus define a continuous mapping a -> h(oc) from 0>N into U by the formula ?(h(a)) = 2<S(1 - ( max «(/c))" 1 «^))^^^. We put «P* = ySO>',(ot), a e ^ , where jff ^ /?0 for some ß0 > 0 which will be chosen in Step 2) below. It is then obvious that (xi>a)aE!f>N satisfies condition (I) of Proposition (19.14). Conditions (II) and (III) will be verified in Step 3) below. 2) Consider the sets Gh ô Ga(0*), heU. For each ß > 0 we let îp = sup t(G*, ß
By (18.8), f(G*,j80*) g

e~ßil2/Xc(Gsl2(&))

for all / j e t / . Hypothesis (19.4)(ii) thus ensures that tß —> 0 as ß —> oo. So we can select a number ß0 > 0 such that 4Nz(tp) < 1 for all ß ^ ß0. 3) According to our choice of e, [/, and Ô, the inclusion G* c GE(
420

Phase transitions without symmetry breaking

t± sup

tiC^^tß

and thus 4Nz(t) < 1. On the other hand, suppose a e ^ and 1 ^ n ^ N are such that a(n) = 0. Then pn(h(a)) = 2ô. Hence ^-inf,0O*<«>) ^ i„ < (1 - £„)/JV. The last inequality follows from the fact that (TV + \)tß < 4Nz(tß) < 1. Consequently, if ß 2: ß0 then the family (xi"')aie^K satisfies the conditions of Proposition (19.14) and thereby those of Proposition (19.13). We set h(ß) = h(a*), where a* is chosen according to the latter proposition. The conclusion of Theorem (19.6) then follows immediately, a

19.3

Models with an entropy-energy conflict

There is a common feature of almost all phase transitions which have been studied so far in this book: the phase transitions originate from a degeneracy of ground states and hence occur at all sufficiently low temperatures. The emphasis on this type of phase transition is justified by the physical interest in this standard case. However, there exist different phenomena which also merit study. In this section we shall discuss just such a different type of transition. Its characteristic feature is the coexistence of a high-temperature phase and one or several low temperature phases at a definite value of temperature. This phenomenon is closely related to what is called a first order phase transition. As we will show, it always occurs in the presence of a (sufficiently strong) conflict between energy and entropy. To give an idea of what we mean by the last statement we consider any C-potential ic(G1). Suppose further that for some reason the configurations in the complement of G0 U G1 are improbable. If ß is small, the entropy will dominate, in that any Gibbs measure for ßQ> will give overwhelming weight to the configurations in G1. For large ß the energetic advantage of the configurations in G0 will come into play, whence these configurations will be favoured by the Gibbs measures for ßQ>. At some intermediate value ß* we should achieve a balance of energy and entropy, giving rise to the existence of both a G0-phase and a G1 -phase. In Subsection 19.3.1 we shall prove an abstract theorem that makes the above ideas precise. Two typical applications will follow. In Subsection 19.3.2

Models with an entropy-energy conflict

421

we shall deal with a specific case of particular interest, namely the Potts model with a large state space. 19.3.1

Competing potential wells

Let (E, S) be a standard Borel space and X a finite a priori measure. For a given integer TV ^ 1, we consider an (TV — l)-parameter family {Q>h)hsaui °f bounded C-potentials which depend continuously on h (relative to the supnorm). (Recall that 3IR+ stands for the boundary of the positive octant in W. In particular, dU.\ = {0}.) We assume that each 0 (0 ^ n ^ TV), as well as constants 0 < a < b < o o , 0 < c < o o , and 0 < <5 < a such that the following assumptions hold: (Al) Shape of the wells. For all h e dRN+, A-infOj.

0, /L-inf
(1 ^ n g TV)

w„

and A-sup «De g ö, /l-sup <££ ^ a + hn + c v„ ^0 (A2) Barrier between the wells. For each h e

( l ^ n g TV).

C

X- inf (D* ^ b, Ec\W

where W = W0 U... U WN. (A3) Stability of the wells. For each face F of C, the projection images are °F(WO)> • • • > °>(Wtf) pairwise disjoint. One should note that condition (A) particularly implies that Gs(
Figure 19.1 An illustration of assumption (A)

422

Phase transitions without symmetry breaking

(19.16) Theorem. Suppose the above condition (A) holds. Suppose further that b is sufficiently large and AC(W0)/ min XC(V„) is sufficiently small; see (19.17) and (19.18) below. Then there exists a parameter h* e dUl and an inverse temperature ß* > 0 such that \ex%(ß*®h*)\ ^N + 1. Specifically, there exist N + 1 Gibbs measures / i 0 , . . . , /iw e %(ß*^h*) such that Hj£\Wn, •) # 0) = 1 and T(/I„) = Un far all x e 0 {Ic,,(<î>h): h e dUN+} with xWn = Wn,0^n^N. Proof. 1) We intend to apply Proposition (19.14) to a suitable family ( x F a ) 0[e ^ +i . To this end we first look at the quantity t(-,-); cf. Comment (18.11). The definition of t( •, ß
R

where the inf extends over all rectangles R in Sc. Letting R = V0 resp. Vn (1 Sn S N) and noting that min hn = 0 for all h e dUN+ we can conclude from assumption (A) that ^"-^ r(ß,h)^

Ußäßc(V0)^ 1 eßia+c)/ m i n

^c^ji/4

2) Next we specify the conditions on b and AC(W0). First of all, we pick a number £ > 0 with 4(N + \)z(t) < 1. (A glance at (18.13) shows that we can take any t ^ 3/(JV + 1)25.) We also fix a number ß0 > 0. We assume that b is so large that (19.17)

ßob

e~

^te-ßoia+c)

min Àc(V„)1/4/À(E).

We also assume that AC(W0) is so small that (19.18)

lc{W0fIA <(N + l)- 1 e-" o(a+c) (l - t) min lc{V„fIA.

Condition (19.17) and assumption (A2) immediately imply that t(WJ®h) S e-ßbX(E)r(ß,h) ^ t for all ß 7t ß0 and h e dUl. On the other hand, condition (19.18) ensures that t(Ec\W0,ß0Oh) for all h. Also, if 1 ^n^N t(E?\W„,ß&)

^ lc(W0)^r(ß0,h)

< (1 - t)/(N + 1)

then ^ Q~ß(a+K)A{E)r{ß,h) g e - « » + t.-«](£)/]C( F o )l/4

Models with an entropy-energy conflict

423

Consequently, we can find a number ß^ > ß0 such that t(Ec\Wn,ß®h)<(l-t)/(N

+ l)

when ß = /?! or hn = ß± and ß ^ ß0. As we will now show, the above estimates are the essential ingredients which allow us to apply Proposition (19.14). 3) Let a = (a(n))oènèN ->• (ß(ot),h(ot)) be a continuous mapping from &>N+l into[j3 o >0i] x (3R+ nCO,^]^) which is such that (i)0(a) = j?owhena(0) = 0, and (ii) ß(a) v /i„(a) = ßx when a(n) = 0 and 1 <^n <^ N. For example, we can take the mapping J8(a) = & + (ft - j30)a(0)/ max oc(fc), hn(a) = ßA max a(fc) — a(n))/ max a(fc),

1 S » ^ N.

(Since max a(fc) ^ (iV + l ) - 1 for all a G &N+1 •> the above mapping is evidently continuous.) We define a family 0F°% e ä V l in ^@ by *¥a = j3(a)0*(a), a e 0>N+1. In view of Step 2), this family satisfies the hypotheses (II) and (III) of Proposition (19.14) (with G„ = Wn and f(-, •) instead of t(-, •)). Hypothesis (19.14) (I) is an immediate consequence of the assumptions that the mapping h ->• O* is continuous and all O* are bounded. The theorem thus follows from Propositions (19.14) and (19.13). • To illustrate the preceding theorem we shall now look at two specific examples. The first deals with wells of self-potentials, whereas the second example is concerned with the case of wells of a nearest-neighbour interaction. For simplicity we confine ourselves to the case N = 1 of the theorem; that is, the parameter h is degenerate. (19.19) Example. Wells of a self-potential. For a given spin dimension p ^ 1 and a large constant r > 0, we let E= {xeW:

\x\ ^ 3r}

be the p-dimensional ball of radius 3r. X is the Lebesgue measure on E. We put u(x,y) = |x — y\2 (x,y e E) and v(x)

= (5 + \x\2) A K(|X| - 2r)2

(x e E),

where K ( ^ 8) is large and 0 < 5 < 6r2. We consider the C-potential O with self-potential v and nearest-neighbour interaction u. That is,

Oc=

£ {i,j}ŒC:\i-j\

«fa,rç) + I "fa)=l

ieC

To apply Theorem (19.16) to this O we set N = 1. (We thus can omit the parameter h.) We introduce the sets

424

Phase transitions without symmetry breaking

A0 = {xe E: (\x\ - 2r)2 ^ r2/i<}, A, = {x e E: \x\ ^ r}, B0 = {xeE:\x-(2r,0,...,0)\2^ô/6K},B1

= { x e £ : | x | ^ s},

where 0 < s :g r. It is easily verified that assumption (A) holds with the sets W„ = Acn, V„ = B^{n = 0,1) and the constants a = 40, b = r2, c = 20s2, and ô as above. We put t = 3/50 and ß0 = ( ^ r 1 log (51/50). A glance at (19.17) and (19.18) then shows that s should be chosen in such a way that e~20ß°s2sp is maximal. Hence s = (p/40ß0)1/2, and we require that r exceeds this value of s. With the abbreviation q = ß0r2, conditions (19.17) and (19.18) then take the form qP/2e~« ^ (p/360e) p/2 /17,

(2 + K~1/2)P - (2 - K " 1 / 2 ) P < (p/40qe)p/247/107. These inequalities certainly hold when r (and thus q) and K are sufficiently large. For example, if p = 1 and ô = 1/4 we can take r = 19.2 and K = 1.5 • 104, and if p = 2 and ô = 1/4 we can choose r = 24.1 and K = 1.2-108. For these values of r and K, Theorem (19.16) gives the following result: There exists an inverse temperature ß* ^ ß0 as well as two Gibbs measures tio,^! e %{ß*0) with n„(Ç°(W„, •) # 0) = 1 (n = 0,1)./ioand/ij are preserved by the lattice rotation r0 and the reflections r1,r2. Moreover, if p ^ 2 then /z0 and /i! are invariant under the spin rotation group SO(p); cf. Example (5.2) (3). (The last statement also follows from Theorem (9.20).) In the case p = 1, /i0 and /ij are preserved by the spin flip. As a matter of fact, if p = 1 then it even follows that |ex %(ß*(U)\ ^ 3. This is because the set A0 then consists of two intervals A% and Aô of distance > 2r, whence W0 can be replaced by (Ac )c U (AQ )C. The symmetry argument of Theorem (18.25) thus shows that n0 = (HQ + nö )/2 for two Gibbs measures ti,liö e %(ß*) with Mo±(^0((^0±)c, •) # 0) = 1. o (19.20) Example. Wells of a nearest-neighbour potential. Let p ^ 2 and E = {x e W: \x\ = 1} be the unit sphere in 1RP. X is the surface measure. We consider any C-potential $ of the form $c=

Z

Z

Moi-ffj)-

Here (p, i/^: [— 1,1] -> IR are two continuous functions, and the dot denotes the inner product. We stipulate that the nearest-neighbour interaction cp contains an energy barrier that forces any two adjacent spins to form an angle which is either obtuse or extremely acute. We also require that the lack of free play in the latter case is compensated by an energetic advantage. The nextnearest neighbour interaction will be needed for a stabilization of the spin behaviour: ip should be so attractive that the spins on C have to decide whether they prefer to form a ferromagnetic pattern ("all spins are extremely

Models with an entropy-energy conflict

425

close to each other") or an antiferromagnetic pattern ("any two adjacent spins form an obtuse angle"). To be specific, we choose three parameters e > 0,0 < ô < 1, and K > e and define (p(s) = e when — 1 <; s ^ —1/2, q>(s) = K + 2(K — e)s when —1/2 ^ s ^ 0, (p(s) = K when 0 ^ s ^ 1 — ô, 1 — ô for all i,j e C with \i — j \ = 1} and Wy = {co e £ c : co;• to; < 0 for all i, j e C with \i -j\

= 1}

be the sets of all ferromagnetic resp. antiferromagnetic configurations on C. A straightforward application of Theorem (19.16) then gives the following result: If K is sufficiently large and ô is sufficiently small, there exists an inverse temperature ß* > 0 with \ex%(ß*®)\ ^ 2. More specifically, %(ß*<&) contains two Gibbs measures which are respectively supported on the disjoint sets {Ç°(W„, •) ^ 0} (n = 0,1) and are invariant under the lattice rotation r0, the reflections r^ and r2, and the continuous group SO(p) of all spin rotations (cf.(5.2)(3)). o 19.3.2

Potts' model with a large state space

Let N > 2 be any integer. Later on, N will be assumed to be large. We set E = {1,..., N}. The a priori measure X on E is counting measure. We consider the Potts potential <ï> e ^ 0 which is defined by (19.21)

X

ô(<jh<jj).

{i,j}
Here [0

otherwise

is Kronecker's function on E x E. Clearly, the local ground states of <S> are the constant configurations on C. Accordingly, we introduce the singletons G„ = {(™)} (n e £), and we write G = Gt U... U GN for the set of all local ground states. We shall also consider the set G0 of all local ceiling states of Q>. By definition, G0 consists of all <x>e Ec which maximize
426

Phase transitions without symmetry breaking

are dominant at high temperatures. Of course, at low temperatures the local ground states will dominate because of their energetic advantage. The configurations in EC\G fail in both entropy and energy. This indicates that these configurations might be negligible at all temperatures. Since the partition {G0,...,GN} of G is stable, we are thus led to conjecture that at a suitable inverse temperature ßN there exist N + 1 distinct Gibbs measures that correspond to the sets G0,...,GN. The theorem below will show that this conjecture is true. In fact, we shall be able to characterize the critical inverse temperature ßN in terms of the pressure P. To this end we introduce the function (19.22)

ß^F0{ß)±P(ßQ)/4.

By Proposition (16.1), F0 is convex. Its left and right derivatives d^F^ and d+ F& are therefore well-defined. (19.23) Theorem. Consider Potts' model with state space {l,...,N}, and suppose N is sufficiently large. Then there exists a threshold value 0 < ßN < oo which is uniquely determined by the condition d-F9(ßN)£l/2£d+F9(ßN), as well as a number 0 < eN < 1 which tends to zero as N -> oo such that the following assertions hold. (I) When 0 ^ ß < ßN, there exists a Gibbs measure [i0 e ^&(ß(5)) which is invariant under r0, rx, r2 and all permutations of E and is "chaotic" in the sense that n0(Ç°(G0, -)*0) = land n0(0 e £°(G0, •)) ^ 1 - e*. (II) Given any ß > ßN, there exist N distinct Gibbs measures /x1, ..., fiN e ^@(ß(U) which are random perturbations of the absolute ground states. That is, fin(t°(Gn,-)^0)=land l-eN^nn(0e^(G„,-))^l

as ß -> oo

for all 1 ^ n ^ N. Moreover, if\-^m,n^N then fim is transformed into \in by each permutation of E which sends m to n. In addition, ii1,...,/iN are invariant under the rotation r0 and the reflections r^,r2. (III) The above-mentioned N + 1 types of Gibbs measures persist at ß = ßN: There exist N + 1 Gibbs measures (i0, ..., fiN e ^&(ßN(5)) which share the properties of the corresponding measures in (I) and (II). In particular, \ex«ßB{ßNQ)\^N

+ l.

As a matter of fact, some additional results are available. Using different methods, Laanait et al. (1986) showed that the theorem holds with ßN = 2" 1 log(l + y/N) for all N > 25. (This result is restricted to the case d = 2 under consideration.) Moreover, in arbitrary dimension d it is known that |#(jSO)| = 1 when 0 ^ ß < ßN, |ex^ @ (;VD)| = N + 1, and \ex%(ß<&)\ = N when ß > ßN; see Martirosian (1986) and the reference above. In other words,

Models with an entropy-energy conflict

427

there are no other pure phases than those described in the theorem. By the way, the theorem readily implies that cL F^ßx) < ô+ F^ißu) when TV is so large that % < 1/2. This discontinuity of the first derivative of the pressure is usually referred to as a first order phase transition. We now turn to the proof of Theorem (19.23). Clearly, an application of Propositions (19.13) and (19.14) would only give us part (III) of the theorem. Fortunately, we can avoid using Sperner's lemma because a more specific argument is available. This argument is based on the strict convexity of the pressure. As before, we let r0 denote the lattice rotation (18.20). (19.24) Lemma. Let N ^ 3. Then there exists a unique number 0 < ßN < oo such that d-F9(ßN)£lß£d+F9(ßN). Moreover, if 0 ^ ß < ßN then n(ai = er,) < 1/2 whenever \i — j \ = 1 and n e y&iß®) is r0-invariant. On the other hand, if ß > ßN then fi(at = er,) > 1/2 whenever |i — j \ = 1 and /i e ^0(/?) is r0-invariant. Proof. 1) First of all, we note that F9 is strictly convex. This follows immediately from Corollary (16.15) provided we can show that the potentials ß<S>, ß^O, are pairwise non-equivalent. Since O is equivalent to the potential

yA = \2N~l - WW) (0

iiA

= iÜh lJ' ~J\ = L

otherwise,

it is sufficient to prove that the potentials /?¥, ß ^ 0, are pairwise nonequivalent. But this follows from Theorem (2.35) because *F is normalized by the equidistribution on E. 2) Next, we combine Remark (16.6)(1) with Theorem (16.14). This yields d.F9(ß)£

-/4gS+F„(/ï)

for all /i e %(ß = -/i(
whenever /i e ^@(&,^) d-F9(ß)£n(
is r0-invariant and \i —j\ = 1. Hence =
for all ßeU, all r0-invariant /i e %(ß). Step 2) then implies that d-F»(0) g nip0 = (x(li0)) = 1//V < 1/2. On the other hand, Lemma (18.10) and Step 2) ensure that d+F9(ß) ^ n(a0 = (x(li0)) £ /i(0 e K(G, )) ^ 1 - t{G,ß
428

Phase transitions without symmetry breaking

for all ß ^ 0 and \x e %{ß%. Remark (18.9)(4) thus shows that d+F^ß) -> 1 as ß -> oo. The existence of ßN as well as its properties now follow readily from Steps 1) and 2). D In Theorem (19.16), we used an energy barrier to ensure that the configurations in EC\W are unlikely to occur; recall assumption (A2). In the present model there is no energy barrier which excludes the configurations in EC\G. Fortunately, we can now use an entropy argument to establish the unlikeliness of these configurations. (19.25) Lemma. Let tN = 7N~1/4. Then ti(Df)V(G,-)

=

0)^tW

for all De y and ne (J %{ßO). ßZO

Proof. 1) To begin, we introduce a classification scheme for the configurations in C. A nearest-neighbour bond {i,j} in C will be called excited for <x> e Ec if coi + coj. Accordingly, we associate to each a>e Ec its excitation pattern 7r(co) =

(^( f t ) (0,0). f f l (l,0))^( ( ( ) (l,0). ( ( ) (U))^( W (l,l). ( l , (0,l)Pho,l).%,0)))'

By definition, G = {coe Ec: n{co) = (0,0,0,0) or (1,1,1,1)}. We split EC\G into the following seven sets of configurations with a prescribed excitation pattern: M, = {TT(-) = (1,0,0,0)}, Mk = r*TiMl (k = 2,3,4), M 5 = {*(•) = (1,1,0,0) or (0,1,1,0)},

M 6 = r02M5,

M 7 = {7r(-) = ( l , 0 , l , 0 ) o r ( 0 , l , 0 , l ) } . (Note that excitation patterns with three one's are impossible.) Clearly,
X

n{oc+i e Mv(i) for all i e D)

C>:D-{1,...,7}

^ X lim sup "yZfa+i = L 11 tN,vU)-

G M„ w for all i e D)

Models with an entropy-energy conflict

429

In the above, A(w) = ] - n , n ] 2 n Z 2 , iN,k = limsupV*(^(«)) 1 / | A ( , , ) l n-*oo

Ak(n) = {(TC o r' o 0*$"> e Mk for all i e A(n)} c £ A(n)

(1 ^ fc £ 7),

n)

and 0^ and r' are given by (17.3) resp. (17.5) and (17.14). Consequently, the lemma will follow once we have proved that tNik ^ iV~1/4 for all 1 ^ k ^ 7. By rotational invariance, it is enough to investigate the cases k — 1,5, and 7. This will be done in Steps 3) to 5) below. We shall consider a fixed square A = A(«) as well as the events Ak = Ak{n). We shall also use the abbreviation °H* = X
for the periodic Hamiltonian which appears in (17.20). 3) The case k = 1. If co e At then 0C(i)(co) = — 1 for all i e A and thus °H%(to) = — |A| = —An2. Moreover,coiuj for all i, j e [0, n] 2 }. If co e B1 then <5C(i)(co) = - 4 whenever C{i) c [0,n] 2 . Thus °H^(co) ^ - 4 n 2 . Since | ß t | = jv 1+4 " 2_(B+1)2 , we end up with the estimate °yrUi)^

I

exp[-/TitfM]/ X

me Ai

exp[-rfl*(û))]

I meBi

^ l^il/IBJ
e

A: h = w, or i\ = u — 1 mod In and i2 = a mod 2, or J'J = u + 1 mod In and i2 = b mod 2}.

That is, co is constant on the union of the u-th column in A with all even (resp. odd) sites on the two adjacent columns. This fact provides us with the estimate \A5\ < 4^iV"+',2. On the other hand, there exists a set B5 of large cardinality such that °H%(co) ^ - 8 n 2 for all co e B5, namely B5 = {tu e EA: cot = coj for all i,j e [0, n] x ] — n, n]}. Since ]B5\ = jvi+2"(«-i>5 w e arrive at the estimate >r(^5)^i^5i/iß5i<4"iv3"-"2.

430

Phase transitions without symmetry breaking

Taking the 4n 2 -th root and letting n-»oowe obtain that tN>5 ^ iV"1/4. 5) The case k = 7. If co e An then °H®(œ) = — 8n2, and œ is constant either on each row or on each column in A. Hence \A-,\ < 2N2n and thus °yr(A1)S\A1\/\B5\<2N*»~2»2. This gives the bound tNil ^ iV~1/2. The proof is thus complete,

o

We are now ready for the proof of Theorem (19.23). (19.26) Proof of Theorem (19.23). Let N be so large that sN 4 8z(7iV_1/4) < 1. (From (18.13) we see that this condition certainly holds when N ^ 2-108.) Also, let ßN be determined by Lemma (19.24). We consider the cases ß ^ ßN and ß ^ ßN separately. 1) Suppose that 0 ^ ß ^ ßN. Then there exists a Gibbs measure n e ^0(/M>) with the properties below: (i) ft is invariant under r0 as well as all T e T which are induced by a permutation of £. (ii) n(D H F(G, •) = 0) ^ (7i\T1/4)lDl for all fle^. (iii) Ufa ^ o}) ^ 1/2 for all i,j e S with \i - j | = 1. Indeed, ii ß < ßN we can take any /i e (S0{ßQ>). This follows from Proposition (18.12), Remark (18.22), and Lemmas (19.25) and (19.24). In the case ß = ßN we let ft be any cluster point of any sequence (^„)„ àl with nne %((ßN — i)). The existence of such cluster points is ensured by Example (4.11) (2). Using Lemmas (18.14) and (18.16), we conclude from property (ii) that M0e£°(G,-))^l-%/2. Together with property (iii), this implies that MO e £°(G0, •)) = MO e £°(G, - U f c o * ffa.o)) £ (1 - eN)/2 > 0. (We have used the fact that the partition {G0, G} of G is stable.) Consequently, we can define the conditional probability fi0 = fj.(-\Ç°(G0, •) ^ 0). By Theorem (7.7)(b), n0 e &(ß
Appendix. Sperner's lemma

431

some n e %(ß<&) which is preserved by r0, r l5 r2 and all permutations of E and satisfies the conditions /x(£°(G, •) # 0) = 1, MO e £°(G, •)) ^ 1 - %• The partition { G ^ . . . ^ ^ } of G evidently satisfies the stability condition (18.23)(a)(ii), and the sets Gl5 ..., GN are related to each other by suitable permutations of E. The argument proving Theorem (18.25) thus shows that H can be decomposed into measures /xl5 ..., ßN which exhibit the required properties. This proves the remaining part of (III) as well as (II), except for the statement that /i„(0 e £°(G„, •)) ->• 1 as ß ->• oo. But this follows from Theorem (18.25); cf. Subsection 18.3.5. D

19.A

Appendix. Sperner's Lemma

Let N ^ 1 and N( = {a e 0>N:
be the rf -grid in i^jy. Two elements a, a' oîâPN^ will be called adjacent if there are two integers 1 ^ m < n ^ JV such that |a(m) — a'(m)| = |a(n) — a'(n)| = if-1 and a(/c) = a'(/c) when k ^ m,n.A subset A of ^ y will be called an n-clique if\A\ = n and any two distinct elements of A are adjacent, I ^ n ^ N. Clearly, each n-clique in @>N>e consists of the vertices of an (n — l)-dimensional simplex in the simplicial decomposition of &N which is determined by &Nti. In the following, it will be convenient to regard the set {1,..., N) as a set of colours. Accordingly, we introduce the term N -colouring for an arbitrary mapping cp: 0>Nj ->• {1,...,N}. Let us agree to call an iV-colouring cp admissible if oc((p(oc)) > 0 for all a e 3PNJ. In other words, cp is admissible if each a is painted with a colour in its support {1 ^ n ^ N: oc(n) > 0}. Finally, we will say that an iV-clique A N^ is panchromatic relative to
1 and suppose the lemma holds with N — 1 instead of N. We introduce the following four sets: the set <& of all TV-cliques A <= 0>N( with (pA => {1,..., N — 1}, the set % c ^ of all panchromatic TV-cliques, the set 3> of all (N - l)-cliques A <= 3PNJ with $_, = {a e 0>N(: a(JV) = 0} oî0>NJ.

432

Phase transitions without symmetry breaking

Since the induction hypothesis implies that \3>0\ is odd. We need to show that |# 0 | is odd. We shall prove this by establishing the identity (19.A2) \%\ + 2|
2\®\%\.

We claim that the right side of equation (19.A2) coincides with the sum (19.A3)

X De®

I

X(D,Q.

Ce«

Here m C )

fl ~ \ 0

ifDcC, otherwise.

Indeed, for each D e 3) the inner sum in (19.A3) is either 1 or 2, according to whether D is contained in a face of é?N or not. The former case occurs if and only if D e 3>0. For suppose D is contained in the face {a e â?n/- a (") = 0}Since cp is admissible, this is only possible when n <£ cpD. By the definition of 3>, we conclude that n = N and thereby D cz &$ with D c C, namely D = CXqT1 {N}. In the alternate case C e ^ 0 there exist precisely two distinct elements a 1( a2 e C with ^ ( a j = e of £?Nj in such a way that a e Kv(a) for all a e &Nif. This is possible because K1ö...'öKN = 0>N. The relations' Kn D {a e 0>N: a(n) = 0} = 0 (1 ^ n ^ N) then ensure that each q>e is admissible. By virtue of Sperner's lemma, each a as /c -> oo for all 1 ^ n ^ iV. By construction, a^, e K„ for all n and £. Since each K„ is closed, it follows that a e K i n . . . f l K J v . a

Chapter 20 Continuous symmetry breaking in iV-vector models

This chapter is devoted to the proof of the existence of low temperature phase transitions in spin systems with continuous symmetries. In contrast to the examples of phase transitions which have been considered in Chapter 18, the breaking of a continuous symmetry group at low temperatures can never be proved by a version of the Peierls argument. This is because in the presence of continuous symmetries there does not exist any useful concept of a contour. Consequently, the methods of proving a continuous symmetry breaking are necessarily completely different from those in the last two chapters. Nevertheless, we shall again use the chessboard estimate as an essential tool. Specifically, we shall be concerned with systems of spins which are located at the sites of a lattice S = Zd and take values in a Euclidean space RN. We shall always think of the case N > 1, although the case N = 1 is not excluded. We shall assume that the a priori measure X on RN is invariant under the rotation group SO(N). We shall assume further that the spins interact via a pair potential $ which is also invariant under SO(N). A spin system of this kind is usually referred to as an N-vector model or, if X is supported on the unit sphere, a classical Heisenberg model In the main theorem of this chapter, Theorem (20.15), we shall provide conditions on $ which imply that for all sufficiently large inverse temperatures ß the SO(iV)-symmetry of ßO is broken. Of course, the extreme boundary ex &(ß is a ferromagnetic nearest-neighbour potential. A guide to the proof of the above result is provided by the intuitive picture that the values of the spins vary continuously and more or less periodically over the lattice and thus form what is called a spin wave. In this picture, the existence of a long range order in the spin configuration is equivalent to the statement that large spin waves can only occur with a small probability. From a mathematical point of view, this picture suggests considering the low frequency part of the (spatial) Fourier transform of the spin-correlation function. It will turn out that this Fourier transform admits an estimate from above, the so-called infrared bound. This bound is the key to the proof of a phase transition. As a matter of fact, the Fourier analytic techniques of this chapter resemble in many respects those which have been exploited in the Gaussian case; cf. Section 13.3. Although the present chapter is formally independent of Chapter

434

Continuous symmetry breaking in N-vector models

13, it is worthwhile to compare its main theorem with that of Section 13.3, viz. Theorem (13.36).

20.1

Some preliminaries

Throughout this chapter, the parameter set S is Zd, the integer lattice of any dimension d ^ 1. The state space E is IR^, the iV-dimensional Euclidean space, or any rotationally invariant Borel subset of IR^. Although the case N = 1 is not excluded, we shall focus primarily on the more interesting case N ^ 2. Let A be a finite a priori measure on the Borel subsets of E. We assume that X is rotationally invariant. (In the case N = 1 this means that X is even.) We also stipulate that X is non-degenerate, in that A(£\{0}) > 0. To have something definite in mind the reader might think of the cases where E is the unit sphere in IR* and X is the surface measure, or E = IR* and X(dx) = exp [ — c|x| 2 + £ ] dx with positive constants c and e. Next, we introduce the interaction potentials which we are going to consider. These are the isotropic (or: Heisenberg) pair potentials $ of the form (20.1)

ibA

=

\ |

J

^ - ^ 0

a

i

ifA = . { ' • . . / M * ; . otherwise.

Here J : S —> M is any even function which satisfies 7(0) = 0 and the summability condition

(20.2)

||7|| 0 4 X |7(*)|
If E is bounded, each such $ belongs to the Banach space 3ft@ of all shiftinvariant and absolutely summable potentials. In particular, O is A-admissible, and the Gibbs specification y* is well-defined. However, if £ is unbounded and J has infinite range then the above even violates the defining condition (2.2) (ii) of a potential. This section is mainly devoted to the technicalities which are necessary to treat this case. A reader who is only interested in the case of a bounded state space E can (and should) skip the following paragraphs. He should continue with Lemma (20.8). Suppose E is unbounded. (Without loss we can assume that E = RN) We then impose the decay condition (20.3)

J X(dx)eaW2 < oo

for all a ^ 0

on X. We also replace (20.2) by the slightly stronger hypothesis that (20.4)

£ \J(i)\a(i) < oo. ieS

Some preliminaries

435

Here a(i) = (1 v log |i|) 1/2 (i e S). The above hypothesis clearly implies that Z \J(i - j)\z(i) < œ

(je S).

Consequently, the Hamiltonians H%(co), Ae £f, are well-defined whenever co belongs to the tail-measurable set Q,,, = <sup |<7il/a(i) < oo Moreover, for each ft) e Q„, and A e ^ there exists a number c(cos\A) < GO such that \H*(co)\ ^ c(coSXA) £ [1 + K | 2 ] . When combined with condition (20.3), the above inequality shows that Z*(to) 4 A A W > ) < oo for all A e y and co e £1^.. (Recall the notation (2.4).) We thus can define a specification y*" setting

(20.5) rfMM = I f «»r1 W * M » h « "' - "•• (KosxM) otherwise. Here A e ^ , y4 e J^, and m e f l , and <50At»sVA is the Dirac measure at the configuration which equals zero on A and co on S\A. Just as in (13.18), the tail-measurability of Q,,, implies that y® = (y^Ae^ is indeed a specification. Moreover,in analogy to(13.19)wealso have that n(Ci*) = lforall/ie ^(O) = ^(y">). In other words, the artificial part of the definition of y9 has no effect on the associated class of Gibbs measures. It is thus natural to call y*" the Gibbsian specification for O. The proof of the main result of this chapter will rest on the fact that the sequence (°y\in))„^ i of Gibbs distributions with periodic boundary conditions has a cluster point in ^(O). This follows directly from Comment (4.14)(1) and Example (4.20)(2) when E is bounded. In the alternate case we shall need the lemma below. For each n ^ l w e let M« — 7A(«)

X

°0sU(n)-

Here A(n) = ] — n,ri]dHS, 5 0 is the Dirac measure at the configuration in S\A(n) which is identically zero, and °y%(n) is the Gibbs distribution in A(n) for $ with periodic boundary condition. The AA-density of °y%(„) is given by formula (17.24). (Condition (20.3) readily implies that °y%{n) is well-defined.) As in Example (5.20)(3) we let %(
436

Continuous symmetry breaking in N-vector models

Q( = {|ff, | ^ tu(i) for all ieS}

(t > 0)

of slowly increasing configurations. Clearly, Q^ = [j Q t . t>o

(20.6) Lemma. Suppose X satisfies (20.3), and J is nonnegative definite relative to the reflections rx, ..., rd; cf. Definition (17.27). Then for each a > 0 there exist three constants b^a), b2(a), b3(a) > 0 such that the following conclusions hold for allne{nf:n^l}\J %{<&): (i) nie^^b^a) for allieS; («) ÀWi • 0)1 l{krayiät}) ^ MaOe""' ^ allt^O (Hi) n(Qt) ^ 1 - b3(a)e-a'2 for all t ^ 1.

and i,j e S;

Proof. Let A = A(n) be given. Theorem (17.29) and the chessboard estimate (17.11) imply that

/ C ( N 2 è 0 ;g ° Ä I

2

^ t for all j 6 A) 1 ^'

for all i e S and t ^ 0. To estimate the expression on the right we first look at the periodic partition function °Z* which was defined below (17.25). Using the inequality \œ{• coj\ rg |a>;|2 + |tOj|2, it is easy to see that °Z* ^ <5|A|, where <5^ j A ( d x ) e x p [ - | x | 2 | | J | | 0 ] > 0 . Thus "^(Iff/^fforaUjeA)1^! /

ô n

- ~ \lb

\1/|A|

2

ho^t}^pU^\ \\J\\0n

^ b(a)e~a'

for all t ^ 0 and a > 0. In the above, b(a) = Ô-1 j A(dx)exp [|x| 2 (a + || J|| 0 )]. By continuity, we can conclude that H(\°i\2 ^t)^

b(a)e-°'

whenever ieS, a > 0, t ^ 0, and ju e {/if: n ^ 1} U ^o($)- This estimate readily implies the lemma. For let /i be as above and a > 0. Then l<Til

/i(e"

f\"A2 \ ) = 1 + a/il j e a s d s ) GO

= 1 + a J e ÛS /i(|ff ; | 2 ^s)ds o ^ 1 + ft(2a) â b t (a)

Some preliminaries

437

for all i e S. This proves (i). Using the Cauchy - Schwarz inequality, we further conclude that *»K\
+ 1) 4 b2(a)

for all t ^ 0 and i,j e S. This gives (ii). Finally,

i - MQ,) ^ X MN ^ *«(0) ^ 6 ( a + d + 1) X (e v |,-|)-(-+"+i)'2 ieS

^ b(a + d + l)e- fl ' 2 X (e v |/|)" ( d + 1 ) ieS

^ fc3(a)e~at2 for all t ^ 1. This proves assertion (iii). D (20.7) Proposition. Suppose J is nonnegative definite relative to the reflections rl9...,rd, and let conditions (20.3) and (20.4) be satisfied. Then

Proof. For each o ^ l w e let y" = p"X. be the Gibbsian specification for the À-admissible finite range potential -JA(n)(i-j)

*'A=<

(Ti • Oj

-{jA(nM\°i\2 0

if A = {/, y ) c A ( / i ) , i ^ j ,

ifA = {i}c otherwise,

A(n),

where

• W O = Z A* + 2iy)

(» e S);

cf. (17.25). By definition, (if = yA(n)(-|0), 0 being the constant configuration with value 0. A straightforward calculation shows that lim sup sup p^(w) < oo for all A e if and t ^ 1. Lemma (20.6) (iii) therefore implies that the sequence (/C)nâ i satisfies the hypotheses of Theorem (4.12). We thus can conclude that

%m* 0. Next we consider an arbitrary (i e %(<&). It is clear that (i is shift-invariant; cf. Example (5.20)(3). A glance at the proof of Theorem (4.17) shows that (i e ^() provided we can show that ßniylf) - Mf)

for

all A e ^ and fe J?

as n runs through a subsequence (or, if you want to avoid Proposition (4.15),

438

Continuous symmetry breaking in N-vector models

a subnet) (nk) with /i* -• p. To check this we take any A e ^ and fe & with 0 ^ / ^ 1. We also pick an arbitrary e > 0. In view of Lemma (20.6) (iii), there exists a number t > 0 such that /i(Q,) ;> 1 - e and /i*(Qr) ^ 1 - e for all n ^ 1. By (20.4) and a short calculation, we can find a null sequence (ô„)ni 1 such that sup |fl?"-*(Cu> SVv )| ^ <3„ X [1 + ICI 2 ] for all £ e £ A and n ^ 1. Mimicking the proof of Proposition (4.19) and using hypothesis (20.3) we then obtain that U(yÂ/-yA7)inJI-o as n -» co. So we can find an integer m such that sup||(tf/-tf/)l0t||:ge and | | ( t f / - t f / ) l ^ | | £ e . Then

sup |/tf(tf/ - tf/)| S 2e nam

and |/z(tf/ - tf/)| ^ 2e. Since y ^ / e JS?, < ( y ^ / ) ? /*(TA/) a n d thereby limsup|<(y^/)-/i(y*/)|^4e. The proof is thus complete, D In the above, we detailed the technicalities which are necessary when E contains arbitrarily large elements. We shall now provide a lemma which deals with the case when E contains arbitrarily small elements. This lemma is trivially true (and thus can be skipped) when E is assumed to be bounded away from zero. We again consider the Gibbs distribution °y* on a cube A = A(n) = ] — n, n] d D S with periodic boundary condition; cf. (17.24). (20.8) Lemma. There exists a number ô > 0 which only depends on the a priori measure À such that

for each cube A = A(n) and each potential O of the form (20.1), provided J is nonnegative and also nonnegative definite relative to the reflections r 1; ..., rd; cf. Definition (17.27). Proof. As X is rotationally invariant and non-degenerate, there exists a Borel set U c E with k{U) > 0 such that u = inf x • y > 0. x.yeU

We put a = À({0})/À({0} U U) and let s > 0 be so small that s ^ u and

Spin wave analysis, and spontaneous magnetization

b±X{0<\-\

439

<e)/X{U)< 1 -a.

Theorem (17.29) and the chessboard estimate (17.11) show that °7A(0

< k 0 1 < e) ^ °y£(0 < \at\ < e for all i e A)1/|A| ^

CZ*)-i/|A| À(o

< | • | < £ ) exp[ £ 2 ||J||o/2].

Since °Z*^A(lDAexp[i

£

J A ( i - j ) u ] ) = {A([/)exp[u||J|| 0 /2]}l A l

we conclude that °y*(0 < |
20.2

Spin wave analysis, and spontaneous magnetization

Let J: S -» R be even and summable and $ the associated potential of the form (20.1). We intend to show that at sufficiently low temperatures /J -1 the SO(iV)-symmetry of ß® is broken. Of course, we need to impose some conditions on J. Since SO(N) is a continuous group (when N ^ 2), the results of Chapter 9 tell us that a breakdown of SO(N) is impossible except when d ^ 3 or J decays sufficiently slowly. In order to motivate what follows we consider the particularly interesting case where X is the surface measure on the unit sphere in RN and J(i) = 1 when \i\ = 1 and J(i) = 0 otherwise. (This case is called the ferromagnetic nearestneighbour Heisenberg model.) The absolute ground states of O are then the constant configurations co" = (u)ieS, u being any unit vector in RN. As we have seen in Example (18.19), there is a sense in which these ground states are approached by ^(/M>) as ß -» oo: If e > 0 is arbitrary and ß is sufficiently large then, with probability one relative to a suitable /J, e ^(/M>), a two-dimensional layer in S contains a random ocean £° which is such that \at —
440

Continuous symmetry breaking in N-vector models

phase transition we need to control the "infrared" part of the spin wave spectrum. We are thus led to investigate the (spatial) Fourier transform of the spin-spin correlation function. Let K = { z e C : | z | = l} denote the complex unit circle and G = Kd the d-dimensional torus. Note that G is the dual ( = character) group of the additive group S = Zd. Also, let n ^ 1 be fixed and A = A(ri) = ] — n,ri]dH S be the associated cube. As in Chapter 17, we think of A as a discrete torus. That is, we equip A with the coordinatewise addition modulo In. This operation turns A into an abelian group. Its dual group is GA = {weK:

w2n = l}d c= G,

the set of all 2n'th roots of unity in G. Note that \GA\ = | A|. For any function J which is admitted in (20.1) we let (20.9)

J{z) = £ z'J(i)

{zeG)

'

ieS

be the associated Fourier transform. (As in (13.32), z* is a short-hand notation for the product zî1 ...z'dd.) Since J is even and summable, J is real and continuous. Moreover, for each z e GA we have (20.10)

J(z) = X z'JA(i). ieA

Here, as in (17.25), •/A(0

= I ^ + 2n^)

(i e A)

is the A-periodized version of J which appears in the definition (17.24) of "y*, the Gibbs distribution in A with periodic boundary condition relative to the potential <S which is associated to J via (20.1). We are interested in the correlation function i -> °y*(a0 • O;) as well as its Fourier transform (20.11)

C*{z) = X zl ° y > o ^ )

(z e GA).

ieA

(If the state space E is unbounded, the existence of °yAI>(o'o ' &i) follows from Lemma (20.6) (ii).) The following estimate for C* is known by the name infrared bound. It allows to control the expected length of spin waves and thereby provides a key to the proof of a phase transition. Throughout the following we let 1 = ( 1 , . . . , 1) denote the neutral element ofG. (20.12) Proposition. Suppose, in addition to the general hypotheses of this chapter, J is nonnegative definite relative to the reflections r1, ..., rd; cf. Definition (17.27). Then 0 g C*{z) g N(J(l) - J(z))-i for all z e GA with J(z) =£ J(l).

Spin wave analysis, and spontaneous magnetization

441

The proof relies on the lemma below. This lemma is the only place in this chapter where we take significant advantage of reflection positivity. (Note that the applications (20.6) and (20.8) of the chessboard estimate are less important because they are trivial when E is bounded and bounded away from zero.) Consider the function /*(Ç,co) = exp

-ï

Z

W-JMi-tjl

(j'eA

A

C, co e (KT) .

(20.13) Lemma. Under the conditions of (20.12), O

^(/A*(C,-))^O^(/A°(0,-))=1

for all C e (UN)A.

Proof. Starting from formula (17.24) we obtain by a short calculation that In the above, exp[\x\2J(l)/2]k(dx),

k(dx) = g = exp -î

Z

^(»"-7)1^

i,jeA

and TC: (UN)A -» {UN)A is defined by TÇCO = (œ, - Ç ; ) ieA , œ e (UN)A. Consequently, in order to prove the lemma we need to compare the measure 1A with its Tç-image T^(XA). To this end we assume first that X is equivalent to the Lebegue measure, i.e., there exists a positive measurable function r on UN such that X(dx) = r(x)dx. The translational invariance of the Lebesgue measure then implies that x^{XA) has the density I l r(at + QlritJi) <e A

relative to XA. Hence WA*(C,

•)) = XA(g _n r(0l +

Qlr(ai)\k\g)

= > A 5 ( n / K + C i )/K0)(The last equality is a by-product of the calculation at the beginning of this proof.) By Theorem (17.29) and the chessboard estimate (17.11), the last expression is at most

442

Continuous symmetry breaking in N-vector models

je A

/

\1/|A|

\ieA

/

From the above we know that this product is equal to n °7A(/A*(CV))I/IAI, JeA

J

where Ç stands for the constant configuration with £/ = £,. for all i e A. But f\(C]> ' ) = 1 for all;'. This completes the proof of the lemma in the special case when X (and thus X) is equivalent to Lebesgue measure. The general case now follows by an iterated limiting procedure: As a first step, we extend the result to the case where X is absolutely continuous with respect to Lebesgue measure and supported on a compact ball B. (The density of such a X can be approximated by positive density functions in such a way that the dominated convergence theorem can be used to interchange limit and integration.) In a second step we extend the lemma to the case that X is any finite measure on B. This is possible because each such X is a weak limit of absolutely continuous measures on B. (The passage to the limit depends on the fact that f*(£, • ) and the exponent on the right side of (17.24) are continuous and bounded on BA.) Finally we let B increase to UN. This gives the lemma in the general case, a Proof of Proposition (20.12). 1) Let £ e (UN)A be given, and consider the function

(tem.

Lemma (20.13) asserts that q> attains its maximum at t = 0. Hence q>"(0) ^ 0, provided the second derivative q>" of q> exists. Clearly, the second derivative (d/dt)2f*(tÇ, •) exists and is given by gtf, -)/A*(t£ •), where

Ui-JMai-ajHti-tt-tKi-tjnl2

0«(C,-) = te I i,je

A

-* I

JA(i-J)\d-Cj\2.

i, je A

In particular, we conclude from Lemma (20.6) (i) that

°y*(sup\{d/dt)2ff{tÇ,-)\) 0. (This is trivial when E is bounded.) The mean value theorem and the dominated convergence theorem therefore imply that
for all C e (UN)A.

Spin wave analysis, and spontaneous magnetization

443

2) We now consider an arbitrary z e GA. Let e{ denote the basis vector of UN in direction £, 1 S t ^ N. Consider the vectors Ç = (^(z')e,f); 6A and r\ = (J'^(zi)e()ieA in (RN)A. (Me and J m stand for the real resp. imaginary part of a complex number.) The inequality

then implies that (20.14) °y«

j A (i-7)k'-^'l 2 .

^ i l

Here, a;y = at-e( is the (f'th coordinate of ah and the vertical bars |-| now denote the complex modulus (instead of the Euclidean norm on UN, as before). Inequality (20.14) readily implies the proposition. Indeed, the sum on the right of (20.14) is equal to

ï I ut - m - J-1 - tl->) i,jeA

= I

JA(i-j)(l-z^)

i,je A

= |A|(J(1)-J(z)); cf. (20.10). As for the expression on the left of (20.14), we note that

JAi-Moij-OvW-z1)

I I i,je A

JAU - j)a«zl(l

= I

- zj~l)

i,jeA

= (.7(1) - J(z)) X ouz\ ieA

We also observe that

°tf

=I -

ieA

|A| I z; °tf K ieA

i - j °^,*

'yÂK^v)

i,;eA ,<x0.,).

The sum over { of the last expression is equal to |A| C*(z). Combining all these facts with (20.14) we conclude that 0 ^ |A|(J(1) - J(z))2C®(z) S |A|iV(J(l) - J(z)). The statement of the proposition is now immediate, D We are now ready to state the main result of this chapter. Let us agree to call a function J: S -> R irreducible if there exists no proper subgroup of S containing the support {i e S: J(i) ^ 0} of J. Of course, there is no loss

444

Continuous symmetry breaking in N-vector models

in assuming that the function J in (20.1) is irreducible. For otherwise the Gibbsian specification y® is a product specification in the sense of (7.18), at least on H,, (cf. (20.5)). We let dz denote the normalized Haar measure on G; cf. the lines above (13.33). (20.15) Theorem. Suppose the a priori measure X is non-degenerate, rotationally invariant, and satisfies (20.3). Consider an isotropic pair potential O of the form (20.1), where J: S ->Misa nonnegative, even, irreducible function which satisfies (20.2) or, if E is unbounded, (20.4). Suppose further J is nonnegative definite relative to the reflections rl,...,rd (cf. (17.27)), and its Fourier transform satisfies the integrability condition (20.16)

J (J(l) - J(z))" 1 dz < oo. G

lfß>0 is sufficiently large, there exists a number c(ß) > 0 and a family {nu: u e UN, \u\ = c(ß)} in ex %(/?) such that ßu{o0) = u for all u. In other words, at sufficiently low temperatures there is spontaneous magnetization and thereby a breaking of the SO(N)-symmetry. Postponing the proof for a while, we first comment on the hypotheses of the theorem and then provide some examples. (20.17) Comments. (1) As J is nonnegative, even, and irreducible, its Fourier transform satisfies J(z) < J (I) for all z ^ 1. The integral in (20.16) is thus well-defined. Of course, we can assume without loss that ||J|| 0 = J (I) = 1. By a criterion of Chung and Fuchs (1951), condition (20.16) is then equivalent to the statement that the random walk ( X ^ à l on S with transition matrix (J(i —j))ijEs i s transient. More precisely, let W = SN, iV the usual
v^Xl^oj^oo,

i.e., the expected number of visits to the origin is finite. By definition, this means that the random walk (X,),^0 is transient. To verify the equivalence of (20.16) and (20.18) we pick any 0 < s < 1. Then we can write X s'v,(^ = 0)= I

s'v,(Jz*'dz)

= X s
Spin wave analysis, and spontaneous magnetization

445

= J (1 - sJ(z))"1 dz. In the limit s f 1, the last integral tends to j ( l — J(z)) _1 dz. This follows from Fatou's lemma and the inequality (1 - sJ(z))- 1 ^ s-^l -

J(z))-\

The integral in (20.16) thus coincides with that in (20.18). (As has been discussed in Comment (13.28), the condition of transience is also fundamental in Gaussian models.) (2) Condition (20.16) is always satisfied when d ^ 3. Indeed, it is not too difficult to show that for each nonnegative irreducible J there is a constant c > 0 such that J(l) - J(zp) ^ c\p\2 for allp e ] — 1, l] d , where zp is defined by (13.33). (A proof of this fact can be found in Spitzer (1976), Section 7, P5; cf. also Example (20.20) below.) This inequality implies (20.16) because the function p->|p|~ 2 is Lebesgue integrable over ] — 1, l ] d when d ^ 3. (3) What happens when J fails to be nonnegative? (The extent to which this is possible has been discussed in Comment (17.28)(3).) The conclusion of Theorem (20.15) can then be shown to remain valid, provided its hypotheses are sharpened as follows. The support of X should be bounded away from zero, and (20.16) should be replaced by the following, slightly stronger, integrability condition of Riemann-type: (20.19)

liminflAMr 1

X

(J(l) - J(z))" 1 < oo

(Here, as before, A(n) = ] — n, n\d 0 S, and in the case J(z) = J (I) we use the convention 0 _ 1 = GO.) Of course, the difference between (20.19) and (20.16) is primarily aesthetical. In specific models it will be just as easy to check (20.19) as (20.16). Incidentally, we shall see below that the proof of Theorem (20.15) is simpler when (20.16) is replaced by (20.19). o Here are some specific cases to which the theorem (or the last comment) applies. (20.20) Example. Short-range interactions in three or more dimensions. Let d ^ 3 and a, b be any two real numbers such that a ^ 2(d — i)\b\ and c = a + 2(d- l)(b A 0) > 0. Let J be defined by a J(i) = ^ b 0

if|i| = l, if|i| 2 = 2, otherwise.

446

Continuous symmetry breaking in N-vector models

Q>, as given by (20.1), is then a ferromagnetic (or, if b < 0, predominantly ferromagnetic) next-nearest neighbour potential. (We emphasize that the ferromagnetic nearest-neighbour case a > 0 = b is included.) In Example (17.31) we have shown that J is nonnegative definite relative to the reflections r1,...,rd.To apply Theorem (20.15) (or, iffo< 0, Comment (20.17) (3)) we thus need only prove that J satisfies conditions (20.16) and (20.19). For each p e ] — 1, l ] d we let zp e G be defined by (13.33). Then

d

= 2a Y, cos npk k= l

+ 2b

C cos n(Pk + Pi) + c o s APk - Peï\

Y

d

= 2a Y,

cos n

Pk + 4è

k=l

Y

cos n

Pk c o s nVt

lgk
and thus J(l) - J(zp) = 2a Y (1 - cos npk) k

+ 2b Y, (1 ~~ cos7rpk)(l + COS7TP,,) k±(

= 2 £ ( 1 -cos7rp k ) a + b Y (1 + c o s nP() k

t +k

^2c£(l

-cosnpk)

k

^4c\p\2. The last inequality was proved in Example (13.43). Since A ^ 3, the function p -> |p|~ 2 is integrable over ] —1, l] d . This implies (20.16) and also (20.19). Theorem (20.15) and Comment (20.17) (3) thus give the result that there is a continuous symmetry breaking in predominantly ferromagnetic next-nearest neighbour A/-vector models at low temperatures, provided the lattice dimension d is at least three. (Recall from Theorem (9.20) and Comment (9.34)(2) that the last proviso is necessary.) o (20.21) Example. Long range interactions in dimensions one and two. Let d = 1 or 2 and J(,) =

|o

in = o.

Hypothesis (20.2) resp. (20.4) requires that a> d. If a^2d

and N ^ 2, the

Spin wave analysis, and spontaneous magnetization

447

above J does not allow for a breaking of the SO(./V)-symmetry, at least when E is bounded; cf. Example (9.23) and Comment (9.34) (2). On the other hand, in the case d < a < 2d there is a breaking of the S0(./V)-symmetry at low temperatures. Indeed, from Example (17.32) we know that J is nonnegative definite relative to the reflections rlt ..., rd. Moreover, in Examples (13.44) and (13.45) we have shown that there exists a constant c > 0 such that J(l) - % ) =

X

\i\-a(i-cosnp-i)^c\Prd

0#ieS

d

for all p e ] - 1, l] . (zp is given by (13.33).) For a < 2d, the function p -> |p|d"° is Lebesgue integrable over ] — l , l ] d . So, if d < a < 2d then the above J satisfies all hypotheses of Theorem (20.15). o Now we supply the proof of Theorem (20.15). Proof of Theorem (20.15). 1) Let J be the er-algebra of all shift-invariant events; cf. (14.2). We intend to show that for sufficiently large ß there exists some [i e ^@{ß
2

li{\li(G0\J)\

)>Q.

(Of course, fj.(a0\cf) = (/i(er0tA^))i^tiNi where er0y is the rf'th coordinate function of the [R^-valued function er0.) This will prove the theorem. Indeed, Theorem (14.10) ensures the existence of a (^©(Q, #"), ^-kernel 7r in the sense of Definition (7.21). Inequality (20.22) can thus be rewritten in the form ju(| 7r"(cr0)|2) > 0. On the other hand, Theorem (14.17) asserts that n e ex^@{ß(^) jU-almost surely. Consequently, there exists an œ e Q. such that n"3 e ex%{ßO) and nm(a0) + 0. We put v = 7r°> 0 ), c(ß) = \v\, and fi" = nm. For arbitrary u e UN with \u\ = c(ß) we let nu be the image of (iv under any rotation which maps v to u. Remark (5.10) as well as the observations in the paragraph above (14.11) imply that all these ju„ belong to ex %(ß<&). They are pairwise distinct because jU„(er0) = u. This is the desired result. 2) We shall now use the infrared bound (20.12) to establish inequality (20.22) for a suitable /j. e %{ß<&) when ß is large. As a matter of fact, the infrared bound can be used in a more direct way when the integrability hypothesis (20.16) is replaced by the stronger condition (20.19). Therefore we will first look at the case when the latter condition holds. (The proof of (20.22) under condition (20.16) will follow in Step 3 below.) Let ß > 0 and A = A(n) = ~\ — n,ri]dC\S be given. Applying Proposition (20.12) to the function ßJ and using the orthogonality of the mappings z-* zl (i e A) on GA we obtain for each j e A

°ynkol2)-0yn*o-°>-) = iAr i zeG

(1-Z--0CH*) A

448

Continuous symmetry breaking in N-vector models

^lAI" 1 X

{\-ms(z-'))Nß-\J(\)-J{z)Y"

Z6GA

g2Nr1|A|-1

(J(l) - J(z))"1.

I l*zeGA

In view of (20.19), there exists a finite number c such that the last expression converges to cß~* when A runs through a suitable subsequence of the sequence (A(n))„ âl . By (the proof of) Proposition (20.7), we can find a subsequence (or, if we don't want to use Proposition (4.15), a subnet) (A(nk)) of the above subsequence such that °yA*k) converges towards some /i e y@(ß<&). (If E is bounded, this follows already from Comment (4.14)(1) and Example (4.20) (2).) Using Lemma (20.6) (ii), we obtain by an obvious truncation argument that H(a0 • oj) = lim °y£*k)(o0 • a}) k

for all j e S. (Again, this is trivial when E is bounded because then a0 • a-} is bounded.) By the same reasons, /x(|c0|2) < oo. Finally, if the support of I is bounded away from zero or J ^ 0 then there exists a constant S > 0 such that °yA*(kol2) ^ <* f o r a11 A a n d ßl cf- Lemma (20.6). W e thus end up with the following result: There exists some ß e ^0(/?) such that /x(|
ii(a0-a})^o-cß-1 for all j e S. By virtue of the L 2 ergodic theorem (14.A3), the last inequality implies that K\ti
^ <5 -

X

j£[-m,m]d

MCTo • ffj)

cß-1.

This gives (20.22) whenever ß is so large that ß > c/ô. In view of Step 1), this result completes the proof of Theorem (20.15) under the integrability condition (20.19). 3) We will now show how inequality (20.22) can be derived from the more elegant integrability condition (20.16) when J is nonnegative. Since any factor of J can be absorbed into ß, we may assume that 1(1) = 1. We then can look at the random walk (Xe)t^0 with increment distribution J; the underlying probability space (W, iV, Vj) was specified in Comment (20.17)(1). As we have shown there, the integrability condition (20.16) is equivalent to the assertion (20.19) that (X()(^0 is transient. We consider an arbitrary cube A = A(n) = ] — n, n\d D S. Since \J\ ^ 1, the infrared bound (20.12) (as applied to ßJ instead of J) implies that

Spin wave analysis, and spontaneous magnetization

(1 - J(zY)Cr(z)

^ (1 - J{zY)Nß~\\

449

- J(z))- 1

= Nß-i ' £ J(z)k k=0

for all / ^ 1 and z e GA. (The case J(z) = 1 in which (20.12) does not apply is trivial.) To exploit this inequality we introduce the random variables X( A = X( mod A, / ^ 1. More precisely, for each w e W we let X^ A(w) be the unique element of A such that Xe>A(w) = X^(vv) + 2ra for some i e S. We will examine the random variables ff

xtJyv, co) = coXfAw)

(weW,coeQ,S^l)

on (W x Q, iV x J^). We will also think of a0 as a random variable on W x Q by putting <70(w, co) = co0. The inequality above then gives the estimate v, x M V o - ^ J = I

v,(X,,A = i ) ° ^ V o - ^ )

ieA

= X |AP1 X z'-^(X,,A = 0°^>o-^) zeGA

1

= lAr i ^rcHz) zeG

A

_

^lAI" 1 Z

a*(z) - Nß-i x 7(z)' k=0

Z E C .

^ H k o l 2 ) - ^ " 1 I vJ(xt>A = o) k=0

^«5-iV/T1 I

v,(Xt = 0 o r X k £ A ) .

k=0

The above (5 is the positive number which is provided by Lemma (20.8). Now let n be any cluster point of the sequence {°y^n))„^. Proposition (20.7) ensures that such a ß exists and belongs to %(ß<&). Applying Lemma (20.6) in the same way as in Step 2) above, and using the fact that Vjft, A = X( for sufficiently large A) = 1, we conclude from the above estimate that v, x ß(a0 • ax) ^ Ô - 7V/T1 ' l Vj(Xk = 0)

for all if ^ 1. (Of course, aX((w,co) 4 coXAw).) In view of (20.18), the expression on the right is at least ô — ß~l c, where c is a finite constant. By Lemma (20.6), n(\a0\2) < °o. The random ergodic theorem (20.23) below thus will imply that

450

Continuous symmetry breaking in N-vector models H(\H(
=

Vjx

n(a0-n(a0\J)) L-l

= lim I T 1 £

V

J x P(ao • °x)

ß-xc.

^ 0 -

Consequently, if ß is large enough then \i satisfies inequality (20.22). The proof is thus complete, D The ergodic theorem along random paths which has been used above reads as follows. (20.23) Proposition. Suppose J is an irreducible probability vector on S = Zd, and let {W,iV,Vj,(X^Z0) be the associated random walk; cf. Comment (20.17)(1). Also, let (Q, #",//) be an arbitrary probability space which is equipped with a family 0 = (ö ; ) ieS of ^-preserving transformations such that Ö; o 9j = 9i+jfor all i,j e S. Let J denote the a-algebra of all ©-invariant events in 2F. Finally, let f be any fi-integrable function on Q. Then lim L" 1 L f / o 0Xtlw)(a>) = n(f\f){a>) L->oo

for Vj x fi-almost all (w, a>) e W x Q and in the L}(\j x n)-norm. If fi(\f\2) < oo, the convergence also holds in the L2(vj x fi)-sense. Proof. 1) Consider the transformation T(W, CO) = (tw, 6Wico)

(weW,ft)eQ)

of W x Q. Here t: W -> W is the left-shift, i.e., (tw)k = wk+l for all k ^ 1. Also, let g: W x Q -» [R be defined by #(w, cu) = f(o>), we W, œ e Q. Then / ° öx,(w) = f° öWl + ...+W/ = / ° % - i w ) l ° ••• ° öWl = 9 ° *'(w> •) for all w e W and *f ^ 1. On the other hand, the product measure v7 x \i is T-invariant. Indeed, for aü A e if and ß e ^ w e can write VJXM(I/1XBOT)=

X

v ^ ^ i j n r M ^ M )

= Vj x /z(yl x ß ) .

In the second step we have used that v, is t-invariant and t _ 1 A is independent of Xt relative to v7. As a result, we can apply the classical (one-dimensional) ergodic theorem; cf. Appendix 14.A or Breiman (1968), e.g.. This theorem implies that the limit

Spin wave analysis, and spontaneous magnetization

451

g à lim L" 1 L f g o t ' = lim L" 1 ^ / o Ö exists Vj x /i-almost surely and in Ll(vj x /i)-norm. If/i(|/| 2 ) < oo, the convergence is also in the L2(vj x /i)-sense. 2) It remains to identify g. Of course, 3 is z-invariant. In fact, the convergence in Ll(vj x /i) immediately implies that g is the conditional expectation of g relative to v, x /i and the a-algebra of all i-invariant events in W x Q. On the other hand, for any fixed œ e Q. the function w -> g(w, co) is invariant under all permutations of finitely many coordinates of w. (This is because each such permutation affects only finitely many X/s.) The zero-one law of Hewitt and Savage (cf. Example (7.16)) therefore implies that g~(-,œ) = f(œ) = Vj(g(-,œ)) Vj-almost surely. To complete the proof we thus only need to show that / = n(f\J) /i-almost surely. To this end we first prove that / is ©-invariant. Take any œ e fi, and let ie S be such that J(i) > 0. Then vJ(X1 = i) > 0, and Vj-almost surely on {Xi = i] the identity ROM = äitmA<») = go T(-,to) = g(-,œ) = f{w) holds. Hence foQi = J when J(i) > 0. As J is irreducible, it follows that / o 9t = / for all i e S. In other words, / is ./-measurable. Finally, we let A e J be any ©-invariant event. Then W x Ais i-invariant, and this implies that M / U = Vj x n(glWxA) Hence / = n{f\J)

= Vj x n(glWxA)

=

n{f\A).

/i-almost surely, and the proof is complete.

•

Bibliographical Notes

Besides references to those papers which have been used as sources, these notes also contain some historical remarks and a number of references to related work. These remarks and references, however, are not intended to provide an exhaustive account of the literature on Gibbs measures, and they are clearly a reflection of the background and personal taste of the author. Introduction The foundational problems of Statistical Mechanics (which are not even touched upon in the text) are thoroughly discussed in the following books and articles. First of all, there is the classical monograph by Gibbs (1902). Two further classical references are Khinchin (1949) and Kac (1959). The more recent literature on the foundations of Statistical Physics includes the lectures of Lanford (1973) and Martin-Löf (1979a) and the article of Penrose (1979). A historical account of some fundamental ideas can also be found in Thompson (1972), Wightman (1979), and Gross (1982). In addition to these sources we mention a few recent textbooks on Statistical Mechanics, namely Reichl (1980), Toda et al. (1983), and Ma (1985). These books contain, in particular, a physical treatment of various models which are also considered here. Chapter 1 Section 1.2. As stated in the text, the concept of a Gibbs measure was introduced independently by Dobrushin (1968a), and Lanford and Ruelle (1969). The abstract formulation in terms of probability kernels was developed later by Dobrushin (1970), Föllmer (1975a), and Preston (1976). In particular, the term specification is due to Preston. In view of the consistency property of a specification one is led to ask whether each random field is a Gibbs measure relative to a suitable specification. For sufficiently well-behaved state spaces (£, S\ this question was answered in the affirmative by Goldstein (1978), Preston (1980a), and Sokal (1981). (In the context of quantum field theory, a similar result was obtained by Röckner (1986).) Section 1.3. In Proposition (1.30), the equivalence of (a) and (b) was already noticed by Dobrushin (1968a); statement (c) appeared first in Preston (1980a). Theorem (1.33) is a direct extension of a remark of Dobrushin (1968a). Flood and Sullivan (1980) studied the more general problem of whether a specification is uniquely determined by a suitable, not necessarily positive, subspecification. General references. There is now quite a number of books and introductory texts on the theory of Gibbs measures. The state of the art just before the invention of Gibbs measures is described in a well-known monograph of Ruelle (1969). The first texts discussing Gibbs measures were Spitzer (1971b, 1974), Georgii (1972,1974), Lanford (1973), Preston (1974a), and Griffeath (1976). A stage in the development of the theory was marked by the appearance of the monographs by Preston (1976), Gruber,

454

Bibliographical Notes

Hintermann and Merlini (1977), Ruelle (1978), and Israel (1979). Preston's book provides an abstract theory in a general measure theoretic setting, Gruber et al. concentrate on specific methods for Ising-type systems, Israel deals with the variational principle and related topics, and Ruelle's emphasis is on the use of the Gibbsian formalism in Ergodic Theory. Later on, Kindermann and Snell (1980) wrote an elementary introduction which concentrates on the case of nearest-neighbour interaction. In addition, Sinai (1982) gave an excellent account of some pearls of the theory. A great variety of concepts and results is summarized in a recent book of Prüm (1986). A nonstandard analysis approach to Gibbs measures can be found in Albeverio et al. (1986). A two-volume monograph by Simon (forthcoming) will appear in the future. Chapter 2 Section 2.1. Gibbsian specifications were introduced by Dobrushin (1968b, c, 1969), and Lanford and Ruelle (1969). It was also Dobrushin who initiated the study of the non-uniqueness problem and proposed its interpretation as a phase transition. However, we remind the reader of the fact that the non-uniqueness phenomenon does not, by any means, exhaust the totality of all possible phase transitions. One prominent example of a phase transition without non-uniqueness of the (shift-invariant) Gibbs measure is the Kosterlitz-Thouless transition which was mentioned in the text. A further example is due to Gruber (1976). Concerning the relation between Gibbs measures and various other concepts of an equilibrium state for an infinite system (including Kirkwood-Salsburg-type equations for the correlation functions) we refer to Lanford and Ruelle (1969), Brascamp (1970), Gruber and Lebowitz (1975), and Fannesetal. (1982). Section 2.2. The notion of a quasilocal specification appeared first (in a more particular setting) in Dobrushin (1968a). The non-quasilocal specification of Example (2.27) belongs to a class of specifications which arise in the study of canonical Gibbs measures. These are defined by replacing the Gibbs distributions in (2.9) (which, in the lattice gas interpretation, are called grand canonical) by the so-called canonical Gibbs distributions. The theory of canonical Gibbs measures and, in particular, their relation to the Gibbs measures in the sense of (2.9) is described in Georgii (1979). Section 2.3. The concept of an a-normalized potential was introduced by Israel (1979) as a common generalization of the (physically natural) notions of gas and spin potentials. The representation theorem (2.30) was proved over and over (partly independently, partly in increasing generality) by many authors, including Averintsev (1970, 1972, 1975, 1978), Spitzer (1971a), Besag (1972), Grimmett (1973), Preston (1973), Sherman (1973), Sullivan (1973a, b), Kozlov (1974), Moussouris (1974), Suomela (1976), Evstigneev (1978), Speed (1979) and Leha (1983). Most of these papers deal with the case where E is finite or countable, S is the integer lattice or a simple graph, and O is a nearest-neighbour gas potential. Our proofs of (2.30) and (2.32) are essentially those of Sullivan (1973b)/Kozlov (1974) and Grimmett (1973). A comparison of different proofs can be found in Moussouris (1974) and Speed (1979). The above papers also contain results which are not covered by (2.30) and (2.32): Kozlov (1974) showed that if ||logp A || < oo for all A then p can be represented by an absolutely summable (but not necessarily normalized) potential. (A similar result was also obtained by Sullivan (1973b).) Moussouris (1974) and Averintsev (1975) weakened

Bibliographical Notes

455

the positivity assumption on p, Averintsev (1978) investigated the conditions under which the potential <1> in (2.32) can be chosen to be a pair potential, Evstigneev (1978) translated the representation problem into an abstract question on finitely additive measures, and Leha (1983) weakened the hypothesis of quasilocality by allowing for potentials with a tail dependence. Many authors also noticed that if S = Zd and p is shift-invariant then <5 is also shift-invariant; we state this fact in (5.9). Finally, we note that Corollary (2.32) proved to be useful in other contexts even when S is finite: It was applied to statistical models of contingency tables by Darroch et al. (1980), and to the restoration problem for digital pictures by Geman and Geman (1984); see also Prüm (1986). Section 2.4. The equivalence of potentials in the sense of (2.33) is often called "physical equivalence". For shift-invariant, absolutely summable potentials over S = Zd there exists a further concept of physical equivalence which appears, for example, in Ruelle (1978); this concept is suggested by the variational principle (15.39). Under the conditions which ensure the implication (2.34)(iv) => (i) one can easily show that this concept of physical equivalence coincides with that of Definition (2.33). Supplement: Correlation inequalities for ferromagnetic systems. Much progress in the theory of Gibbs measures is based on the assumption that the interaction potential is ferromagnetic, i.e., that the spins attract each other. This is because under this assumption one can derive various extremely useful integral inequalities which provide information on the spin-spin correlations and their monotone dependence on the interaction parameters. These inequalities are referred to as correlation inequalities. In spite of its importance, the field of ferromagnetic systems is not discussed in the text; to do so properly would require an entire book. To atone, in part, for this sin of omission we will at least provide some references. We shall concentrate here on the main inequalities and a few applications. Further applications of these and other inequalities will be mentioned later in these notes. The condition of attractiveness can be specified in several ways and may be combined with various additional hypotheses on the potential. Therefore we shall only give some vague indications of the type of condition which will be needed for the results below. We may distinguish three classes of correlation inequalities. A. Order inequalities. This name hints at the fact that the inequalities in this class require a linear ordering of the state space E and take advantage of the induced partial order of the configuration spaces EA. The prototype in this class was discovered by Fortuin, Kasteleyn, and Ginibre (1971) and is thus referred to as FKG inequality. A general version of this inequality implies that n{fg) ^ n{f) n{g) whenever / and g are nondecreasing (bounded measurable) functions on the finite product EA of a linearly ordered set E and \x is a Gibbs distribution on JEA relative to a potential which satisfies a condition of supermodularity. The original result of FKG covers the case |E\ = 2; the particular case of no interaction is already due to Harris (1960). If g is a probability density relative to \x (which can be assumed without loss) then the FKG inequality just asserts that the probability measure v = g\i is stochastically larger than \i, in that fj,(f) ^ v(f) for any nondecreasing/. Holley (1974) extended the FKG inequality by providing a sufficient condition on \i and v for the last property to hold. Holley's proof involved a coupling of /i and v by means of an auxiliary time evolution. This approach, though being natural when considered in the context of attractive infinite particle processes (cf. Liggett (1985)), was considered as a challenge to find more

456

Bibliographical Notes

elementary proofs which proceed by induction on |A|. Such induction proofs were found by several authors, including Brunei (1976), Hansel (1976), and Preston (1974c). Preston extended Holley's result from the case | £ | = 2 to the case of an arbitrary linearly ordered E. A great simplification of the induction step was achieved by Batty and Bollmann (1980) who proved a more general inequality which involves four probability measures and was suggested by earlier work of Ahlswede and Daykin (1978, 1979). A survey of several distinct proofs can be found in den Hollander and Keane (1986). Holley's idea of a dynamical coupling was taken up by Thomas (1980) who proved a version of Holley's inequality for continuous state spaces £ under a differential condition of attractivity. At this point we mention only one main application of the FKG-Holley inequality: If £ is a compact subset of IR with its natural order and ê is generated by monotone sets, and if is attractive in the appropriate sense, then there exist two (necessarily extreme) Gibbs measures /if, /i? f° r $ which are stochastically maximal resp. minimal in ^($). /if and /i? are the infinite volume limits of the Gibbs distributions for Q> with maximal resp. minimal boundary condition, and they are invariant under each ^-preserving transformation of S. The stochastic extremality of /if and /if implies that |^(0)| = 1 if and only if/if = /i?, and this occurs if and only if/if (ff;) = /i?(fff) for all i. In particular, if S = Zd and )\ = 1 /if (a0) = n-{o0), and the last property is equivalent to a differentiability property of the pressure. These results are due to Lebowitz and Martin-Löf (1972) and Ruelle (1972); see also Section 9 of Preston (1976). Similar results in the case E = U were obtained by Lebowitz and Presutti (1976) and Bellissard and Hoegh-Krohn (1982). B. Symmetry inequalities. This term indicates that the inequalities in this class are based on certain symmetry properties of the state space and the potential. A typical sufficient condition is that £ = { — 1,1} and O be a spin potential (cf. Example (2.38)) which is ferromagnetic in the sense that J (A) = 0 for all Ae£f. The basic symmetry inequalities are those of Griffiths (1967a). In the generalized version of Kelly and Sherman (1968), these inequalities (which are often referred to as GKS inequalities) assert that n{oA) = 0 and n(aAaB) _ fi{qA)fi{qB) when £ and O are as above, \i is a finite volume Gibbs distribution for O (with free or positive boundary condition), and A and B are finite; here aA = n,-ej4 1, the second GKS inequality can be deduced from the FKG inequality; this was observed by Baumgartner (1983). Kasteleyn and Boel (1979) showed that the second GKS inequality is a non-extreme element of a certain convex set of inequalities. Just as with the FKG inequality, the second GKS inequality can be used to prove the existence of the limit /if = lim y®(-\a>+), where m+ stands for the constant configuration with value + 1; details can be found in the survey articles of Griffiths (1971, 1972). Slawny (1974) noticed that /if maximizes the correlations /if (aA) and is thus extreme in ^(O). In the case where S = Zd and O is shift-invariant, /if is also shift-invariant. Moreover, the magnetization /if (CT0) is a nonnegative nondecreasing function of the coupling constants J (A) _ 0 which define . In particular, there exists a threshold value ßc e [0, oo] such that / i f K ) = 0 when 0 g ß < ßc and / i f K ) > 0 when ß > ßc. When combined with the above-mentioned consequences of the FKGHolley inequality, this result implies that in some cases (including ferromagnetic spin pair potentials with zero external field) |^(/JO)| = 1 when ß < ßc and |^(/M>)| > 1 when ß > ßc.

Bibliographical Notes

457

We now turn to a brief discussion of some further symmetry inequalities. First, there are several Griffiths-type inequalities for ferromagnetic systems of vector-valued spins. Such inequalities were derived first by Ginibre (1970) for plane rotor models. Further references in this context are Monroe (1975), Dunlop (1976), Kunz et al. (1976), and Messager et al. (1978). Monroe and Pearce (1979) gave a survey, and Sylvester (1980) showed that Ginibre's inequalities cannot be extended in general to TV-vector models with N > 2. A different type of symmetry inequality is the so-called GHS inequality of Griffiths, Hurst, and Sherman (1970). This inequality asserts that the third cumulant (or Ursell function) of a spin system with ferromagnetic pair interactions is always nonpositive. (One should note that the case \A\ = \B\ = I of Griffiths' inequalities just means that the first and second cumulant are nonnegative.) In physical terms, the GHS inequality just asserts that the magnetization is a concave function of a positive external field. Monroe (1980) showed that the assumption of pairwise interaction is essential. Simple proofs and generalized versions of the GHS inequality can be found in the articles of Lebowitz (1974) (who used the inequalities of GKS and FKG), Ellis and Monroe (1975), Ellis, Monroe and Newman (1976), Ellis and Newman (1978), Ellis et al. (1981), and Graham (1982b). Two related references are Sylvester (1975) and Kondo et al. (1985). Finally, we will briefly mention three further symmetry inequalities: those of Lebowitz (1977, 1978) (which were extended to certain vector-valued spin systems by Kalus (1981)), those of Messager and Miracle-Sole (1977) (which express a monotonicity of Ising-model correlations under reflections and translations and were extended by Hegerfeldt (1977)), and those of Wells (which can be found in the appendix of Bricmont, Lebowitz and Pfister (1981a) and allow to estimate the correlations of real-valued spin systems with even a priori measures in terms of the corresponding correlations of { — 1, l}-valued systems). We also note that there are some excellent surveys on symmetry inequalities and their applications, namely Sylvester (1976), Bricmont (1977a), and Shlosman (1981). A concise discussion of correlation inequalities can also be found in Chapter 4 of Glimm and Jaffe (1981). C. Geometric inequalities. As the name indicates, the correlation inequalities in this class are characterized by the fact that graphical, or geometric, arguments play a dominant rôle. In most cases, however, these inequalities also require some symmetry properties, and graphical methods were also used in the proofs of symmetry inequalities. The distinction between symmetry inequalities and geometric inequalities is therefore somewhat arbitrary. What we have mainly in mind here are the inequalities which result from random walk representations or random current representations of correlation functions. Such representations were investigated and exploited by Brydges, Fröhlich and Spencer (1982), Brydges, Fröhlich and Sokal (1983), Dynkin (1983), Aizenman (1982, 1985a), Graham (1982a), Shlosman (1986a), and others. The results include bounds on cumulants of fourth or higher order, estimates of critical temperatures and critical exponents, and the exponential decay of correlations. A further correlation inequality which may be included in the class of geometric inequalities is the Gaussian inequality of Newman (1975). This inequality compares the even correlations of ferromagnetic spin systems with those of centered Gaussian variables which have the same covariance; see also Sylvester (1975), Bricmont (1977b), Dunlop (1979a, 1985), and Brydges et al. (1982). Graphical methods were also used by Simon (1980a) to establish an inequality for pair correlations which is an efficient tool for proving an exponential decay of correlations. Simon's inequality was refined and extended to plane rotor models by Lieb (1980) and Rivasseau (1980).

458

Bibliographical Notes

Chapter 3 Section 3.1. Theorem (3.5) and its proof are taken from Spitzer (1971b). Brascamp (1971) independently obtained a similar result in the case when \E\ = 2 and y is a positive homogeneous specification with an arbitrary finite range. The uniqueness part of Theorem (3.5) is a special case of earlier results of Dobrushin (1968a) and Ruelle (1968), cf. Theorem (8.39). Section 3.2. The Ising model was proposed by Lenz (1920) and first studied by his student Ising (1924, 1925) in his doctoral dissertation. Brush (1967) gave an excellent historical account of both the physical origins and the later developments in the investigation of the model. The analysis of the Ising chain in terms of the largest eigenvalue of a so-called transfer matrix is now standard. The origins of this technique can be traced back to Kramers and Wannier (1941). A similar analysis of the case when the external field is allowed to vary from site to site can be found in Perçus (1977). References on the Ising model in two dimensions will be given later. Appendix 3.A. A standard reference for the theory of positive matrices is the book of Seneta (1973). The entropy-proof of the ergodic theorem (3. A3) is due to Rényi (1961). In fact, the idea of using relative entropy for proving limit theorems is already due to Linnik (1959). Kendall (1964) extended Rényi's proof to the case of a countably infinite state space. We also note that Rényi's argument can easily be refined to obtain exponential convergence; this can be read off from Problems 17 and 18 on pp. 58/59 of Csizâr and Körner (1981). Chapter 4 Sections 4.1 and 4.2. In our context, the topology of local convergence was first used by Preston (1976). He also used some general compactness criteria of the type (4.9) and (4.12). These, of course, are intimately related to the classical compactness theorems of Prohorov and Dunford/Pettis; cf., for example, Dellacherie and Meyer (1978). Proposition (4.15) is a slight modification of a result of Gänssler (1971). Sections 4.3 and 4.4. Example (4.16) seems to be new. Theorem (4.17) and Proposition (4.19) are due to Dobrushin (1968a, b), at least in the case of a finite state space. In this case, he also obtained the existence theorem (4.22) and the closed graph result (4.23) (c). Our formulation of (4.22) is essentially that of Preston (1976). Kotecky and Preiss (1981) and Winkler (1985) showed that Preston's existence theorem can be deduced from an abstract existence result for inverse limits. There are numerous papers which deal with the problem of existence of Gibbs measures for systems of genuinely unbounded spins (including the lattice models of quantum field theory). On the one hand, there is a general existence theorem of Dobrushin (1970); its proof exploits ideas which are similar to the contraction technique which underlies Dobrushin's uniqueness theorem, cf. Section 8.1. This existence theorem can also be found in the book of Sinai (1982) and, in an extended form, in Shlosman (1986d). On the other hand, there is a notable existence condition called superstability. This condition was introduced by Ruelle (1976) on the basis of his earlier work on classical particle systems. The superstability condition was used by Lebowitz and Presutti (1976), Cassandro, Olivieri et al. (1978), and Benfatto, Marchioro et al. (1980) to prove both the existence and the compactness of Ud-valued

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459

Gibbs measures which satisfy a condition of temperedness. Results of this type were also obtained by Royer (1977), Khanin (see Sinai (1982)), and Bellissard and H0eghKrohn (1982). The existence of Gibbs measures for systems of unbounded spins can also be proved by means of the chessboard estimate (17.11). This approach was suggested by Dobrushin and Shlosman (1981) and Shlosman (1984, 1986c); we use it in (18.12) and (20.7). To conclude, we note that the existence problem becomes trivial as soon as we are willing to confine ourselves to finitely additive (rather than c-additive) random fields; this was pointed out by Accardi and Pistone (1982). We also note that assertion (4.23)(c) has a converse which was established by Hugenholtz (1982): Under suitable conditions, the convergence of a sequence of Gibbs measures relative to certain potentials (<£")„ ^ i towards a Gibbs measure relative to some implies that " -> . Appendix 4.A. Two standard references for the isomorphism theorem (4.A6) are Kuratowski (1966) and Parthasarathy (1967). We follow the exposition of Dynkin and Yushkevich (1979). The notion of a countable core is equivalent to Dynkin's concept of a "support system" which can be found in the last reference. The second half of Theorem (4.A11) was noticed independently by Ramakrishnan and Rao (1980) and von Weizsäcker and Winkler (1980); see also p. 119 of Winkler (1985). Chapter 5 In the particular case where / is a finitely generated abelian group, the existence of /-invariant Gibbs measures was proved first by Dobrushin (1968a). He used the averaging argument of the Markov-Kakutani fixed point theorem; cf. Dunford and Schwartz (1958), p. 456. This argument (which is the heart of Theorem (5.15)) can also be found in Dobrushin (1970) and, for example, Preston (1976). A reader who is familiar with the concept of an amenable group will notice that Theorem (5.15) admits a natural extension which takes adventage of this concept; a standard reference for the theory of amenable groups is Greenleaf (1969). Theorem (5.19) is a formalization of the common idea of the examples in (5.20). These examples are "folklore". It is natural to ask whether there is a unique Gibbs measure which is invariant under the full symmetry group of the potential. This question can be answered in the affirmative in the case of lattice spin systems with a ferromagnetic shift-invariant interaction. For, in this case one can combine various powerful tools (including the variational principle (15.39) and correlation inequalities) in order to give a fairly explicit description of the class of all shift-invariant Gibbs measures, at least for most values of temperature. Some relevant references are Slawny (1973), Gruber, Hintermann and Merlini (1975, 1977), Gruber, Hintermann et al. (1977), Gruber and Lebowitz (1978), and Pfister (1982, 1985, 1987); see also the Notes on Section 16.2, Paragraph B. Chapter 6 Section 6.1. Theorem (6.4) is "common knowledge". I learned of it from Brascamp in a discussion in 1975. This theorem also appears as a starting-point in the papers of Sullivan (1975) and Sullivan and Vanheuverzwijn (1983). The Ising chain on Z with inhomogeneous interaction and inhomogeneous external field was studied by Miyamoto (1984) and Tejero (1987). An analysis of the inhomogeneous Widom-

460

Bibliographical Notes

Rowlinson chain can be found in Miyamoto (1985). For a general analysis of onedimensional models with inhomogeneous nearest-neighbour interaction we refer to Winkler (1981). As a matter of fact, Theorem (6.4) implies the existence of a phase transition in a number of further models. On the one hand, a phase transition occurs for any potential of the form 0 + *?, where *? is an even spin potential; this follows from the GKS inequality, cf. Griffiths (1971). On the other hand, an inequality of Wells (cf. Bricmont et al. (1981a)) shows that a phase transition still occurs when the state space { — 1,1} is replaced by any interval \_ — a,a] and

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461

before that generating functions for dimer configurations can be computed with the aid of Pfaffians. A simplified account of this method can be found in Thompson (1972) and McCoy and Wu (1973). A further approach to Onsager's formula was discovered quite recently by Plechko (1985). Considered as a function of the inverse temperature ß, Onsager's expression for the pressure exhibits a single singularity, namely at ßc = ^log(l + y/2). Kramers and Wannier (1941) had already shown that this ßc is the unique fixed point of an involution of ]0, oo[ which establishes a high temperature - low temperature duality of the Ising model on I2. It is far from obvious that this critical value of ß coincides with the threshold value for the non-uniqueness of the Gibbs measure. Nevertheless, this turned out to be the case. In view of the FKG inequality, the non-uniqueness of the Gibbs measure is equivalent to the phenomenon that the pressure, considered as a function of an external field h, has a kink at h = 0; this was shown by Lebowitz and Martin-Löf (1972) and Ruelle (1972). As a matter of fact, the existence of such a kink is precisely the traditional definition of a (first order) phase transition. Unfortunately, Onsager's formula is limited to the case h = 0 and thus fails to provide any information about the existence of such a kink; one thus resorts to various other concepts of spontaneous magnetization which are all physically meaningful but not automatically equivalent. A discussion of such concepts can be found in Schultz et al. (1964) and Griffiths (1966). Adopting a very particular definition of spontaneous magnetization, Yang (1952) derived the remarkable expression [1 - ( s i n h 2 / ? r 4 ] 1 / s for the magnetization at inverse temperature ß. This result had been claimed without proof by Onsager (1949). Using a different concept of magnetization which is based on the long range pair correlations at zero external field, Montroll, Potts, and Ward (1963) and Schultz, Mattis, and Lieb (1964) arrived at the same expression. (Montroll et al. followed the combinatorial route of Kasteleyn; an exposition of their method can be found in McCoy and Wu (1973).) In spite of this stupendous success, the problem of reconciling the different concepts of magnetization still remained open. This problem was finally solved by Martin-Löf (1972) for large ß, and then independently by Benettin et al. (1973) and Abraham and Martin-Löf (1973) for all ß. To conclude this section we note that some additional information on the abovementioned results can be found in the survey articles of Durrett (1981), Section 8, and Leha (1986). We also emphasize that no explicit formulas are known for the pressure and the magnetization of the Ising model in three or more dimensions. However, there are many results concerning the differentiability or analyticity of the pressure (cf. the Notes on Chapter 16), and the qualitative behaviour of the magnetization near the critical point (ß, h) = (ßc, 0) was revealed by Aizenman and Fernandez (1986) and Aizenman, Barsky and Fernandez (1987). C. The number of pure phases. The first result on the number of pure phases for the Ising ferromagnet at zero external field was obtained by Gallavotti and MiracleSole (1972). They showed that %(ß$>) = [>*./x£] whenever ß is large enough; the dimension of the lattice is arbitrary. Capocaccia et al. (1973) extended this result to small perturbations of the Ising model, and Borgs (1984) showed that the same argument can be used to prove the uniqueness of the shift-invariant Gibbs measure for Yang-Mills gauge models. In two lattice dimensions, Messager and Miracle-Sole (1975) were able to show that %(ßQ>) = [>0 ,/x?] for all ß > ßc. The same conclusion also follows from Onsager's formula for the pressure and a result of Lebowitz (1977) for more general ferromagnetic systems.

462

Bibliographical Notes

D. The ( + ) boundary condition. In view of the results of paragraph C above, it is natural to ask whether or not the Ising ferromagnet also admits some Gibbs measures which fail to be shift-invariant. In three or more dimensions this is indeed the case and was shown by Dobrushin (1972a). He considered the Gibbs distributions with the ( + ) boundary condition which is equal to 1 on the upper half-space {i e Zd: id ^ 0} and equal to - 1 on the lower half-space {i e Zd: id < 0}. Assuming that the inverse temperature ß is large enough, he showed that these Gibbs distributions converge to an infinite volume limit n% which resembles \i%(resp. fit) on any part of the upper (resp. lower) half-space which is sufficiently far from the boundary plane. In particular, \x% fails to be invariant under translations in direction d. Physically speaking, \i% describes the coexistence of a ( + ) phase and a ( — ) phase with a rigid interface. The threshold value of ß below which this phenomenon disappears is called the inverse roughening temperature. In a subsequent paper, Dobrushin (1973a) extended the above result to a general class of predominantly ferromagnetic spin systems. He also showed that \i% satisfies an exponential mixing property; this property implies that \x% is extreme in ^(ß<S>). A simple approach to the existence of non-shift-invariant Gibbs measures was found by van Beijeren (1975). He introduced a coupling of a Gibbs distribution in a three-dimensional cube with ( + ) boundary condition and a Gibbs distribution in a square with ( + ) boundary condition. This coupling gives the inequality \i% (<x0)\i= 3 ^ \i% (a0)\d=2• In particular, the inverse roughening temperature is not larger than ßc = ^log(l + ^/l), the critical inverse temperature in two dimensions. Bricmont, Lebowitz, Pfister, and Olivieri (1979) noticed that van Beijeren's argument can be extended to ferromagnetic spin systems of a fairly general type; they also established the existence of interface Gibbs measures for the Widom-Rowlison model in three dimensions. Gibbs measures with oblique interfaces were shown to exist in at least four dimensions by Messager and Miracle-Sole (1977). Borgs (1984) obtained an analogue of Dobrushin's result for Yang-Mills gauge models in four dimensions, and Holicky et al. (1988) developed a theory of interface states in the spirit of the Pirogov-Sinai theory; the same kind of result for unbounded spin systems with double-well potentials can be found in Zahradnik (1986). It is a major open question whether or not the roughening temperature coincides with the critical temperature for non-uniqueness of the Gibbs measures. We mention only two papers which deal with this problem, namely Bricmont, Fontaine, and Lebowitz (1982) and Graham (1984). What about the ( + ) boundary condition in two dimensions? This question was first studied by Gallavotti (1972a). He began with the observation that every configuration on a square A = [ — N, N~\2 fl Z 2 with ( + ) boundary condition defines a broken line, the phase separation line, which runs from ( — N — j , —j) to (N + j , — j) and is such that each elementary segment of this line separates a ( + ) spin (on the "upper" side) from a ( —) spin (at the "lower" side of the line). Gallavotti showed that this line is subject to large fluctuations as N -> oo when ß is large. Specifically, he proved a central limit theorem for the (lowest) point at which the phase separation line meets the vertical axis. This result was sharpened by Higuchi (1979): For sufficiently large ß, a suitably rescaled version of the phase separation line converges in distribution to a Brownian bridge. An outline of the proof can be found in Higuchi (1980). In addition, Abraham and Reed (1976) calculated the magnetization profile of the spins along a vertical axis, and Bricmont, Lebowitz and Pfister (1981b) made a detailed analysis of the intrinsic structure of the interface. The fluctuations of the phase separation line suggest that in two dimensions any limit ni of Gibbs distributions with ( + ) boundary condition is shift-invariant. In fact, for large values of ß the above mentioned result of Gallavotti readily implies that the

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limit n% exists and is equal to (fill + nt)/2. Messager and Miracle-Sole (1977) proved this assertion for all ß > ßc. (Bricmont, Lebowitz and Pfister (1979) gave a short argument that /i£ cannot be extreme and non-shift-invariant, as it is in three or more dimensions.) More generally, Messager and Miracle-Sole showed that all limiting Gibbs measures for boundary conditions with antisymmetric minorants or majorants are necessarily shift-invariant. In the same spirit, Higuchi (1978) studied the question of how many minuses may be contained in a boundary condition to obtain the limit fiß+. The next step towards a complete determination of all Ising-model Gibbs measures in two dimensions was a result of Russo (1979). He showed that any such Gibbs measure which is invariant under horizontal shifts is also invariant under vertical shifts (and vice versa). The final answer was given independently by Aizenman (1980a, b) and Higuchi (1981) on the basis of Russo's ideas: For all ß > ßc, the set of all Gibbs measures for the two dimensional Ising ferromagnet at inverse temperature ß is precisely the interval [ / I ? , / I £ ] . For large ß, this remarkable result is in fact a particular case of a recent general theorem of Dobrushin and Shlosman (1985c). Finally, we need to mention a completely different (and, in fact, earlier) approach to the phenomenon of phase separation in the Ising model. This approach is due to Minlos and Sinai (1967b). They considered an Ising ferromagnet for large ß in a large box A with ( + ) boundary condition under the restriction that the average magnetization |A| _ 1 YJ at takes a fixed value m in the open interval ]/i^(<70),/i£(<70)[. They ïe A

proved that, with probability close to 1, there exists a large connected set (the "droplet") whose average magnetization is close to nt{a0) and whose complement has an average magnetization close to fi+{(r0). (The size of the droplet is then determined by m.) Moreover, the shape of the droplet is nearly that of the solution of the isoperimetric problem on Z2, viz. a square. Kuroda (1983a) obtained a similar result in the case of the free boundary condition: There exist again two complementary regions whose averaged magnetization is close to nt{q0) r e s P- ^+(°o)> but now each of these regions is adherent to the boundary, and their shape is determined by the rule that the length of the separation line is minimal. The phase separation phenomena of lattice gases with several types of particles were studied by Kuroda (1983b). To conclude this section we note that there exist several excellent survey articles on phase separation in the Ising model, including Gallavotti (1972b), Pfister (1983), and Abraham (1986). These articles may also be consulted for a review of results on the surface tension. E. Miscellanea. There are various results on the Ising ferromagnet on a subset S of Zd. For example, Dobrushin (1972b) investigated the limiting behaviour as A | S, and its dependence on the shape of S, of the Gibbs distributions in finite regions A c S with free boundary condition. Chayes and Chayes (1986) studied the case where S has the shape of a wedge. Under the condition that the area of any cross-section through S increases at least logarithmically with its distance from the vertex, they established the existence of a phase transition at sufficiently low temperatures. The picture changes as soon as we impose a boundary condition by attaching fixed spins to all sites in Zd\S which are adjacent to S. In the case when S is a half-plane in Z2 with ( + ) boundary condition, Messager and Miracle-Sole (1977) and Russo (1979) showed that the Gibbs measure is unique for all ß. Rather than fixing the spins at the boundary of S, one can introduce a particular coupling parameter for adjacent boundary spins which differs from the coupling of the spins in the bulk. This setting is appropriate for a study of various surface effects. One such study is in Fröhlich and Pfister (1987).

464

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Next, we point out that the experimental behaviour of certain ferromagnetic or antiferromagnetic compounds coincides surprisingly well with the mathematical properties of the Ising model (or of plane rotor or Heisenberg models) in two or three dimensions. A review of experiments which support this statement was given by de Jongh and Miedema (1974). Ikeda and Hirakawa (1974) found that the staggered magnetization of the antiferromagnet K 2 CoF 4 is perfectly described by the OnsagerYang formula for the magnetization of the two-dimensional Ising model. Further information of this sort can be found in Birgeneau et al. (1984). Finally, we note that the Ising model can also be used to describe cooperative phenomena outside the realm of Statistical Physics. Applications of the Ising model to the biology of hemoglobin and DNA can be found in Thompson (1972), and an application to the theory of equilibrium prices in large economies was given by Föllmer (1974). Section 6.3. Theorem (6.21) is due to Shlosman (1983). The low temperature behaviour which is described by this theorem contrasts with the behaviour at high temperatures. For small ß, Fröhlich and Spencer (1981) established the existence of unbounded fluctuations, in that y;T(k; - o)l2|0) ^ c(/3)log(l + \i -j\) for all A e y and i,j e A and some constant c(ß) > 0; here 0 is the zero configuration. One might ask whether the low-temperature phase transition of the discrete Gaussian model persists in the presence of a cutoff, i.e. when the state space Z is replaced by {— N,..., N} for some N. This is not the case, as can be shown by means of the Pirogov-Sinai theory; we refer to Bricmont, El Mellouki and Fröhlich (1986) for more details. We note further that the discrete Gaussian model has a close relative, the so-called solid-on-solid model, which is defined by replacing the expression (a, — o))2 on the right side of (6.16) by |<7, — Oj\. This model plays a significant rôle in the study of random surfaces; see, for example, the last reference, Dobrushin and Shlosman (1985c), Abraham (1986). As a supplement to the definition (6.18) of a ground state, we note that a thorough discussion (as well as some refinements and modifications) of this important concept can be found in the papers on the Pirogov-Sinai theory (cf. the Notes on Chapter 19) and, in particular, in Slawny (1986) and Dobrushin and Shlosman (1985c). Among other things, the last article contains an analysis of the ground states of the Ising model in one, two and three dimensions. Moreover, this article contains a discussion of the more general notion of a ground random field: A specification y with a finite state space E is called a ground specification relative to a potential C> if each yA(- |cu) is the equidistribution on the (finite) set of all configurations ( e Q which minimize //* under the constraint that Cs\\ = ffls\A; an Y Gibbs measure relative to this y is called a ground random field (or random ground state) for $. The significance of this concept comes from the fact that the set of ground random fields for a given $ contains (though, in general, is different from) the set of all possible limits of sequences ( / i j , ^ with \i„ e ^(jß„$) and ß„ -» oo. For example, it is easy to see that the set of all ground random fields for the one-dimensional Ising potential Q>Jh (cf. (3.13)) is given by the table J= l h=0 h>0

tf-M

{
J -= - 1 \h\ <2

h = 2

h>2

[<5_+,<5+_]

(M

{<M

cf. the discussion in Section 3.2. In particular, the Markov chain jiF is an example of a ground random field which is really random, and the cases J = \,h = 0 and J = — 1,

Bibliographical Notes

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\h\ < 2 show that not every ground random field is a limit of Gibbs measures for finite /Ts. It is a major open problem to characterize the ground random fields which are stable, in that they are limits of Gibbs measures for positive temperatures. A natural conjecture of Dobrushin and Shlosman (1985c) was recently disproved by a counterexample of Miekisz (1987b).

Chapter 7 Section 7.1. In the particular case where each n e Ff is induced by a measurable transformation of Q, Corollary (7.4) is a classical result of Ergodic Theory which is to be found, for example, in Farrell (1962) and Section 10 of Phelps (1966). Its origins can be traced back to Kryloff and Bogoliouboff (1937). Proposition (7.3) is essentially due to Hopf (1954); cf. also Section 9.11 of Jacobs (1963). An extension of Corollary (7.4) can be found in Kerstan and Wakolbinger (1981). Farrell (1962) already gave an example of a family II for which Jn(fi) is different from the /i-completion of Jn for all /i e ^ n ; his n consists of two non-commuting transformations of Q. Assertion (a) of Theorem (7.7) was discovered by Lanford and Ruelle (1969) and independently, in a somewhat weaker form, by Dobrushin (1968a, b). Statements (c), (b), and (d) appeared in Georgii (1973) and Preston (1974a, 1976). Proposition (7.11) is a result of Dobrushin (1968a, b), and Theorem (7.12) is a generalized version of an observation of Georgii (1973). In the case S = Z, it is a traditional practice in Probability Theory to study the triviality of the right-sided tail «r-field $~+ = f] ^j„,„ + i,...}. It might thus be worthwhile to note that the triviality of ST for S = Z is by no means equivalent to the triviality of ,T+, even under the condition that the random field is shift-invariant. Quite in the contrary, it may happen that 9~+ is trivial whilst 9~ = ^ almost surely; cf. Gurevich (1973), Ornstein and Weiss (1975), and Bradley (1986). Section 7.2. Example (7.16) - the Gibbsian approach to exchangeable distributions - can be found in various references, including Georgii (1975, 1979), Dynkin (1978), and Accardi and Pistone (1982). The proof of (7.17) is classical and can already be found in Meyer (1966). Example (7.18) combines ideas of Georgii (1973) and Ledrappier (1974); it appeared first in Georgii (1974). Section 7.3. Theorem (7.26) and its proof are due to Föllmer (1975a). He exploited a technique which had been developed by Dynkin (1971) in his investigation of the entrance boundary of a Markov process. In particular, the concept of a (#, j/)-kernel was coined by Dynkin. The roots of this concept, however, are to be found in the earlier work on ergodic decompositions. The basic rôle of property (7.21)(i) - the sufficiency of the «r-algebra se for 3P - for ergodic decompositions was brought to light by Farrell (1962) and Varadarajan (1963), and a result which is very similar to the identity (7.23) can already be found in the article of Kryloff and Bogoliouboff (1937). On the basis of this tradition but apparently independently of Dynkin, Maitra (1977) used the same kind of technique as Dynkin (1971). Similar ideas were also exploited in connection with de Finetti's theorem for exchangeable distributions; see, for example, Dynkin (1953) and Hewitt and Savage (1955). The universal applicability of (#, «s/)-kernels to various problems of extreme decomposition was popularized by an article of Dynkin (1978). For further work in this direction we refer to Kerstan and Wakolbinger (1980, 1981) and Mauldin, Preiss and von Weizsäcker (1983).

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If &(y) is compact relative to a suitable locally convex topology, the extreme decomposition theorem (7.26) follows also from the well-known Choquet's theorem of convex analysis; a classical account of this theorem is Phelps (1966). To obtain the uniqueness of the extreme decomposition one then needs to know whether the cone of all nonnegative multiples of all Gibbs measures for a given y is a lattice in its own order. Lanford and Ruelle (1969) already observed that this is indeed the case. At this point we should also mention a paper of Miyamoto (1974); he proved Theorem (7.26) in the compact case independently of Föllmer by means of a combination of Choquet's theorem and Dynkin's technique. We conclude this paragraph with three remarks. First, we point out that Theorem (7.26) can be derived from a general theorem on the existence of inverse limits of standard Borel spaces. This was shown by Kotecky and Preiss (1981) and Winkler (1985). Next, we note that Corollary (7.30) was already obtained (in the compact case) by Dobrushin (1968b) using the techniques of convex analysis. Finally, we emphasize that Example (7.31) is just a representative of a whole class of examples which are known as canonical Gibbs measures or repetitive statistical models, cf. Georgii (1979) and Lauritzen (1982). Section 7.4. Theorem (7.33) may be ascribed to common knowledge. A particular case appeared explicitly in Georgii (1981a). A weaker result of the same type was already obtained by Shiga (1977). The idea of the proof comes from Proposition (8.38); some comments on this proposition will follow in the Notes on Section 8.3. Chapter 8 Section 8.1. The notes on this section fall into four groups. A. The contraction argument. The first statement of Theorem (8.7) - the uniqueness of Gibbs measures for specifications which satisfy the weak dependence condition (8.6) - is perhaps the most outstanding result of Dobrushin's fundamental paper (1968a). The original contraction argument of Dobrushin was essentially that which is presented in the text, but differed somewhat in appearance. The formulation of this argument in terms of the local oscillations öj(f) is due to Lanford (1973). The more general comparison theorem (8.20) was stated and proved by Dobrushin (1970). A reformulation of the proof in Lanford's style appeared in Kiinsch (1982). We follow the simple presentation of Föllmer (1982). The last three references also deal with the more general case that the total variation distance (8.1) is replaced by the more general Vaserstein-Kantorovich-Rubinstein distance relative to some metric on the state space. (If this metric is the discrete metric, the VKR distance coincides with (8.1); a general discussion of the VKR distance can be found in Dudley (1976).) We emphasize that this generalization is particularly well-suited for a treatment of genuinely unbounded spin systems, whereas Theorem (8.7) and (8.20) in our formulation can only be applied to systems with bounded Hamiltonians. Three further references which treat Dobrushin's uniqueness theorem in the context of unbounded spin systems are Royer (1977), Cassandro et al. (1978), and Dobrushin and Pecherski (1983). In a recent paper of Dobrushin and Shlosman (1985a), the uniqueness theorem (8.7) was extended in a different direction: The underlying parameter set is S = Zd, and Dobrushin's condition is replaced by a weaker condition which is stated in terms of the subspecification {yA+i: i e Zd}, where A is any cube in Zd. (In the case | A| = 1, this condition coincides with condition (8.6).) There are some indications that this condition is not too far from being necessary. Namely, the condition holds for all Gibbs

Bibliographical Notes

467

specifications relative to a class of potentials which are distinguished by various natural (and, in fact, equivalent) properties. These potentials are called completely analytical; cf. Dobrushin and Shlosman (1985b, 1987). On the other hand, this condition can only hold if any restriction y s° of y to a subset S0 c Zd admits a unique Gibbs measure on S0. This is not necessarily the case: There is a class of threedimensional models which admit a unique Gibbs measure on the whole lattice Z 3 but exhibit a phase transition on a half-space of Z 3 ; cf. Shlosman (1986b). Three additional remarks are in order. First, we mention that a friend of Nonstandard Analysis can find a nonstandard approach to Dobrushin's theorem in the book by Albeverio et al. (1986). Next, we note that Dobrushin's condition was used by Föllmer (1984) to prove almost sure convergence of certain multiparameter martingales. Finally, we refer to the work of Deuschel (1985, 1986, 1987) who used Dobrushin's technique to prove the uniqueness and some decay properties of a tempered solution of certain infinite-dimensional stochastic differential equations. B. The range of Dobrushin's condition. As is pointed out in the text, there are basically two different types of conditions on a potential which imply that the associated Gibbs distribution satisfies Dobrushin's condition. The conditions of type I mean that the total interaction is small. If the potential is equipped with a temperature factor ß, the type-I conditions essentially require that the temperature ß"1 is sufficiently high. By way of contrast, the conditions of type II are characterized by the presence of an energy well which is sufficiently deep or wide. A typical representative of a type-I result is Proposition (8.8). This proposition essentially goes back to Dobrushin (1968c, 1969); in this form it is due to Simon (1979). The result of Example (8.9)(1) was obtained first by Lanford (1973). Levin (1980) improved Proposition (8.8) in the case of N-vector models with N > 5. Type-I results for unbounded spin systems were provided by Royer (1977), Cassandro et al. (1978), and Dobrushin and Pecherski (1983). Using different methods, Bassalygo and Dobrushin (1986) obtained a type-I result for nearest-neighbour potentials on possibly inhomogeneous graphs. We now turn to sufficient conditions of type II. This type of condition is illustrated by the three examples in (8.13). Example (8.13)(1) - the case of a large external field is already essentially due to Dobrushin (1968c, 1969). Example (8.13)(3) - the case of an energy well of large entropy - was communicated to the author by R. Kotecky. Example (8.13) (2) - the case of a low temperature system with a unique ground state is a particular case of a theorem of Dobrushin and Shlosman (1985a). A forerunner of this theorem can already be found in Dobrushin and Pecherski (1981), and a further discussion in Dobrushin and Shlosman (1985b, 1987). As a specific application of this theorem, Dobrushin, Kolafa and Shlosman (1985) obtained some detailed results on the shape of the uniqueness region of the two-dimensional Ising antiferromagnet in an external field at low temperatures; see also Radulescu and Styer (1987). C. Mean field bounds. In Example (8.9) (2) it is shown that, for some ferromagnetic spin systems, the mean field condition for the absence of spontaneous magnetization implies Dobrushin's condition and thereby the uniqueness of the Gibbs measure. In the case of unbounded spin systems, a similar observation was made by Cassandro et al. (1978). A different, more common way of studying the relationship between a ferromagnetic spin system and its mean field approximation is the use of correlation inequalities. Using this approach, one can show in various kinds of models that (i) the mean field magnetization is an upper bound of the real magnetization, or at least that (ii) the mean field critical temperature is an upper bound of the real critical temperature. Statement (i) (as well as some improvements) can be found in the papers

468

Bibliographical Notes

of Pearce (1981), Slawny (1983), Tasaki and Hara (1984), and Siu (1985). In the context of Example (8.9) (2), statement (ii) is due to Griffiths (1967b). Proofs of statement (ii) in other cases and related results are to be found in the papers of Simon (1980b), Brydges et al. (1982), Sokal (1982a), Aizenman (1985b), and Monroe (1985); see also the references in these papers. D. Other proofs of uniqueness. Besides the contraction argument of Dobrushin, there exist several further techniques for proving the uniqueness of Gibbs measures. The most prominent technique is the method of correlation equations of KirkwoodSalsburg or Mayer-Montroll type. Roughly speaking, this method consists in showing that the correlation function of a Gibbs measure is the unique fixed point of a suitable contraction operator on some Banach space. The origins of this method lie in the theory of point particles in Euclidean space, cf. Minlos (1967) and Ruelle (1969) and the references therein. In the case of lattice systems, this method was used by Gallavotti and Miracle-Sole (1968) and Lanford and Ruelle (1969) to prove the uniqueness of Gibbs measures for lattice gases at low densities or high temperatures; an exposition of these results can be found in the texts of Spitzer (1971b, 1974) and Preston (1974a). When applied to suitable ensembles of contours, the method of correlation equations (or, more generally, the so-called technique of cluster expansion) is also useful for a study of lattice systems at low temperatures. In fact, this technique is a main ingredient of the remarkable theory of Pirogov and Sinai which will be discussed in more details in the Notes on Chapter 19. Using these kinds of ideas, it is possible to establish the uniqueness of the Gibbs measures (or, at least, the shift-periodic Gibbs measures) at low temperatures for certain lattice systems with a unique periodic ground state, cf. Malyshev et al. (1983) and Zahradnik (1984). Related ideas can also be used to prove the uniqueness of tempered Gibbs measures for unbounded spin systems which are small perturbations of suitable Gaussian systems; see Malyshev (1980) and Malyshev and Nickolaev (1984), as well as Dobrushin and Zahradnik (1985) and Zahradnik (1986). Finally, we refer to two further approaches to uniqueness. In the case of spin systems which are ferromagnetic in the sense of the FKG-Holley inequality, the uniqueness of the Gibbs measures follows from the absence of spontaneous magnetization or the differentiability of the pressure with respect to an external field; see Ruelle (1972) and Lebowitz and Martin-Löf (1972) and the related references in the Notes on Chapter 2. Some applications of this result will be mentioned in the Notes on Section 16.2; a further interesting application is in Sokal (1982a). A quite different approach to uniqueness consists in showing that a suitable stochastic process which leaves the Gibbs measures invariant admits only one invariant random field. This kind of argument was used by Holley and Stroock (1976) and Doss and Royer (1978). Section 8.2. The results and techniques of this section are basically due to Dobrushin (1968a, 1970). They were developed further in the papers of Gross (1979, 1981), Kiinsch (1982), and Föllmer (1982). The last two papers were our main sources. In particular, Proposition (8.34), in this form, is due to Föllmer, and Corollary (8.37) on the differentiability of the functions 0 -+ fi^g) on Q) is Künsch's improvement of a result of Gross (1981). Prakash (1983) showed that also the higher-order derivatives of these functions exist. As a matter of fact, the literature on the decay of correlations in Gibbs measures is enormously extensive. So we will only give a few cursory directions. Some recent papers which are still close in spirit to the ideas of Section 8.2 are Dobrushin and Pecherski (1981), Dobrushin and Shlosman (1985a, b, 1987), and Dobrushin (1987).

Bibliographical Notes

469

Another classical and rather universal technique for studying the decay of correlations is the method of cluster expansions; see, for example, Duneau et al. (1974,1975) and the expositions of Malyshev (1980), Glimm and Jaffe (1981), and Malyshev and Minlos (1985). In the case of ferromagnetic spin systems of Ising, plane rotor, or Heisenberg type, the decay of correlations can be investigated by means of correlation inequalities. Some of the relevant references are Lebowitz (1972, 1975), Martin-Löf (1973), Aizenman and Simon (1980), Simon (1980a), Lieb (1980), Rivasseau (1980), Brydges et al. (1982), Bricmont and Fröhlich (1985), and Aizenman et al. (1987). Still other methods were used by Holley and Stroock (1976), McBryan and Spencer (1977), Messager et al. (1984), and Araki (1986). We emphasize further that the problem of decay of correlations is intimately connected with the problem of analyticity of the pressure. A number of papers which deal with the latter (and thus also the former) problem will be listed in the Notes on Chapter 16. We will now provide some references on two specific topics, namely the central limit problem for Gibbs measures, and the global Markov property. A. Central limit theorems. It is a classical problem of Probability Theory to find reasonable conditions under which a shift-invariant random field JX over S = Zd satisfies the central limit theorem, in that the random variables

S?±\A(n)r'2

X

fo-Mfo))

i e A(n)

converge in distribution to a Gaussian random variable; here A(n) = [0, n — l ] d fl Zd. More generally, one seeks to find all functions / o n f i which are such that |A(«)I"1/2 I

(f°8-i~Kf))

isA(n)

converges in distribution to a Gaussian limit, or, even stronger, that the random field

(\\(n)r12 \

X isA(n)+nj

(/° 0-i - M/))) JjeZd

converges in distribution towards a Gaussian random field. It is evident that these questions were studied extensively both in general and in the special case when fi is a Gibbs measure. As the one-dimensional case is classical, we will concentrate on a few references which deal with the case d^2. There are various different ways of obtaining multidimensional central limit theorems. First, there is the classical Bernstein-method of decomposing the random field into nearly independent blocks and a comparatively small remainder. This method requires a sufficiently rapid decay of the uniform mixing coefficients
470

Bibliographical Notes

Fourier transforms of the S*'s; see, for example, Iagolnitzer and Souillard (1979) and DeConinck (1987b). A fifth method applies to lattice systems which are ferromagnetic in the sense of the FKG-Holley inequalities. Newman (1980) discovered that these inequalities allow to control the Fourier transform of S* rather efficiently and thus permit us to prove a central limit theorem for S* under the only additional assumption that the covariance series

converges. This technique can also be adapted to more general cases, see Newman (1983). It can also be refined to prove functional limit theorems in dimensions one and two (see Newman and Wright (1981, 1982)) and a local central limit theorem in one dimension (Wood 1985). Three further references which take advantage of positive correlations in the sense of the FKG inequalities are Hegerfeldt and Nappi (1977), Neaderhouser (1978a), and DeConinck (1987a). A sixth method (which is well-known in the one-dimensional case) is the method of approximating the partial sums of a random field by a martingale. Neaderhouser (1981) used this method to derive a multidimensional functional central limit theorem. The above-mentioned references are mainly devoted to integral central limit theorems. Local central limit theorems for multidimensional random fields were obtained by Del Grosso (1974), Dobrushin and Tirozzi (1977), Campanino, Capocaccia and Tirozzi (1979), and Campanino, Del Grosso and Tirozzi (1979). The last three papers provide conditions under which the local central limit theorem is a consequence of the integral limit theorem. Last but not least, there are some references which establish central limit theorems for the Ising ferromagnet in two or more dimensions. These include Martin-Löf (1973), Gallavotti and Martin-Löf (1975), Gallavotti and JonaLasinio (1975), and Pickard (1976). B. The global Markov property. In Example (8.24) it is shown that, under Dobrushin's condition, the global Markov property of a Gibbs measure is a consequence of the local Markov property. This fact was observed independently by Föllmer (1980) and Albeverio et al. (1981). There are various further results on the global Markov property. For a Markov potential Q> which is ferromagnetic in the sense of the FKG-Holley inequality, Föllmer (1980) and Goldstein (1980) showed that the maximal Gibbs measure p+ (and similarly /x?) exhibits the global Markov property; see also the related papers of Zegarlihski (1986,1987) and Kessler (1987). Goldstein (1980) and Winkler (1981) provided examples of non-extreme Gibbs measures which exhibit the local but not the global Markov property. In view of Theorem (10.21) one might expect that an extreme Gibbs measure with the local Markov property also enjoys the global Markov property. This, however, is in general not the case; some counterexamples were found by Kessler (1985) and Israel (1986a). On the other hand, Higuchi (1984) showed that, in the case of the Ising ferromagnet, any convex combination of pf and p® satisfies the global Markov property. Further examples of non-extreme Gibbs measures with the global Markov property are provided by Corollary (11.33) and Theorem (11.39). Finally, there is a result of Kuik (1988a) which is related to Theorem (10.25): If €> is a shift-invariant bounded nearest-neighbour potential on Zd with %(Q>) = {p} then p. satisfies the global Markov property relative to all halfspaces V a Zd. See also Kuik (1988b). Section 8.3. Theorem (8.39) was obtained independently by Ruelle (1968) and Dobrushin (1968a, c, 1969). In fact, Dobrushin proved a uniqueness theorem which is related

Bibliographical Notes

471

but not equivalent to Proposition (8.38). A generalized version of his result can be found in Dobrushin (1974a). Ruelle's proof rests on a study of a transfer operator and is thus similar in spirit to our proof of Theorem (3.5); Walters (1975) extended this argument to a more general, ergodic theoretic setting. The proof via Proposition (8.38) which is presented here was designed by H. Rost in a discussion in 1973. It appeared first in Georgii (1974,1976) and was rediscovered independently by Bricmont, Lebowitz and Pfister (1979). Still another proof of Theorem (8.39) can be found in Ledrappier (1973). Theorem (8.39) admits various improvements and extensions. Minlos and Natapov (1975) established the uniqueness of the Gibbs measure for a shift-invariant potential over Z under a decay condition which is weaker than (8.24). Miyamoto (1985) proved a uniqueness theorem for one-dimensional systems with finite state spaces and spatially inhomogeneous nearest-neighbour potentials. Uniqueness results for onedimensional systems of unbounded spins were obtained by Dashyan and Sukhov (1984) and Papangelou (1984a), see also the report of Papangelou (1984b). In particular, Papangelou proved the uniqueness of "regular" Gibbs measures (and thus the uniqueness of shift-invariant Gibbs measures) for shift-invariant potentials which are superstable in the sense of Ruelle (1976) and satisfy a condition analogous to (8.42). Incidentally, there also exist several interesting techniques of proving the absence of spontaneous magnetization in one-dimensional Ising-type spin systems with ferromagnetic long range interactions, see Dyson (1969b), Rogers and Thompson (1981), and Simon and Sokal (1981). We will now refer to some further results on the unique Gibbs measure, say p., for a shift-invariant potential which satisfies (8.42). ß exhibits fairly good mixing properties. In particular, these mixing properties imply that n, together with the shifttransformation, is isomorphic (in the sense of Ergodic Theory) to a Bernoulli scheme. In the case of a finite E this was shown by Gallavotti (1973) and Ledrappier (1973), and for certain unbounded spin systems by Dashyan and Sukhov (1984). If the potential is nearest-neighbour and E is finite, Akcoglu et al. (1979) showed in addition that \x can be obtained from any Bernoulli scheme of larger specific entropy by means of a finitary code. The intrinsic randomness of p. also becomes apparent from the following result of Lalley (1986) (for the case of a finite E): ß can be represented as the distribution of a two-sided random sequence which consists of infinitely many independent identically distributed blocks of finite expected length. Chapter 9 Section 9.1. Theorem (9.5) was proved independently by Fannes et al. (1982) and Burkov and Sinai (1983). Our proof is identical with that in the latter reference. Dobrushin and Shlosman (1985c) obtained a two-dimensional counterpart of Theorem (9.5). Using the assumptions and techniques of the Pirogov-Sinai theory as well as Proposition (9.1), they showed that - in two dimensions and at sufficiently low temperatures - each Gibbs measure is shift-periodic with a period which is the least common multiple of the periods of the periodic ground states. Theorem (9.11) is an obvious one-dimensional analogue of the more prominent Theorem (9.20); it should be ascribed to "common knowledge". Section 9.2. The pioneering work on the absence of a continuous symmetry breaking in two-dimensional systems is due to Mermin and Wagner (1966). They established the absence of spontaneous magnetization for quantum Heisenberg models in two

472

Bibliographical Notes

dimensions. (By the way, the name of these models refers to the work of Heisenberg (1928).) A classical (i.e., non-quantum) version of the Mermin- Wagner result appeared in Mermin (1967). Theorem (9.20), in the case of a finite range interaction, was obtained first by Dobrushin and Shlosman (1975). Their proof takes advantage of an asymptotic equidistribution property of products of independent SO(2)-valued random variables; see also the exposition by Sinai (1982). Shlosman (1977) extended this proof to potentials of infinite range with an exponential decay. In the form stated, Theorem (9.20) is due to Pfister (1981). In fact, Pfister's decay condition is slightly weaker than (9.21). Up to one simplification, our proof of Theorem (9.20) is also due to Pfister. This proof admits a variant which is based on the concept of relative entropy; see Fröhlich and Pfister (1981). Entropy arguments were also exploited by Simon and Sokal (1981) who obtained related but weaker results. The method of Mermin and Wagner was developed further by Klein et al. (1981), Bonato et al. (1982), and Simon (forthcoming). In the last two references, the conclusion of Theorem (9.20) is deduced from the hypothesis that the integral in (20.16) diverges. In the setting of Theorem (20.15), condition (20.16) is therefore both necessary and sufficient for a continuous symmetry breaking. As for the non-existence theorems (9.16) and (9.24), we note that this kind of result was obtained first (under different technical conditions and with different proofs) by Dobrushin and Shlosman (1980) and Fröhlich and Pfister (1981); a related result appeared already in Brascamp et al. (1975). Comment (9.34) (2) on the absence of continuous symmetry breaking in one-dimensional systems is taken from Simon (1981). In connection with Comment (9.34)(1), we refer to a paper of JonaLasinio et al. (1978) which contains a discussion of the problem of the conservation of continuous symmetries in three or more dimensions. We conclude the notes on this section with a few comments on the ferromagnetic plane rotor model on Z 2 with a shift-invariant, say nearest-neighbour, coupling. Using correlation inequalities, Bricmont, Fontaine and Landau (1977) showed that in this model there exists only one shift-invariant Gibbs measure fi, and this n is extreme; see also Messager et al. (1978). In addition, Fröhlich and Pfister (1983) showed that ft is the unique shift-periodic Gibbs measure for almost all values of temperature. (This result holds in any lattice dimension.) So it is natural to expect that ß is the unique Gibbs measure at all temperatures, i.e., there is no phase transition in the sense of Definition (2.10). Nevertheless, the above fi exhibits a critical phenomenon of a different sort: Its pair correlations show an exponential decay at high temperatures, but only a power decay at low temperatures; this is the so-called Kosterlitz-Thouless transition which was established by Fröhlich and Spencer (1981). As is shown in Example (20.20), the Gibbs measure is non-unique in three or more dimensions. The plane rotor model on a three-dimensional half-space combines some features of the two-dimensional and the three-dimensional case; see Fröhlich and Pfister (1986). Chapters 10 and 11 Sections 10.2 and 10.3. The analysis of one-dimensional Markov fields with infinite state spaces was initiated by Spitzer (1975b). His paper laid the ground work for all the results of Chapters 10 and 11. In particular, Spitzer proved Theorem (10.21) in the special case of a countable state space. The general version of this result is due to Dang-Ngoc and Royer (1978). Spitzer also derived a precursor of Theorems (10.25) and (10.35). In the form (11.13), these theorems were first proved by Kesten (1976). Using different techniques, Gurevich (1984) extended Kesten's result to the case where

Bibliographical Notes

473

Q is irreducible (rather than positive). Theorems (10.25), (10.34), and (10.35) - as they stand - are due to Papangelou (1983). In fact, Papangelou's results are not limited to the case when y is a A-specification relative to some a priori measure X. Except for some simplifications which are mainly due to this additional hypothesis, our proofs of Theorems (10.25) and (10.35) are those of Papangelou. (By the way, Papangelou (1984a) extended the basic ideas of these proofs to obtain a uniqueness result for Gibbs measures on Z with long-range interaction.) Under rather strong hypotheses, Theorem (10.35) was also obtained by Dang-Ngoc and Yor (1978). This paper includes an investigation of Markov fields having the real line (rather than Z) as parameter set. For further comments concerning Theorems (10.21) and (10.25) we refer to the paragraph on the global Markov property in the Notes on Section 8.2. Section 11.1. The setting of Chapter 11 was developed by Spitzer (1975b). He introduced the equivalence relation " ~ " between positive matrices and noticed the validity of Remark (11.4). He also introduced the concept of a boundary law and proved the fundamental Theorem (11.9). (Part (c) of this theorem appeared later in Cox (1977a).) In connection with this theorem it is interesting to note that the extreme boundary of &(y) is completely determined by the extreme boundaries of the associated Markov specifications on Z + and Z_ (which are governed by the right and left tail cr-fields, cf. Example (7.34)); see Föllmer (1975b) and Winkler (1981). We also note that Theorem (11.9) has a counterpart in the theory of entrance laws for Markov chains; see Cox (1977b). Theorem (11.15) and Corollary (11.17) are due to Kesten (1976). The simple proof of Theorem (11.15) was found by Cox (1979). Corollary (11.19) is a result of Spitzer (1975b). Spitzer's proof rests on the classical fact that the extreme space-time harmonic functions for random walks are exponentials. Our proof here - the construction of an equivalent homogeneous matrix - follows an argument of Kesten (1963). For an extension of Corollary (11.19) to the case when E is an arbitrary locally compact but non-compact abelian group we refer to Dang-Ngoc and Yor (1978). Sections 11.2 to 11.4. The Markov fields for the matrix Q of Section 11.2 were first investigated by Spitzer ( 1975b). He discovered the Markov chains \iu'° (u ïï 0) in ^{Q). The complete description of ^ ( 0 , Theorem (11.31), as well as Corollary (11.33) are due to Cox (1977a). The rôle of the random variables U and V seems to have been noticed here for the first time. Incidentally, Corollary (11.33) is merely a special case of an abstract characterization of all Markov chains in &(y) for an arbitrary Markov specification y on Z; see Föllmer (1975b) and Winkler (1981) (for the case of a countable state space) and Dang-Ngoc and Yor (1978) (for general E). The matrix Q of Section 11.3 was first considered by Kalikow (1977) (for a special choice of the parameters p and q). He discovered the remarkable semi-stationary entrance law (11.38), and thereby the Markov chains pj(j e ZU {—oo})in ^{Q). The complete analysis of ^(Q) in Theorem (11.39) seems to be new. The third example of phase transition, Theorem (11.46), is taken from Spitzer (1975b). To obtain further examples of phase transition it is sufficient to note that | ex ^(Q) | > 1 for any stochastic matrix Q which admits a non-stationary entrance law; the conditions under which such entrance laws exist were analyzed by Cox (1977b). On the other hand, we know from Section 6.1 that it is quite easy to find examples of inhomogeneous Markov models on N or Z which exhibit a phase transition; such examples can also be found in the paper of Winkler (1981).

474

Bibliographical Notes

Chapter 12 Section 12.1. The literature on Markov systems on trees may be split into two, rather unconnected, parts: the work of mathematicians and that of physicists. The following notes refer mainly to the first part. The mathematical literature on Markov fields on trees began with the work of Preston (1974a) and Spitzer (1975a). These authors were concerned with the case \E\ = 2, S = ^^(d). They introduced the concept of a completely homogeneous Markov chain and used it to study the phase transition problem for Markov random fields; see the notes on Section 12.2 below. A general study of countable-state Markov fields on trees was made later by Zachary (1983). He proved a version of Theorem (12.6). (A weaker version of this theorem appeared already in Spitzer (1975a).) Zachary also introduced the concept of a boundary law (in the sense of Definition (2.10)) and deduced the characterization (12.12) of the Markov chains in <&(y). Corollaries (12.17) and (12.18) are also due to him. The case of an arbitrary compact state space was investigated by Miyamoto (1982). He found sufficient conditions for the existence, reversibility, and uniqueness of Markov chains in y{y), and established the breaking of a continuous symmetry in a particular model. In the case of a countably infinite state space, the existence problem for Markov chains in <&(y) was studied by Zachary (1985). A variational characterization of &(y) for homogeneous Markov specifications y on homogeneous graphs (like Zd or ^^"(d)) can be found in the article of Föllmer and Snell (1977). Section 12.2. The existence of a phase transition for the Ising model on the Cayley tree ^^(d) for d ^ 2 was established independently by Katsura and Takizawa (1974) and Preston (1974a). The work of these authors was based on an analysis of (a version of) the fundamental equation (12.22). In particular, Preston showed that ^{J,h) contains one, two, or three completely homogeneous Markov chains. In the ferromagnetic case J > 0, he used the FKG-Holley inequality to show that \&(J,h)\ = 1 if and only if there is only one such Markov chain. Zachary (1983) showed that either \^(J,h)\ = 1 or \ex^(J,h)\ = oo; an extension of this result to general ferromagnetic or anti-ferromagnetic systems can be found in Zachary (1985). For J > 0 and h = 0, Higuchi (1977) showed that the boundary condition which is equal to + 1 on a Cayley subtree and equal to — 1 otherwise leads to a limiting Gibbs measure which is extreme and has a positive magnetization at the origin. A further interesting phenomenon can be observed at h = 0: Besides the critical point J = J(d) (or w = d~l) for the nonuniqueness of the Gibbs measure, there is a second critical point w = d~v2. This critical point involves the intermediate Markov chain \i# (which coincides with the limit of Gibbs distributions with free boundary condition). Falk (1975) observed that the magnetic susceptibility lim jA^I"1 X //#(ö'/ö'j) (with ^N = {i e S: d(i,i0) ^ N} N-œ

UjsAn

for some fixed i0 e S) is finite if and only if w < d ' . By an application of a general limit theorem for multitype branching processes, Moore and Snell (1979) established a central limit theorem for \i# when w < d~1/2; this limit theorem breaks down when w 2Î d~lj2. In this reference and in Higuchi (1977) it is also shown that / i # is non-trivial on ST, and thus not extreme, when w 5: d~lß. Nevertheless, fi# is a limiting Gibbs measure; cf. Higuchi (1977). It is an open question of whether ^ # is extreme for d~l < w < d~112. The antiferromagnetic case J < 0 was analyzed by Spitzer (1975a) and later by Zachary (1983). In particular, Spitzer introduced the alternating Markov chains and showed that \&(J,h)\ = 1 if and only if equation (12.35) (which in his setting takes a different form) has a unique solution. He also computed the uniqueness region for d = 2. Molchanov (1978) studied the uniqueness region in the limit d -> oo.

Bibliographical Notes

475

The analysis of the Cayley-tree Ising model can be extended in several directions. On the one hand, there is the Potts model which extends the Ising model to the case |£| ^ 3. A study of this model can be found in Miyamoto (1982), Zachary (1983), Peruggi (1984), and Peruggi et al. (1983, 1985, 1987). On the other hand, one can examine Ising-type models with next-nearest neighbour interaction; for a preliminary discussion of such a model see Mariz et al. (1985). Chapter 13 The study of multidimensional Gaussian fields with prescribed conditional distributions was initiated by Dobrushin in a talk in 1966 and began with the work of Rozanov (1967a). Rozanov considered the case of homogeneous interactions J of finite range and obtained a result which is similar to (and includes) our Corollary (13.42). (In the one-dimensional case, the linear interpolation problem had already been investigated before; see Rozanov (1960, 1967b).) This work was continued and improved by Chay (1972) and Pitt (1975) who were particularly concerned with the consequences of the Markov property (13.20)(ii) when condition (13.20)(i) fails. The case of homogeneous nearest-neighbour interactions was studied in detail by Spitzer (1974). In particular, Spitzer showed how the potential theory of Markov chains can be used to solve the linear interpolation problem; cf. Comments (13.14) and (13.28). This kind of ideas was also exploited by Williams (1973). The main results of this chapter were obtained independently by Dobrushin (1980) and Kiinsch (1979,1980; see also 1981b). Dobrushin's main result is Theorem (13.36). Kiinsch proved the same result and also considered the inhomogeneous case. In fact, his thesis (Kiinsch 1980) was our main source for the results and proofs of Sections 13.1 and 13.2. A different approach to Theorems (13.20) and (13.26) can be found in the diploma thesis of Bodden (1981); he used the technique of reproducing kernel Hilbert spaces. The same technique was exploited by Preston (1979) in order to construct Gaussian specifications for Gaussian fields with general (possibly continuous) parameter set. As a matter of fact, (generalized) Gaussian fields with parameter set Ud constitute a complete subject in their own right which plays a major role in Mathematical Physics, especially in quantum field theory, but cannot be discussed here. We just mention the paper of Holley and Stroock (1980) which contains an analogue of Theorem (13.26) for £f'(Ud)-valued Gaussian fields. Concerning the additional results of Section 13.3, we note that Remark (13.39) is taken from Dobrushin (1980) and Corollary (13.40) is taken from Kiinsch (1980). The uniqueness of tempered Gaussian fields can also be deduced from general uniqueness theorems for unbounded spin systems; cf. the Notes on Section 8.1. The Gaussian model with homogeneous nearest-neighbour interaction has, in fact, been studied for a long time, especially by physicists. Two early references are Kac and Berlin (1952) and Montroll (1956). As a sample from the more recent literature on the nearestneighbour case we mention the following few articles. Brascamp et al. ( 1975) discussed the rôle of the lattice dimensionality for harmonic (and certain anharmonic) oscillators. Moran (1973) considered the case S = Z2, J(l) > 0 and constructed the unique element of &e(yJ,°) as a limit of Gibbs distributions with periodic boundary conditions. Benfatto et al. (1978) gave a complete description of ex^{yJ'°) in the case S = Z, J(l) > 0. Finally, in the case of three lattice dimensions, Bricmont, Lebowitz and Maes (1987) showed that the unique centered Gauss field [ic e ^(yJ-°) exhibits a critical percolation behaviour: There is a finite threshold value tc ^ 0 such that, for all t < tc, the random set {;' e J?: cr; §; t} contains an infinite cluster /^-almost surely, whereas for t > tc this is not the case.

476

Bibliographical Notes

Since Gaussian models are relatively easy to handle, it is natural to ask whether there is any relationship between Gaussian and certain non-Gaussian models. One attempt in this direction was made by Kac and Berlin (1952) by introducing the so-called spherical model. Roughly speaking, this model is conditionally Gaussian, with a condition which was intended to imply some similarity to the Ising model, namely that £ of = |A| for the finite volume A at hand. In fact, Stanley (1968) discovered that the spherical model can be viewed, in some sense, as a limit of classical Heisenberg models when the spin dimensionality tends to infinity; see also Kac and Thompson (1971). The (limiting) Gibbs measures of the spherical model were investigated by Molchanov and Sudarev (1975) and Brankov and Danchev (1987). For further results we refer to the recent paper of Fisher and Privman (1986) and the references therein. A more concrete relationship between Gaussian and non-Gaussian models can be established by comparing the correlation functions of these models. One inequality of this type is the Gaussian inequality of Newman (1975); see the Notes on Chapter 2, Supplement C. A different Gaussian inequality can be found in the papers of Sokal (1982a) and Tasaki (1985); this inequality compares the correlations of Gaussian systems with those of ferromagnetic systems which have the same quadratic pair interaction but different a priori measures. Chapter 14 Section 14.1. Theorems (14.5) and (14.10) are standard results of Ergodic Theory. Some relevant comments are already contained in the Notes on Section 7.1 and 7.3. So we will only refer again to the classical paper of Kryloff and Bogoliouboff (1937) (which treated the case of a single transformation) and to the articles of Blum and Hanson (1960), Farrell (1962), Varadarajan (1963), Maitra (1977), and Dynkin (1978). Our approach to the ergodic decomposition theorem (14.10) coincides with that in the last reference. Theorem (14.12) can be traced back to Parthasarathy (1961); he considered the case d = 1, |£| < oo. In the same case, Kieffer (1974) obtained the following stronger result: For each JX s 3P@^l, 3?) which is positive on all cylinder events and any A e 9* there exists some vA Ë ex ^ e (£î, &) such that /j = vAon J \ . In the same year as Parthasarathy, Poulsen (1961) constructed a different example of a compact metrizable simplex with a dense extreme boundary. Such simplices turn out to have many remarkable properties. For example, all such simplices are affinely homeomorphic to each other, they contain all compact metrizable simplices as faces, and all affine homeomorphisms between two such faces can be extended to the whole simplex; cf. Lindenstrauss et al. (1978) and the survey paper of Olsen (1980). Section 14.2. Theorems (14.15) and (14.17) should be ascribed to Lanford and Ruelle (1969). For, these authors proved the variational principle (15.39), and this was known to imply that %($) is a face of SP&{Q., .
Bibliographical Notes

477

3T). Lemma (14.19) can already be found in Hunt (1966), p. 47, and Theorem (14.20) appeared for the first time in Georgii (1973). As a final remark, we note that in some cases it is possible to give a complete description of 4ß0(y); see the Notes on Section 16.2. Appendix 14.A. The multidimensional ergodic theorems (14.A3) and (14.A8) are due to Wiener (1939). Our proof of (14.A8) and, in particular, the covering argument in the proof of Lemma (14.A6) follow Wiener's ideas. The effectiveness of these ideas was demonstrated by Akcoglu and Krengel (1981) who used the same arguments to prove an ergodic theorem for superadditive processes. The simplest proof of the one-dimensional ergodic theorem seems to be that of Katznelson and Weiss (1982). A standard reference for ergodic theorems for more general group actions and "regular" sequences (A„)„g 1 (rather than sequences of cubes) is the paper of Tempelman (1972); see also Ornstein and Weiss (1983) and (for the case of Reactions) Becker (1981). Further references and a comprehensive treatment of all kinds of ergodic theorems can be found in the book of Krengel (1985). Chapter 15 Section 15.1. As is well-known, the concept of entropy (of a probability measure) was introduced by Shannon (1948) in his fundamental work on Information Theory, but the origins of this concept lie, of course, in Statistical Mechanics; see, for example, the report of Wehrl (1978). The concept of relative entropy is due to Kullback and Leibler (1951). Its properties are, by now, standard. A few relevant references are Perez (1957a), Pinsker (1964), and Ochs (1976a). There are various axiomatic characterizations of entropy and relative entropy which may be used to justify these concepts; see Aczél and Darôczy (1975) and Ochs (1976b). Particular interest in relative entropy comes from its use as a distance between probability measures and its significance for the probabilistic Large Deviation Theory; see, for example, Csiszär (1967, 1975, 1984), Groeneboom et al. (1979), Varadhan (1984), and Ellis (1985). Proposition (15.6) is a result of Perez (1957a); our proof follows Nguyen andZessin (1979). For the case of an infinite X see Tempelman (1984). Section 15.2. The existence of the specific entropy was proved first by Shannon (1948) (in the particular case when S = Z, |£| < oo, and A is counting measure). McMillan (1953) and Breiman (1957) obtained the stronger result that the integrands in the definition of \A„\~1J^An(fi) converge in L 1 ^ ) and ^-almost surely. The L 1 (^-convergence result of McMillan was extended by Perez (1957b) to the case of general E and A. The more recent literature on the one-dimensional case includes the papers of Derrienic (1983) and Barron (1985). The multidimensional version of Shannon's result, Theorem (15.12), is due to Robinson and Ruelle (1976). Multidimensional extensions of the theorems of McMillan and Breiman to various settings (including point processes on Ud) and various degrees of generality can be found in the papers of Fritz (1970), Conze (1972), Thouvenot (1972), Kieffer (1975), Nguyen and Zessin (1979), Moulin Ollagnier (1983), Ornstein and Weiss (1983), and Tempelman (1984). It should be emphasized here that specific entropy is the guiding example of a closely related concept, the Kolmogorov-Sinai entropy, which plays a central rôle in Ergodic Theory. In fact, many of the references above deal with this (in one respect more general) notion of entropy. For an introduction to this subject we refer to Parry (1969) and Walters (1982).

478

Bibliographical Notes

The first two assertions of Proposition (15.14) can already be found in Robinson and Ruelle (1967). In the case where £ is a complete separable metric space with Borel ff-algebra S, the arguments of this paper even imply the stronger result that Â{ • ) is upper semicontinuous relative to the weak topology; see Israel (1979) for an explicit proof of this fact. The last assertion of Proposition (15.14) is a slight modification of an auxiliary result of Groeneboom et al. (1979). The one-dimensional version of Proposition (15.16) is classical and can be found in many texts on Ergodic Theory or Information Theory. In the multidimensional case, this result was derived by various authors, including Conze (1971), Föllmer (1973), Nguyen and Zessin (1979), and Tempelman (1984). Particular cases of Theorem (15.20) appeared already in Parry (1969) and Varadhan (1984). Section 15.3. The existence of the specific energy, Theorem (15.23), is physical "folklore" and was stated explicitly, for example, by Dobrushin (1968b) and Ruelle (1969). Theorem (15.30)(a), the existence of the pressure, is due to Gallavotti and Miracle-Sole (1967). The analogous result for point systems in fRd, though being much more difficult, had already been investigated before by van Hove (1949), Yang and Lee (1952), and numerous further authors; see Ruelle (1969) for the relevant references. The idea of proving Theorem (15.30) by means of Lemma (15.28) is due to Föllmer (1973). Extensions of Theorem (15.30)(a) to unbounded spin systems were obtained by Lebowitz and Presutti (1976), Kiinsch (1980, 1981a), and Bellissard and H0eghKrohn (1982). Corollary (15.35) asserts in particular that the pressure P(O) is equal to the maximum of h(ii) — as /i varies in 0*e(fi, ÏF). This statement (which is also called the variational principle) is due to Ruelle (1967). A similar result was obtained a few years later in Ergodic Theory: Dinaburg (1970) and Goodman (1971) showed that the so-called topological entropy of a continuous transformation T is the lowest upper bound of the Kolmorov-Sinai entropies of all t-invariant probability measures. This result implies the special case d = 1, O = 0 of Ruelle's variational principle. In the sequel, the concept of pressure found its way into Ergodic Theory, and it has indeed been possible to prove an ergodic theoretic variational principle which contains the results of Ruelle and Dinaburg-Goodman as special cases. We give here only a few relevant references, namely the papers of Ruelle (1973b), Misiurewicz (1977), Moulin Ollagnier and Pinchon (1979), and Pesin and Pitskel (1984), and the texts of Bowen (1975), Ruelle (1978), and Walters (1982). A more recent exposition of this theory is the book of Keller (1998). Section 15.4. The variational principle (15.39) is due to Lanford and Ruelle (1969). The first half, Corollary (15.35), was also obtained by Dobrushin (1968a, 1969). A transparent proof (which revealed the significance of relative entropy) was given later by Föllmer (1973). The proof given here is taken from Preston (1976). There are numerous further papers on the variational principle. Sullivan (1973b) used the Gibbsian representation theorem (2.30) to give a short proof of the weaker result that any shift-invariant Gibbs measure (for an arbitrary potential in â$@) which minimizes 4(-|0) is a Gibbs measure for O. In the case of dimension one, the variational principle was proved in various degrees of generality by Spitzer (1972), Ledrappier (1974), Gurevich (1984), and Nachtergaele and Siegers (1986). The case of unbounded spin systems was treated by Pirlot (1980) and Kiinsch (1980, 1981a, b). Tempelman (1984) extended the variational principle to the case when Zd is replaced by any amenable group.

Bibliographical Notes

479

As a matter of fact, the variational principle (15.39) has a dynamical counterpart: For certain Markov processes on Q which are such that all Gibbs measures for a given $ are time-invariant, Holley (1971) showed that the function t -> â(p.Pt\<&) is strictly decreasing for all \i e ^&(Cl,#")\^(0); here (F,),fe0 stands for the transition semigroup of the process. This, of course, immediately implies that â{fi\<&) can only be minimal when \x e ^©( 1. In such situations it is more natural to consider conditional free energies relative to fixed boundary conditions. A discussion of this point, and a suitable modification of the variational principle, can be found in the papers of Sewell (1974) and Föllmer and Snell (1977); see also Tempelman (1984). Section 15.5. The theory of large deviations has independent origins in Statistical Mechanics and Probability Theory. In traditional Statistical Mechanics, interest focused on the analysis of the thermodynamic functions in the various Gibbs ensembles and their interrelation in terms of Legendre transforms. Among the first publications which also considered the probabilistic implications were the lectures by Minlos (1968) and Lanford (1973). In Probability Theory, the subject of large deviations remained confined to some particular areas of Statistics and Information Theory, until in the late 1970s the work of Donsker and Varadhan initiated a rapid development of the field which is summarized in the monographs of Deuschel and Stroock (1989), Dembo and Zeitouni (1998), and den Hollander (2000). Probabilistic techniques were then also brought into contact with the concepts of Statistical Physics; examples of this interplay are the works of Ellis (1985), Comets (1986), Olla (1988), and Föllmer and Orey (1988). The discussion here, which forms a new section of this second edition, is a specialized and simplified version of Georgii (1993). The proof of the upper bound (15.46) was inspired by the approach of Csiszâr (1984) to large deviations of empirical distributions for i.i.d. random variables. The argument for the lower bound (15.47) appeared (in slight variants) in Comets (1986), Olla (1988) and Föllmer and Orey (1988). Corollary (15.48) corresponds to the large deviation result of Lanford (1973), which was also recovered by Olla (1988). An alternate approach to the material of this section, which proceeds from Lanford's ideas, has been developed by Lewis et al. (1995) and Pfister (2002); see also the examples in Lewis et al. (1994) and the exposition in Section 3.4 of Olivieri and Vares (2005). For some work on surface-order large deviations see p. 490. The equivalence of ensembles is one of the most central subjects of Statistical Mechanics. Its aim is the justification of grandcanonical Gibbs measures as limits of microcanonical Gibbs distributions, which are generally considered to be more fundamental. Classically, this equivalence is expressed by a thermodynamic relation of the form (15.49) between the (grand canonical) pressure and the microcanonical entropy per volume as a function of the energy level; the latter is defined in terms of microcanonical partition functions in the same way as the pressure in (15.31) in terms of grand canonical partition functions. In the setting of continuum models, this kind of

480

Bibliographical Notes

equivalence is thoroughly discussed in the books of Ruelle (1969), Martin-Löf (1979a) and Bovier (2006). The proof of the minimum free energy principle (15.58) can be understood as a way of converting these thermodynamic relations into limit theorems for conditional Gibbs distributions; it builds on the work of Csiszâr (1984). Similar conditional limit theorems are stated and proved in Deuschel et al. (1991), Dembo and Zeitouni (1998, Section 7.3), Lewis et al. (1995), and Pfister (2002). Convergence estimates in special cases are provided by Diaconis and Freedman (1988) and Johnson (2003). An analogous theory for point systems in Rd can be found in Georgii (1995). Chapter 16 Section 16.1. Proposition (16.1) is due to Gallavotti and Miracle-Sole (1967). For a discussion of the thermodynamic significance of the convexity of the pressure we refer to Wightman (1979). Proposition (16.3) can be traced back to Mazur (1933). An analogous result can also be shown for differentiability in the sense of Fréchet, provided 3S& is replaced by a sufficiently well-behaved subspace; see Phelps (1983) and Preiss and Zajicek (1984). On the other hand, there exist some natural subspaces of 3S& on which the pressure is nowhere Fréchet-differentiable; cf. Daniels and van Enter (1980). Proposition (16.7) is a theorem of Israel (1975); the particular case C = 3S& can be traced back to Bishop and Phelps (1963). Section 16.2. The main result of this section, Theorem (16.14), is due to Lanford and Robinson (1968). In fact, since the concept of a Gibbs measure was not yet known at that time, there authors established the one-to-one correspondence between all tangent functionals and all shift-invariant random fields which minimize the specific free energy. Gallavotti and Miracle-Sole (1967) had noticed before that any limiting Gibbs measure defines a tangent plane to the pressure. The proof of Lanford and Robinson was based on Proposition (16.32) and can also be found in the texts of Lanford (1973) and Preston (1976). Theorem (16.13) was hidden in the work of Lanford and Robinson (1968). The explicit statement appeared in an article of Ruelle (1973b). He showed by a direct argument that a P-bounded functional is necessarily the negative specific energy functional for a shift-invariant random field; this argument can also be found in Israel (1979). The combined proof of Theorem (16.13) and equation (16.12b) via Proposition (16.11) was discovered by Pirlot (1985). It was his idea to introduce the function A(L); cf. the first lines of the proof of (16.11). The remaining argument - the use of the separating hyperplane theorem in order to establish the duality theorem for conjugate convex functions - is quite standard and had already been used before to derive the conjugate variational principle (16.12b); see, for example, Ruelle (1978) and Israel (1979). In fact, this argument can be traced back to Fenchel (1949); he proved the duality theorem for convex functions on W. The general case is due to Brondstedt (1964); a modern exposition can be found in Ekeland and Temam (1976). Corollary (16.15) is a result of Griffiths and Ruelle (1971). An alternative approach to the strict convexity of the pressure was developed by Dobrushin and Nahapetian (1974). Concerning Corollary (16.17) we refer to the Notes on Corollary (8.37). We will now provide a number of references on two related subjects which lie beyond the scope of this book. A. The analyticity of the pressure. It is a central task of Statistical Mechanics to study the differentiability, or even analyticity, of the pressure (and other thermodynamic functions) with respect to some distinguished thermodynamic parameters such as temperature or external field. In fact, the analytic behaviour of the pressure is the classical thermodynamic indicator for the absence or existence of a phase

Bibliographical Notes

481

transition. This paragraph serves to give a short glimpse into the extensive literature on this field. The most famous result on the analyticity of the pressure is the circle theorem of Lee and Yang (1952). This theorem asserts the following. Consider a {— 1, l}-valued spin system with ferromagnetic pair interaction and external field h, and regard the quantity z = eh as a complex parameter. Then all zeroes of all partition functions (with free boundary condition), considered as functions of z, lie on the complex unit circle. This theorem readily implies that the pressure is an analytic function of h in the regions h > 0 and h < 0. Heilmann (1970) showed that the assumption of pair interaction is necessary. A transparent approach to the circle theorem was found by Asano (1970) and developed further by Ruelle (1971a, 1973a), Slawny (1973), and Gruber et al. (1975). Griffiths (1969) and Simon and Griffiths (1973) found a method of extending the Lee-Yang theorem to real valued spin systems with a particular type of a priori measure. Newman (1974) proved the Lee-Yang theorem for every a priori measure which satisfies this theorem in the particular case of no interaction. Dunlop (1979a, b) studied the zeroes of the partition functions for the plane rotor model. A general Lee-Yang theorem for multicomponent systems was finally proved by Lieb and Sokal (1981). For further references on this subject we refer to this paper and the book of Glimm and Jaffe (1981). The only reference we will add is the paper of Shlosman (1986c) who investigated a particular model with strange analytic properties. The Lee-Yang theorem and its variants depend on the ferromagnetic character of the interaction. There are various other ways of proving the infinite differentiability, or even the analyticity, of the pressure for (ferromagnetic, or non-ferromagnetic) systems at high temperatures, or at low temperatures, or at large external fields. Most of these take advantage of a sufficiently rapid decay of correlations and/or cluster expansion methods. Here is a small sample of relevant references in alphabetical order: Bricmont, Lebowitz and Pfister (1980), Dobrushin (1987), Dobrushin and Shlosman (1985b, 1987), Duneau et al. (1973, 1974, 1975), Glimm and Jaffe (1981, 1985), Israel (1976), Kotecky and Preiss (1986), Kunz (1978), Lebowitz (1972, 1975), Malyshev (1980), Malyshev and Minlos (1985), and Prakash (1983). Much work has also been devoted to the differentiability and analyticity of the pressure (and the correlation functions) of the Ising ferromagnet. We mention here the articles of Minlos and Sinai ( 1967a), Lebowitz and Penrose (1968), Gallavotti and Lebowitz (1973), Martin-Löf (1973), Isakov (1984), and Araki (1986). The pressure of the Ising antiferromagnet was investigated, for example, by Brascamp and Kunz (1973). Our final remark refers to one-dimensional systems. In this case, the pressure can be shown to be analytic under mild decay conditions on the potential; see Dobrushin (1973b, 1974), Cassandro and Olivieri (1981), Campanino et al. (1983), and Dobrushin and Shlosman (1985b, 1987). B. Consequences of differentiability for the structure of^@(<S>). Remark (16.6)(1) and Theorem (16.14) imply that ^©(O)! = 1 whenever the pressure is differentiable at ) is equivalent to the statement that the pressure, as extended to a space of "averagable" potentials which contains ^ 0 , is differentiable at $. Unfortunately, the differentiability of the pressure in all directions of a large space of interactions is difficult to verify in general. For some ferromagnetic interactions, however, the differentiability with respect to a single parameter already implies the uniqueness of the shift-invariant Gibbs measure or, at least, allows us to give a complete description of ^&('P).

482

Bibliographical Notes

The first result of this kind was obtained independently by Lebowitz and MartinLöf (1972) and Ruelle (1972). For { —1,1}-valued systems which are ferromagnetic in the sense of the FKG-Holley inequality, these authors showed that ^ ( O ) ! = 10(0)1 = 1 if and only if the derivative dhP() exists and is different from zero then |0 e (O)| = 1. We refer the reader to the papers of Lebowitz (1977,1978), Bricmont (1977a), Bricmont, Fontaine and Landau (1977), and Messager et al. (1978). In a number of cases (including the Ising ferromagnet in a non-zero external field), the existence of dhP(Q>) can be deduced from a Lee-Yang theorem. Alternatively, one can take advantage of the GHS inequality which implies the concavity, and thereby the continuity, of the right derivative d£ P() as a function of h for h > 0; see Preston (1974b). For a large class of ferromagnetic spin models it turns that out that the differentiability of the pressure with respect to the inverse temperature ß has a remarkable consequence. (By convexity, the derivative 8ßP(ßQ>) certainly exists for almost all values of ß, and it will usually exist when ß is small or large.) Namely, if dßP(ß<S>) exists then ex tS&(ß^>) consists of all images of a distinguished Gibbs measure fil® E ex g(ßQ>) under the action of the underlying symmetry group. In some particular cases (including the Ising ferromagnet) this implies that | ex 0 0 (/?) | ^ 2. The above statement has been proved for various kinds of models of different degrees of generality with the help of various symmetry inequalities. Here is a list of relevant references: Slawny (1973, 1974, 1976), Gruber, Hintermann and Merlini (1975, 1977), Bricmont (1977a), Lebowitz (1977,1978), Gruber and Lebowitz (1978), Messager et al. (1978), Holstynski and Slawny (1979), Bricmont, Lebowitz and Pfister(1981a), Pfister(1982,1985,1987), Fröhlich and Pfister (1983), Bricmont and Lebowitz (1986), Miekisz (1987a). Section 16.3. This section is based on the work of Israel (1975,1979) and Ruelle (1977). Using the same techniques, Israel (1975, 1979) derived some further interesting facts. To state these results we note first that in order to define the pressure and the specific energy for some shift-invariant it is not necessary to assume that
ii*iu^ z Mmoj ,430

is finite. Let 38g denote the Banach space of all shift-invariant 's with ||0|| # < oo. For a given $ e 38g, it is in general impossible to define a Gibbs specification y*. However, one can introduce the set 0@() of all "equilibrium states", i.e. of all H e ^ 0 (Q,.^) with P(0) = â{n) - O , 0 > . (Theorem (15.39) just states that %{$>) = 0 0 (O) when e 3S&.) The results of Israel now read as follows: (i) For any N 2: 1 and any nx,..., nN e ex ^@(Q, J*) there exists some <S> s ^ | s u c h t h a t / ^ , . . . , / ^ E e x ^ ( O ) . In particular, each /i E ex ^@(Q, #") is an equilibrium state for infinitely many distinct O's. (ii) The set of all e 36% with uncountably many ergodic equilibrium states is dense in 36%. As a matter of fact, the concept of an equilibrium state can also be introduced in the setting of Ergodic Theory; cf. the comments on Ruelle's variational principle in the Notes on Section 15.3. This concept has received considerable attention. For a

Bibliographical Notes

483

first orientation we refer the reader to the papers of Hofbauer (1977a, b) and Ledrappier (1977) and the texts of Bowen (1975), Ruelle (1978), and Walters (1982). Section 16.4. The basic result of this section, Proposition (16.32), was noticed first by Gallavotti and Miracle-Sole (1967). Theorem (16.33) is essentially due to Lanford and Robinson (1968). They proved the slightly weaker assertion that %() is contained in the closed convex hull of all limits lim fin with fin E ^0(J and 4>„ -> <£. Step 1 in the proof of (16.33) is taken from Ruelle (1978), Section 3.7. A functional analyst might prefer to prove the result of Step 2 by means of the separating hyperplane theorem; this can be done by a slight refinement of the argument in the last reference. Proposition (16.34) is taken from Israel (1979). Theorem (16.35) is a result of Sokal (1982b). A different approach to this theorem can be found in Israel (1986b). This paper also contains a remarkable improvement of Proposition (16.32): For any $ E ^ 9 and TV ^ 1, the set of all vectors (<S>1,...,Q>N)e@g with %(+ N

t M>") =1 for

n

all {h1,...,hN) e IR \{0} is a dense GS subset of aft®. ^ particular, a typical finitedimensional subspace of<M%does not contain any *F with |^ e 0F)| > 1. A key to this result is provided by the following fact which was noticed first by van Enter (1981, 1982) in some particular cases: For each short-range potential $ 6 3S@ and any observable /E£C, there exists a long range potential * P e J ä ( / ) such that \{fi(f): H e ^e(0 + t*¥)}\ = 1 for all t > 0. In the case where / is an order parameter for O, this fact may be expressed by saying that the long-range perturbations $ +1*¥, t > 0, of 4> are order-destroying. Chapter 17 The concept of reflection positivity was developed in quantum field theory in connection with the construction of relativistic fields in terms of Euclidean fields, a problem which was studied first by Nelson (1973). In a fundamental paper, Osterwalder and Schrader (1973) introduced a condition of positivity relative to time reflections. This so-called OS-positivity proved to play a central rôle in the passage from Euclidean space to Minkowski space. A slight variant of this condition was proposed by Hegerfeldt (1974) and Fröhlich (1974). When translated into the framework of classical lattice systems, this variant is nothing other than the condition of reflection positivity as defined here. The lattice version of reflection positivity appeared first in a paper of Hegerfeldt and Nappi (1977) and was exploited systematically by Fröhlich and Lieb (1978) and Fröhlich, Israel, Lieb and Simon (1978). The results of Section 17.2 on the reflection positivity of Gibbs distributions were developed in these three references. Just as with the concept of reflection positivity, the chessboard estimate (17.11) grew out of work in quantum field theory. Estimates of a similar kind were, in fact, used in several papers in this field, including those of Guerra et al. (1975), Glimm, Jaffe and Spencer (1975), and Fröhlich and Simon (1977). Fröhlich and Lieb (1978) derived the chessboard estimate in the version of Theorem (17.11). In particular, these authors found the nice argument in the proof of Lemma (17.9). The usefulness of this estimate for proving phase transitions in various kinds of models became definitively clear from the two remarkable articles of Fröhlich, Israel, Lieb and Simon (1978, 1980); cf. also the report of Fröhlich (1978).

484

Bibliographical Notes

Chapter 18 Sections 18.1 and 18.2. The results of these sections are taken from Georgii (1981b). Of course, this paper had many precursors. In the case of a finite state space E, a theorem very similar to Theorem (18.25) was already proved by Gertsik (1976), and the Peierls condition which appears in the hypothesis of Gertsik's result was related to the existence of a stable partition of the local ground states by Holsztynski and Slawny (1978). The idea of using the chessboard estimate for the contour estimate in the Peierls argument was initiated by Fröhlich, Israel, Lieb and Simon (1978, 1980). The notion of a ground-state ocean was already implicitly contained in the original paper of Peierls (1936). The close relationship between percolation and phase transitions became particularly evident in the case of the Ising ferromagnet; cf. the Notes on Subsection 18.3.1 below. As a general reference on Percolation Theory we mention here the book and a survey article of Kesten (1982, 1987). As a matter of fact, the proof of Lemma (18.16) takes advantage of ideas which have been used first by Harris (1960) in a pioneering paper on independent percolation. The idea of using the chessboard estimate as a compactness estimate in an existence proof for Gibbs measures (cf. Proposition (18.12)) appeared first in a paper of Dobrushin and Shlosman (1981) and was exploited systematically by Shlosman ( 1984,1986d). Asa final remark, we emphasize again that Lemma (18.10) is the only place in Chapters 18 and 19 where we make use of reflection positivity. Consequently, the results of these chapters carry over to any other model which satisfies a suitable version of Lemma (18.10). One possible substitute for Lemma (18.10) is the contour estimate of Bricmont and Fontaine (1981) which applies to certain ferromagnetic systems. Section 18.3. 18.3.1 Ising ferromagnet. As a complement to the Bibliographical Notes on Sections 3.2 and 6.2, we add here some references on the percolative behaviour of the Ising ferromagnet. The intimate connection between percolation and phase transition in this model was realized first by Coniglio et al. (1976). They showed that |^(jßO)| > 1 if and only if, with probability 1 relative to /if®, there exists an infinite cluster of ( + ) spins. The "if" part only holds in two dimensions; an alternative proof can be found in Higuchi (1985). A nice proof of the "only if" part was given by Russo (1979). Russo also showed that relative to any // e ex^(/?) which is different from /if® and nßJ" there exists both an infinite ( + ) cluster and an infinite ( —) cluster, and he obtained some additional information on these clusters. Using this information, Aizenman (1980a) and Higuchi (1981) were able to show that such a coexistence of infinite clusters of different signs is impossible in two dimensions. In this way they proved the long-standing conjecture that ex^(/JO) = {/^®,/4®} for all ß > ßc. In three or more dimensions, the coexistence of two infinite clusters of different signs is possible and does indeed occur for small ß; this follows from work of Campanino and Russo (1985). In two dimensions, it is also known that the islands in the infinite ( + ) ocean are typically small. The best result of this type was recently obtained by Chayes, Chayes and Schonmann (1978): For all ß > ßc, the /^-probability of the event that the origin belongs to an island of diameter n decays exponentially fast as n -> oo. A slightly weaker result was established by Higuchi (1986,1987) by more general arguments. There is also some interest in the percolative region of the Ising ferromagnet in an external field h ^ 0. For each h =£ 0 and any inverse temperature ß, the associated Gibbs measure fiß,h is unique (cf. the notes on Section 16.2), and the percolative region is defined as the set of all pairs (ß, h) which are such that there exists an infinite ( + )

Bibliographical Notes

485

cluster (resp. an infinite ( —) cluster) //•''-almost surely. If h is sufficiently large, the existence of an infinite ( + ) cluster can be shown by a simple adaptation of the Peierls argument; see, for example, Molchanov and Stepanov (1983). Some further information on the percolative region can be deduced from the FKG-Holley inequality, see Coniglio et al. (1977), Higuchi (1987), and Aizenman, Bricmont and Lebowitz (1987). The precise shape of the percolative region is still unknown. 18.3.2 Ising antiferromagnet. The existence of a phase transition for the Ising antiferromagnet in an external field was first proved by Dobrushin (1968c) by means of a version of the Peierls argument. Di Liberto (1973) showed that for sufficiently large ß there is only one shift-invariant Gibbs measure, namely (//f® + /uf*)/2. Heilmann (1974) established the existence of phase transitions for Ising antiferromagnets on triangular, hexagonal, and diamond lattices. 18.3.3 The Gertsik-Dobrushin model. This model bears its name from the work of Gertsik and Dobrushin (1974) who derived the phase diagram and established the four kinds of phase transition. Holsztynski and Slawny (1978) showed that these results are an immediate consequence of the general theory of Pirogov and Sinai (1976). The three-dimensional version of this model was analyzed by Kashapov (1978). 18.3.4 Hard-core lattice gases. It was again Dobrushin (1968c) who proved the existence of a phase transition for the hard-core lattice gas with potential (18.30). He even showed that the same result holds in the presence of a sufficiently small longrange interaction. Hard-core models on more general lattices were studied by Runnels (1975). The name of the potential (18.31) refers to the work of Widom and Rowlinson (1970) who introduced and studied the analoguous model of point particles in Ud. (As a matter of fact, this model is the only continuous model which has been shown to exhibit a phase transition; see Ruelle (1971b).) Lebowitz and Gallavotti (1971) introduced the lattice version of the Widom-Rowlinson model and established the existence of a phase transition. For an extension of this result to a hard core model with more than two types of particles see Runnels and Lebowitz (1974). In three dimensions, the Widom-Rowlinson lattice model admits non-shift invariant Gibbs measures which describe the coexistence of two phases with a rigid interface; see Bricmont, Lebowitz, Pfister and Olivieri (1977). Lattice gas models with exclusion between nearest and next-nearest neighbours were studied by Huckaby (1983) and Fröhlich and Huckaby (1985). 18.3.5 Potts' model. This model was introduced by Potts (1952) (with cp — — Ifo})On the basis of a duality transformation, he conjectured that (in two dimensions) the pressure ß -» P(ß) should exhibit a single singularity at ßc = ^log(l + ^/JV). Baxter (1973) and Hintermann et al. (1978) showed that this is indeed the case, at least when JV 2; 4. The partition function of the Potts model can be extended in a natural way to the case when JV is any positive real number, and this extension is intimately related to certain percolation functions. For this and further properties of the Potts model we refer to the survey article of Wu (1982). The existence of JV distinct phases for large ß seems to have been proved first by Pirogov (1975) who obtained a much more general result. As a matter of fact, if ß is sufficiently large then there exist no other extreme Gibbs measures, at least when JV is large or a product of distinct primes; see Laanait et al. (1986), Martirosian (1986), and Miekisz (1987a). An interesting phenomenon was observed by Pfister (1982) in the so-called Ashkin-Teller model which is isomorphic to the Potts model with JV = 4 and a particular cp: There exist two distinct critical temperatures which correspond to the breaking of two different symmetries. In fact, Pfister even provided a scheme for the construction of models with an arbitrarily large number of critical temperatures.

486

Bibliographical Notes

18.3.6 Anisotropic Heisenberg ferromagnet. The first proof of a phase transition in this model is due to Bortz and Griffiths (1972), though only in the case when the parameter <x of anisotropy is very small. The general case of an arbitrary <x < 1 was first treated successfully by Malyshev (1975a). An extension of his argument to abstract models of a similar type can be found in Kuroda (1978). Fröhlich and Lieb (1978) reobtained Malyshev's result using reflection positivity and the chessboard estimate; see also Fröhlich et al. (1980). By a refinement of this technique, Bricmont and Fontaine (1982) were able to show that the critical inverse temperature diverges at most logarithmically as a -> 1. In addition, it is possible to compare the anisotropic Heisenberg models with different spin dimensions by means of various correlation inequalities; see Kunz et al. (1976) and Dunlop (1985). 18.3.7 Anisotropic Heisenberg antiferromagnets. The phase transition in these models was first established by Fröhlich and Lieb (1978) and Fröhlich et al. (1980). Using different methods, Calheiros (1983) treated an abstract version of this kind of antiferromagnet. 18.3.8 Shlosmari's plane rotor. This model was invented by Shlosman (1980) as an example of a model with an unbroken continuous symmetry which exhibits a breaking of an additional discrete symmetry. 18.3.9 Double-well potentials. For models of continuous spins with a double-well self-potential, the non-uniqueness of the Gibbs measures was first proved by van Beijeren and Sylvester (1978) and (by means of the chessboard estimate) Fröhlich and Lieb (1978). In quantum field theory, an analoguous result was obtained before by Glimm, Jaffe and Spencer (1975). 18.3.10 Slawny's u-model. This model was first studied by Slawny (1974). His proof of Proposition (18.41) exploits the fact that $ is ferromagnetic in the sense of the inequalities of Griffiths; cf. the Notes on Chapter 2. As we have noted there, these inequalities imply the existence of an extreme Gibbs measure which is shift-invariant and has maximal magnetization. Taking this fact for granted, one can simplify our proof of Proposition (18.41) considerably. For a further discussion of this model we refer to Fröhlich et al. (1980) and Slawny (1986).

Chapter 19 Sections 19.1 and 19.2. Theorem (19.6) is similar in spirit to the main result of the Pirogov-Sinai theory which will be discussed below. In fact, Theorem (19.6) corresponds to an early version of this theory, namely that of Pirogov (1975). Pirogov's result only covers the case of a finite space but is not limited to C-potentials. Fröhlich, Israel, Lieb and Simon (1980) were the first who obtained this kind of result in the case of an infinite state space. They used the technique of reflection positivity together with Sperner's lemma. Our treatment in Section 19.2 combines these ideas with those of Georgii (1981b) and is published here for the first time. A further approach to phase transitions without symmetry was developed recently by Gawedzki et al. (1987). The model of Example (19.8) was introduced independently by Blume (1966) and Capel (1966). Lebowitz and Gallavotti (1971) showed that |ex%(/?($ + hO1))! ^ 2 for all h when ß is large enough, cf. also Bortz (1972). For recent work on this model we refer to Bricmont and Slawny (1986) and Bricmont and Lebowitz (1987). The existence of three phases for the stabilized critical Ising antiferromagnet of Example (19.9) was first proved by Pirogov (1975) and again by Fröhlich et al. (1980). The analogous result for the lattice gas model in Example (19.9) is due to Cassandro et al. (1975).

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487

The Pirogov-Sinai theory. This subject is a highlight of the theory of phase transitions and one of the major omissions of this book. So we will at least give a few cursory comments. The main theorem reads as follows. Let E be finite and O be a shift-periodic potential of finite range. Suppose O admits N distinct periodic ground states cu 1 ,..., coN e Q which satisfy the so-called Peierls condition. [Roughly speaking, this condition establishes a relationship between energy and geometry which expresses the stability of the ground states. Namely, there exists a constant c > 0 such that, for each 1 ;g n :£ N and any local perturbation £ of co", H%(Q - H%{w") > c|<3Ç|. <3Ç, the "boundary" of £, is defined as the union of all cubes (of afixed,sufficiently large size) on which Ï, is different from each a>k, 1 ^ k ^ N.] Finally, let O 1 , ..., O^ - 1 be any periodic potentials which resolve the partition {{co 1 },..., {coN}} in a sense very similar to (19.4). Then the following conclusion holds. If ß is sufficiently large, there exists a neighbourhood U of the origin in R ^ 1 and a homeomorphism cp: h -» ((p^h),..., (pN(h)) from U onto a neighbourhood of the origin in dUl such that lex^/SO*)! ^ I{1 ^ « ^ JV: q>n(h) = 0}| for all he U. More precisely, for each he U and any n with q>n(h) = 0 there exists a shift-periodic extreme Gibbs measure n„ for /SO'1 which is a random perturbation of co". In fact, /i„ is the limiting Gibbs measure relative to the boundary condition co". Stated briefly, this result means that the qualitative picture at temperature zero (which associates to each h the set of all periodic ground states for $>h) remains valid at any sufficiently low temperature ß~l. The theorem above is the result of Pirogov and Sinai (1976). To see the development of ideas one should also consult their earlier papers, viz. Pirogov and Sinai (1974) and Pirogov (1974, 1975). A full exposition can be found in the book of Sinai (1982). There exist several variants, developments, and applications of the Pirogov-Sinai theory. A few relevant references are the papers of Kotecky and Preiss (1984), Zahradnik (1984), Bricmont, Kuroda and Lebowitz (1985), and Gruber and Sütö (1987). Galeb (1982), Malyshev et al. (1983), and Dobrushin and Zahradnik (1985) extended the theory to certain systems with infinite state spaces, Park (1988) extended it to interactions of infinite range, Dinaburg and Sinai (1985) used the technique of this theory to analyze the so-called axial next-nearest neighbour Ising model in three dimensions, and Imbrie (1981) obtained a Pirogov-Sinai type result for the P(
488

Bibliographical Notes

has already been obtained before in quantum field theory by Gawedzki (1978); his theorem corresponds to the case p = 1 of Example (19.19). The case of two competing wells for the self-potential was also treated by Dobrushin and Zahradnik (1985) using different methods. In particular, it is shown in this paper that there exist at most two shift-invariant phases. In three or more dimensions, there exist infinitely many nonshift-invariant Gibbs measures which describe the coexistence of two phases (which correspond to the two potential wells) with a rigid interface, see Zahradnik (1986). Finally, Pecherski and Shlosman (1987) studied the case of a self-potential which exhibits a sequence of local minima that converge to a unique absolute minimum. They showed that the Gibbs measure is non-unique at arbitrarily low temperatures, even though there is only one ground state. The main result of Subsection 19.3.2, Theorem (19.23), is due to Kotecky and Shlosman (1982). Our proofs of Lemma (19.24) and (19.25) are taken from this reference. Further proofs of Theorem (19.23) were provided by Dinaburg and Sinai (1984) and Bricmont et al. (1985). Baxter (1973,1982b) showed that the specific energy [d+/dß)P{ß<&) and the "magnetization" n„{a0 = n) are discontinuous in ß at ßN when d = 2 and N > 4. See also the Notes on Subsection 18.3.5. Chapter 20 The main ideas of this chapter are due to Fröhlich, Simon and Spencer (1976). These authors proved the infrared bound (20.12), and thereby the existence of a continuous symmetry breaking, in the three-dimensional classical Heisenberg model. Their proof of the crucial "Gaussian domination lemma" (20.3) differs from that given here; cf. also Section III.3 of Sinai (1982). Fröhlich, Israel, Lieb and Simon (1978) observed that this lemma can be extended to general reflection positive models by means of the chessboard estimate. This paper, together with the reports of Fröhlich (1978) and Lieb (1978), is our main source. For an analoguous treatment of the quantum plane rotor model see Dyson, Lieb and Simon (1978) and the comments of Lieb (1978). The random ergodic theorem (20.23) can be traced back to Kakutani (1950); cf. also the additional references in the book of Krengel (1985). The existence of a phase transition in long-range models like those of Example (20.21) can also be proved by different methods. In fact, it was Dyson (1969a) who gave the first proof of a phase transition in a one-dimensional long-range model. He introduced an auxiliary long-range model, the so-called hierarchical model, which is particularly well-suited for a study of its large-scale behaviour, and showed that this model exhibits a phase transition. In view of Griffiths' inequalities, this phase transition carries over to arbitrary ferromagnetic models with a sufficiently slow decay of interaction. The potentials J(i) = \i\~", 1 < a < 2, are included. As a matter of fact, the hierarchical model turned out to be interesting in its own right, partly because it permits a detailed study of its behaviour near the critical temperature. A few relevant references are Dyson (1971), Bleher (1976, 1985), Bleher and Major (1987), and Chapter IV of Sinai (1982). Penrose (1986) studied the hierarchical model on so-called fractal lattices. The decay condition on J was slightly weakened by Dyson (1971) and Kolomytsev and Rokhlenko (1978), but the boundary case J{i) = \i\~2 remained open until the papers of Simon (1981) and Fröhlich and Spencer (1982). Simon showed that this J can never break an SO(N) symmetry with N > 1; cf. Comment (9.34)(2). On the other hand, Fröhlich and Spencer showed that the |!|" 2 -model exhibits a phase transition when E = {— 1,1}. A more detailed study of this case was made recently by Aizenman

Further Progress

489

et al. (1988). They obtained the remarkable result that the magnetization of the maximal Gibbs measure, considered as a function of ß, exhibits a jump discontinuity at ßc, the threshold value for non-uniqueness. They also showed that, in the case ß < ßc, the unique Gibbs measure has a quadratic falloff of pair correlations. The same conclusion holds for the maximal Gibbs measure when ß is large; see Imbrie (1982). A quite different approach to the phase transition problem in one-dimensional systems can be found in a paper of Hofbauer (1977a). He found a class of functions with several equilibrium states in the sense of Ergodic Theory; cf. the Notes on Section 16.3. Long-range models in two dimensions were first studied by Kunz and Pfister (1976). They established the existence of a continuous symmetry breaking in the two-dimensional plane rotor model with interaction J(i) = \i\~", 2 < a < 4. Their proof depends on an analysis of the two-dimensional hierarchical model. The nearestneighbour plane rotor model in d 5: 3 dimensions was investigated further by Bricmont, Fontaine et al. (1981). Among other things, these authors showed that the pair correlations of the pure phases decay like ß~l\i — j \ ~ d + 2 as \i —j\ -> oo. An analysis of the same model on a three-dimensional half-space can be found in Fröhlich and Pfister (1986).

Further Progress In the more than two decades since 1988 when the first edition appeared, the Gibbs theory has developed considerably in various directions. Here I can and will only hint at some main lines of research. The selection of references is necessarily incomplete and reflects my personal bias; its main purpose is to provide the reader with some key sources from which he or she can go on further. Let me begin with referring to a few general expositions. The first source to be mentioned is the monograph of Simon (1993), which was already announced in the first edition. It offers a large variety of results and techniques, many of which cannot be found here; the quantum case is also included. A smooth introduction to many topics of the present book can be found in the first part of Föllmer ( 1988). The basic principles of the theory are thoroughly discussed in the first part of van Enter et al. (1993). The survey of Georgii et al. (2001) puts its emphasis on geometric aspects of the theory (percolation and random cluster representation) as well as the use of FKG inequalities, and is therefore fairly disjoint to the material here. Further aspects of the Gibbs theory can be found in the recent volumes by Bovier (2006) and Presutti (2009) which have specific scopes (systems with random interaction resp. the interplay of micro-, meso- and macroscopic structures) but also contain much of what could naturally be included here. In particular, these sources provide a detailed discussion of the conceptual foundations of Statistical Mechanics. I will now briefly describe some main areas of recent research. The random-cluster representation of Ising and Potts models. In a series of three papers, Fortuin and Kasteleyn (1972) and Fortuin (1972) introduced a dependent per-

490

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eolation model, the so-called FK or random-cluster model, which allows a (slightly reduced) description of the Ising and Potts model at vanishing external field. Its extreme usefulness was only recognized in the late 1980s, especially in the work of Aizenman, Chayes et al. (1988). The key fact is that spontaneous magnetization in Ising and Potts ferromagnets corresponds to percolation in the associated randomcluster model. Moreover, the model allows a joint analysis of Bernoulli percolation, Ising and Potts models, and the FKG-Holley inequalities apply to it. It is also interesting in its own right; for example, the random-cluster measures admit a Gibbsian description for non-quasilocal specifications. An introduction to random-cluster models can be found in Chapter 6 of Georgii et al. (2001), and a detailed account of all achievements up to 2006 is Grimmett's monograph (2006). In particular, the latter source contains a random-cluster proof of Theorem (19.23). It is also shown that the existence of non-translation-invariant "Dobrushin states" for the Ising ferromagnet in space dimensions d > 3 (as stated in Part D of the Notes on Section 6.2) extends to the framework of random-cluster measures. On the other hand, Bodineau (2006) used the random-cluster representation to show that, in dimensions d > 3, all translationinvariant Gibbs measures for the Ising ferromagnet are mixtures of the two phases ix+ and ix_. Finally, we mention that Biskup et al. (2000) established a random-cluster representation also for Potts models in non-vanishing external fields, but so far this did not prove to be equally useful. We also add that the random-cluster representation carries over to multitype particle systems in Rd with intertype repulsion of WidomRowlinson type, see Chayes et al. (1995) and Georgii and Häggström (1996). Surface-order large deviations and the Wulff droplet. In Section 15.5 we considered events with a probability of order exp[—û( | A | ) ]. The volume order &{ | A | ) was correct because these events were untypical for the whole class of Gibbs measures. In the presence of phase transition, however, it is natural to ask for events which are untypical for one phase, but typical for another. In fact, such events are intimately connected with the phenomenon of phase separation: the formation of a droplet showing a "wrong" phase within a given phase. The probability of such events decays with an exponential rate of surface order. The precise rate is determined by the the approximate shape of the droplet, the so-called Wulff shape, viz. the shape which (for given volume) minimizes the surface tension along its boundary. This program has been fully carried out for the Ising ferromagnet at zero external field, first in two dimensions and at low temperatures. After the qualitative estimates of Minlos and Sinai (1967b), Schonmann (1987) and Föllmerand Ort (1988), a breakthrough came with the work of Dobrushin et al. (1992) who brought the Wulff shape into play and so identified the precise surface-order rate. A variation of their arguments, including various new tools and non-perturbative estimates, was developed by Pfister (1991). A next landmark was achieved by using the random-cluster representation and FKG inequalities: Ioffe and Schonmann (1998) established the Wulff droplet picture in two dimensions for all temperatures up to the critical temperature. In three or more dimensions, it was Pisztora (1996) who laid the groundwork for the random-cluster representation of the Ising model, and Cerf (2000) solved the analogous problem for Bernoulli percolation. Based on this, Cerf and Pisztora (2000) finally derived the complete Wulff picture under assumptions which hold for all ß > ßc, as implied by the later work of Bodineau (2005, 2006). Following an independent line of thought stemming from the theory of Kac potentials discussed below, Bodineau (1999) established the Wulff picture for large ß. Detailed reviews of these approaches are provided by Bodineau et al. (2000) and Cerf (2006).

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Further percolation methods. Besides the random-cluster representation of Ising and Potts models, there is a wealth of further possibilities of exploiting some geometric features of Gibbsian systems with nearest-neighbour interaction. Many of these are discussed in the above-mentioned survey by Georgii et al. (2001). To select only a few, we refer here to the so-called disagreement percolation as a way of deriving the uniqueness of Gibbs measures from the non-existence of infinite clusters of disagreement sites for any two configurations; this technique was worked out by van den Berg and Maes (1994). On the other hand, one can ask for the structure of the set of sites on which a random configuration agrees with a given ground configuration. The usefulness of such an analysis, which for the Ising ferromagnet on Z 2 was first demonstrated by Russo (1979), has been propagated further by Giacomin et al. (1995). In particular, Georgii and Higuchi (2000) used this approach to give a simplified proof of the Russo-Aizenman-Higuchi (RAH) result that the Ising ferromagnet in two dimensions admits only two distinct phases, namely ß+ and ß_; recall Part D of the Notes on Section 6.2. This proof applies also to the hard-core lattice gas and to planar lattices other than Z 2 . A quite different, finite-volume approach to the RAH theorem has been obtained quite recently by Coquille and Velenik (2010). The Ising ferromagnet on trees and nonamenable graphs. Theorem (12.31) on the Ising ferromagnet on Cayley trees can be complemented as follows: There is a second critical value J*(d) = arcoth *Jd such that ß# is extreme in
492

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Phase transitions predicted by mean-field models. Mean-field models are quite popular toy models of Statistical Mechanics; they assume that all spins interact in the same way, independently of their relative position. Due to this lack of spatial structure it is quite easy to compute all relevant quantities, but the free energies in the canonical ensemble are non-convex in general. This shows that mean-field systems are unphysical. Nevertheless, the existence of well-separated minima of the free energy in mean-field models is a strong indicator that a phase transition might also occur in realistic models when the interaction is sufficiently wide-spread or the dimension is sufficiently high. It is quite remarkable that this program could be carried through successfully in some particular cases. One class of interactions are the so-called Kac potentials, which are obtained by rescaling a function of compact support with a spreading-out factor so that its integral is kept fixed. The idea is then to introduce a suitable mesoscopic scale on which the local spin averages must fluctuate between the above minima in a way which is accessible to Pirogov-Sinai contour estimates. It requires a technical tour de force to accomplish this for finite-range Ising ferromagnets and - this is the real achievement - also for systems of point particles in M.d. All this is presented in full detail in the recent monograph of Presutti (2009). An interesting complement to Theorem ( 19.23) is the result of Gobron and Merola (2007), which states that the Af-state Potts model with Kac interaction exhibits a first-order phase transition in the sense of (19.23)(III) when N > 3andc? > 2. A quite different approach to phase transitions in lattice models with approximate mean-field behaviour relies on reflection positivity and chessboard estimates. It applies in particular to the Potts model in high dimensions or with infinite-range interactions, see Biskup and Chayes (2003), Biskup et al. (2006), Chayes (2009) and the review paper of Biskup (2009), which also gives a general overview of various applications of reflection positivity. Absence of continuous-symmetry breaking in two dimensions. Recall from Theorem (20.15) that a continuous spin symmetry can be broken in a two-dimensional system with pair interaction when the associated random walk is transient. It is therefore natural to ask if a continuous-symmetry breaking is impossible for interactions defining a recurrent random walk. Merkl and Wagner ( 1994) were the first who established such an extension of Theorem (9.20). They considered the case when the parameter set S is a locally finite graph and the interaction is nearest-neighbour, but they still required the smoothness assumption (9.20)(i). This latter condition, which was so far essential for all results on the absence of continuous-symmetry breaking, could be removed by Ioffe et al. (2002). Their principal idea is to consider the interaction as a small perturbation of a smooth potential. This suggestion was taken up by Richthammer (2009) who showed that, for point particle systems in R2, a breaking of translation-invariance is impossible even in the case of hard-core interactions. Almost and weakly Gibbsian measures. Although the class of Gibbs measures is large enough to contain a large variety of models that are significant for Statistical Physics, in a sense it turned out to be too small, at least if one insists on the quasilocality of the underlying specification: There are natural transformations which turn a Gibbs measure for a nice potential into a measure which is not consistent with any quasilocal specification. A detailed study of this phenomenon was carried out in the seminal paper of van Enter, Fernandez and Sokal (1993); a short survey is van Enter (1996). So the question arose for a larger class of measures which still keeps some flavour of the usual Gibbs theory. Originally, only the case of a finite state space E was considered, and two concepts emerged. A measure /J. is called "almost Gibbsian" if ix e Sf (j/) for a specification y which is quasilocal in a pointwise sense at all points of a tail measurable set of full measure; ß is "weakly Gibbsian" if ß e %?(<$>) for

Further Progress

493

a potential <$> for which the local Hamiltonians can be defined only pointwise on a tail event of full measure. A pointwise version of Theorem (2.30) implies that every almost Gibbsian measure is weakly Gibbs, but the converse is false in general (Maes et al. 1999a). Examples of non-Gibbsian but almost Gibbsian measures are the maximal and minimal random-cluster measures; see Theorems (4.31) and (4.34b) of Grimmett (2006). Typical examples of only weakly Gibbsian measures are the projections of a

the two-dimensional Ising plus-phase ß^_ for ß > ßc onto one-dimensional layers (Dobrushin and Shlosman 1999), and block spin transformations of low-temperature phases in the Pirogov-Sinai theory (Maes et al. 2000). General orientation can be found in the surveys by van Enter et al. (2000) and Fernandez (2006). For particular classes of weakly Gibbsian measures, a variational principle (or, at least, the first part of it) was established by Lefevere (1999) and Maes et al. (1999b). In general, however, the class of weakly Gibbsian measures is too broad for the second part of the variational principle to hold; this was shown by Külske et al. (2004). In view of this, van Enter and Verbitskiy (2004) proposed a class of "intuitively weakly Gibbsian" measures, which lies strictly between the almost and weakly Gibbsian measures and for which the second part of the variational principle can be shown to hold. A recent review of non-Gibbsianness for systems of continuous spins is van Enter et al. (2010). Unbounded spin systems. The prime examples of unbounded spin systems are harmonic oscillators, i.e., the Gaussian fields of Chapter 13. The variational principle and large deviation results for such models have been investigated in great generality by Caputo and Deuschel (2000, 2001). For anharmonic oscillators (with confining self-potentials), a smooth functionalanalytic approach to the existence of (tempered) Gibbs measures has been provided by Albeverio et al. (2000). The techniques there can also be applied to systems of "spins" that consist of Brownian loops; in view of the Feynman-Kac formula, such systems provide a Euclidean description of quantum crystals. The development in this direction has been summarized in the monograph by Albeverio et al. (2009). Another direction of research are the Ginzburg-Landau interface models; these are obtained from the anharmonic oscillators by dropping the confining self-interaction and taking a nearest-neighbour interaction of the form V(<7, — Oj), \i—j\ = 1. In view of the non-existence result (9.24), the real object of interest is then the gradient field (07 — CT/)|I-;I=J on the nearest-neighbour bonds of the lattice. Funaki and Spohn (1997) introduced a Gibbsian description of these gradient fields and showed that for each average tilt of the surface there exists a unique shiftinvariant gradient Gibbs measure, provided V is strictly convex. For non-convex V, however, the situation is different. If V has a suitable double-well, there exist different gradient Gibbs measures with the same tilt (Biskup and Kotecky 2007). A general theory of gradient Gibbs measures and solid-on-solid models, including a variational principle and large deviation results, can be found in Sheffield (2005). A further nice survey of recent progress in interface models is Velenik (2006). Further directions. Osada and Spohn (1999) introduced Gibbs measures in a quite different setting: the configuration space is the path space C(R, M), the reference measure along an interval is Brownian bridge, and the interaction consists of a singletime potential and a pair interaction between distinct time points. This setup shows up when one applies the Feynman-Kac formula to a single quantum particle that interacts with a free Boson field. The progress in this direction is documented in the forthcoming book of Lörinczi et al. (2011). A quite different class of objects are the determinantal point processes on Zd or d R , which are characterized by an explicit formula for their correlation functions in terms of determinants and have received quite a lot of attention recently. Under certain

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conditions on the underlying integral kernel, these processes turn out to be Gibbsian, see Shirai and Takahashi (2003) for the discrete case and Georgii and Yoo (2005) for the continuous case. Next, there is the huge field of disordered systems, namely of Gibbsian systems with random interactions. These include randomly diluted ferromagnets, spin glasses, the Hopfield model, and various other models, which now form a lively research area of considerable physical interest. Here we confine ourselves to only referring to three key expositions: Newman (1997), Bovier (2006), and Talagrand (2011). A further subject of research is, of course, the vast field of stochastic time evolutions that admit the Gibbs measures for a certain interaction as stationary measures. This lies beyond the scope of this book. But, to single out one type of result, we refer to the fascinating phenomenon of metastability. In the case of the kinetic Ising model, this phenomenon was analyzed in full detail by Schonmann and Shlosman (1998), who also revealed its intimate connection to the Wulff shape. A general exposition of metastability can be found in the monograph of Olivieri and Vares (2005). Finally, we cannot conclude without pointing to a particular highlight, namely the work of Smirnov (2010) on the conformai invariance of the phase separation line in the two-dimensional Ising model at the critical temperature. Namely, consider a bounded domain D c K2 and the phase separation line r$ (defined in terms of the random-cluster representation) for the critical Ising model on D f~\8Z2 with ± boundary condition for prescribed separation points a, b € 3D. Then, in the scaling limit as S -» 0, the probability that Fg hits a point z (modified by a complex "fermionic" term involving the winding number around z) is shown to be preserved under conformai maps of D (and thus of a, b and z). This is a key step for showing that the phase separation line approaches a Schramm-Loewner evolution (SLE). The class of SLEs turns out to describe the asymptotics of various random curves in the plane at criticality. For general information on SLE see Werner (2004), for example.

References

Abbreviations of frequently quoted journals and series: CMP JSP LNM LNPh PTRF ZWvG

Commun. Math. Phys. J. Statist. Phys. Lecture Notes in Mathematics, Berlin etc.: Springer Lecture Notes in Physics, Berlin etc.: Springer Prob. Th. Rel. Fields (Continuation of ZWvG) Z.Wahrscheinlichkeitstheorie verw. Gebiete

Abraham, D.B. 1986 Surface structures and phase transitions - exact results. In: C. Domb and J.L. Lebowitz (eds.), Phase transition and critical phenomena Vol. 10, London etc.: Academic Press, pp. 1-74. Abraham, D.B., and A. Martin-Löf 1973 The transfer matrix for a pure phase in the two-dimensional I sing model. CMP 32, 245-268. Abraham, D.B., and P. Reed 1976 Interface profile of the I sing ferromagnet in two dimensions. CMP 49, 35-46. Accardi, L., and G. Pistone 1982 De Finettïs theorem, sufficiency, and Dobrushin's theory. In: G. Koch and F. Spizzichino (eds.), Exchangeability in probability and statistics, Amsterdam etc.: North Holland. Aczél, J., and Z. Darôczy 1975 On measures of information and their characterizations. New York: Academic Press. Ahlswede, R., and D.E. Daykin 1978 An inequality for the weights of two families of sets, their unions and intersections. ZWvG 43, 183-185. 1979 Inequalities for a pair of maps S x S -> S with S a finite set. Math. Z. 165, 267-289. Aizenman, M. 1980a Translation invariance and instability of phase coexistence in the twodimensional Ising system. CMP 73, 83-94. 1980b Instability of phase coexistence and translation invariance in two dimensions. In: K. Osterwalder (ed.), Mathematical problems in theoretical physics, LNPh 116, pp. 143-147. 1982 Geometric analysis of
496

References (eds.), Statistical physics and dynamical systems. Boston etc.: Birkhäuser, pp. 453-481.

Aizenman, M., D.J. Barsky, and R. Fernandez 1987 The phase transition in a general class of Ising-type models is sharp. JSP 47, 343-374. Aizenman, M., J. Bricmont, and J.L. Lebowitz 1987 Percolation of the minority spins in high-dimensional I sing models. JSP 49, 859-865. Aizenman, M., J.T. Chayes, L. Chayes, and C M . Newman 1988 Discontinuity of the magnetization in one-dimensional l/\x — y\2 Ising and Potts models. JSP 50, 1-40. Aizenman, M., and R. Fernandez 1986 On the critical behavior of the magnetization in high-dimensional Ising models. JSP 44, 393-454. Aizenman, M., and E. Lieb 1981 The third law of thermodynamics and the degeneracy of the ground state for lattice systems. JSP 24, 279-297. Aizenman, M., and B. Simon 1980 Local Ward identities and the decay of correlations in ferromagnets. CMP 77, 137-143. Akcoglu, M.A., A. del Junco, and M. Rahe 1979 Finitary codes between Markov processes. ZWvG 47, 305-314. Akcoglu, M.A., and U. Krengel 1981 Ergodic theorems for superadditive processes. J. reine u. angew. Math. 323, 53-67. Albeverio, S., J.E. Fenstad, R. Heegh-Krohn, and T. Lindstrom 1986 Nonstandard methods in stochastic analysis and mathematical physics. Orlando etc.: Academic Press. Albeverio, S., R. Hoegh-Krohn, and G. Olsen 1981 The global Markov property for lattice systems. J. Multivar. Anal. 11, 599-607. Araki, H. 1986 Analyticity of correlation functions for the two-dimensional Ising model. CMP 106,241-266. Asano, T. 1970 Theorems on the partition functions of the Heisenberg ferromagnets. J. Phys. Soc. Jap. 29, 350-359. Averintsev, M.B. 1970 A way of describing random fields with a discrete argument. Probl. Inform. Transmiss. 6, 169-175. 1972 Description of Markovian random fields by Gibbsian conditional probabilities. Th. Prob. Appl. 17, 20-33. 3975 Gibbs description of random fields whose conditional probabilities may vanish. Probl. Inform. Transmiss. 11, 326-334.

References 1978

497

On some classes ofGibbsian random fields. In: R.L. Dobrushin, V.l. Kryukov and A.L. Toom (eds.), Locally interacting systems and their application in biology. LNM 653, pp. 91-98.

Barron, A.R. 1985 The strong ergodic theorem for densities: generalized Shannon-McMillanBreiman theorem. Ann. Prob. 13, 1292-1303. Bassalygo, L.A., and R.L. Dobrushin 1986 Uniqueness of a Gibbs field with random potential - an elementary approach. Th. Prob. Appl. 31, 572-589. Basuev, A.G. 1985 Hamiltonian of the phase separation border and phase transitions of the first kind I. Theor. Math. Phys. 64, 716-734. Batty, C.J.K, and H.W. Bollmann 1980 Generalized Holley-Preston inequalities on measure spaces and their products. ZWvG53, 157-174. Bauer, H. 1981 Probability theory and elements of measure theory. New York: Academic Press. Baumgartner, B. 1983 On the group structure, GKS and F KG inequalities for I sing models. J. Math. Phys. 24, 2197-2199. Baxter, R.J. 1973 Potts model at the critical temperature. J. Phys. C 6, L445-L448. 1982a Exactly solved models in statistical mechanics. London-New York: Academic Press. 1982b Magnetization discontinuity of the two-dimensional Potts model. J. Phys. A 15, 3329-3340. Becker, M.E. 1981 Multiparameter groups of measure-preserving transformations: A simple proof of Wiener's ergodic theorem. Ann. Prob. 9, 504-509. van Beijeren, H. 1975 Interface sharpness in the Ising system. CMP 40, 1-6. van Beijeren, H., and G.S. Sylvester 1978 Phase transitions for continuous spin Ising ferromagnets. J. Funct. Anal. 28, 145-167. Bellissard, J., and R. Hoegh-Krohn 1982 Compactness and the maximal Gibbs state for random Gibbs fields on a lattice. CMP 84, 297-327. Benettin, G., G. Gallavotti, G. Jona-Lasinio, and A.L. Stella 1973 On the Onsager-Yang value of the spontaneous magnetization, CMP 30, 45-54. Benfatto, G., C. Marchioro, E. Presutti, and M. Pulvirenti 1980 Superstability estimates for anharmonic systems. JSP 22, 349-362. Benfatto, G., E. Presutti, and M. Pulvirenti 1978 DLR measures for one-dimensional harmonic systems. ZWvG 41, 305-312.

498

References

Berezin, F.A., and Ya. G. Sinai 1967 Existence of a phase transition for the lattice gas with interparticle attraction. Trans. Moscow Math. Soc. 17, 219-236. Berg, C , J.P.R. Christensen, and P. Ressel 1984 Harmonic analysis on semigroups. Berlin etc.: Springer. Berg, C , and P.H. Maserick 1984 Exponentially bounded positive definite functions. Illinois J. Math. 28, 162179. Besag, J.E. 1972 Nearest-neighbour systems and the auto-logistic model for binary data. J. Royal Statist. Soc. B 34, 78-83. Birgeneau, R.J., R.A. Cowley, G. Shirane, and H. Yoshizawa 1984 Phase transitions in diluted magnets: critical behaviour, percolation, and random fields. JSP 34, 817-848. Bishop, E., and R.R. Phelps 1963 The support functionals of a convex set. In: AMS Proceedings of Symposia in Pure Mathematics Vol. 7, Providence, R.I.: Amer. Math. Soc, pp. 27-35. Blackwell, D., and D. Freedman 1964 The tail a-field of a Markov chain and a theorem of Orey. Ann. Math. Stat. 35, 1291-1295. Bleher, P.M. 1976 Second order phase transitions in several models of ferromagnetism. Trans. Moscow Math. Soc. 33, 153-217. 1985 Theorem on large deviations in a neighborhood of the critical point of the (pA-hierarchical model. Th. Prob. Appl. 30, 529-541. Bleher, P.M., and P. Major 1987 Critical phenomena and universal exponents in statistical physics. On Dyson's hierarchical model. Ann. Prob. 15, 431-477. Blum, J.R., and D.L. Hanson 1960 On invariant probability measures I. Pacific J. Math. 10, 1125-1129. Blume, M. 1966 Theory of the first-order magnetic phase change in U02. Phys. Rev. 141, 517-524. Bodden, H.P. 1981 Gaußspezifikationen und Gibbsmaße auf einem Gitter. Diploma thesis, Universität Bielefeld. Bolthausen, E. 1982 On the central limit theorem for stationary mixing random fields. Ann. Prob. 10, 1047-1050. Bonato, CA., J.F. Perez, and A. Klein 1982 The Mermin-Wagner phenomenon and cluster properties of one- and twodimensional systems. JSP 29, 159-175. Borgs, C. 1984 Translation symmetry breaking in four-dimensional lattice gauge theories. CMP 96, 251-284.

References

499

Bortz, A.B. 1972 Proofs of spontaneous magnetization using the Peierls method. Amer. J. Phys. 40, 1524-1528. Bortz, A.B., and R.B. Griffiths 1972 Phase transitions in anisotropic Heisenberg ferromagnets. CMP 26, 102108. Bourgoyne, P.N. 1963 Remarks on the combinatorial approach to the Ising problem. J. Math. Phys. 4, 1320-1326. Bowen, R. 1975 Equilibrium states and the ergodic theory of Anosov diffeomorphisms. LNM 470. Bradley, R.C. 1986 A bilaterally deterministic p-mixing stationary random sequence. Trans. Amer. Math. Soc. 294, 233-241. Brankov, J.G., and D.M. Danchev 1987 On the limit Gibbs states of the spherical model. J. Phys. A 20, 49014913. Brascamp, H J . 1970 Equilibrium states for a classical lattice gas. CMP 18, 82-96. 1971 Equilibrium states for a one-dimensional lattice gas. CMP 21, 56-70. Brascamp, H.J., and H. Kunz 1973 Analyticity properties of the Ising model in the antiferromagnetic phase. CMP 32, 93-106. Brascamp, HJ., E.H. Lieb, and J.L. Lebowitz 1975 The statistical mechanics of anharmonic lattices. Bull. Int. Stat. Inst. 46, 393-404. Breiman, L. 1957 The individual ergodic theorem of information theory. Ann. Math. Stat. 28, 809-811. Correction note: Ann. Math. Stat. 31, 809-810. 1968 Probability. Reading, Mass.: Addison-Wesley. Bricmont, J. 1977a Les inégalités de correlation et leur applications aux systèmes de spins classiques. Dissertation, Université Catholique de Louvain. 1977b The Gaussian inequality for multicomponent rotators. JSP 17, 289-300. Bricmont, J., A. El Mellouki, and J. Fröhlich 1986 Random surfaces in statistical mechanics: roughening, rounding, wetting.... JSP 42, 743-798. Bricmont, J., and J.R. Fontaine 1981 Correlation inequalities and contour estimates. JSP 26, 745-753. 1982 Infrared bounds and the Peierls argument in two dimensions. CMP 87, 417-427. Bricmont, J., J.R. Fontaine, and L.J. Landau 1977 On the uniqueness of the equilibrium state for plane rotators. CMP 56, 281-290.

500

References

Bricmont, J., J.R. Fontaine, and J.L. Lebowitz 1982 Surface tension, percolation, and roughening. JSP 29, 193-203. Bricmont, J., J.R. Fontaine, J.L. Lebowitz, E.H. Lieb, and T. Spencer 1981 Lattice systems with a continuous symmetry, III. CMP 78, 545-566. Bricmont, J., and J. Fröhlich 1985 Statistical mechanical methods in particle structure analysis of lattice field theories II. CMP 98, 553-578. Bricmont, J., K. Kuroda, and J.L. Lebowitz 1985 First order phase transitions in lattice and continuous systems: extension of Pirogov-Sinai theory. CMP 101, 501-538. Bricmont, J., and J.L. Lebowitz 1986 On the continuity of the magnetization and energy in Ising ferromagnets. JSP 42, 861-869. 1987 Wetting in Potts and Blume-Capel models. JSP 46, 1015-1029. Bricmont, J., J.L. Lebowitz, and C. Maes 1987 Percolation in strongly correlated systems: the massless Gaussian field. JSP 48, 1249-1268. Bricmont, J., J.L. Lebowitz, and CE. Pfister 1979 On the equivalence of boundary conditions. JSP 21, 573-582. 1980 Low temperature expansion for continuous spin Ising models. CMP 78, 117-135. 1981a Periodic Gibbs states of ferromagnetic spin systems. JSP 24, 269-277. 1981b On the local structure of the phase separation line in the two-dimensional Ising system. JSP 26, 313-332. Bricmont, J., J.L. Lebowitz, C E . Pfister, and E. Olivieri 1979 Non-translation invariant Gibbs states with coexisting phases, I—III. CMP 66, 1-20 & 21-36 & 69, 267-291. Bricmont, J., and J. Slawny 1986 First order phase transitions and perturbation theory. In: T.C Dorlas et al. (eds.), Statistical mechanics and field theory, Groningen 1985, LNPh 257, pp. 10-51. Brendsted, A. 1964 Conjugate convex functions in topological vector spaces. Mat.-Fys. Medd. Danske Vid. Selsk. 34, 1-26. Brunei, A. 1976 Une preuve élémentaire du lemme de Holley. In: J.-P. Conze and M.S. Keane (eds.), Théorie Ergodique, Rennes 1973/1974, LNM 532. Brush, S.G. 1967 History of the Lenz-Ising model. Rev. Mod. Phys. 39, 883-893. Brydges, D., J. Fröhlich, and T. Spencer 1982 The random walk representation of classical spin systems and correlation inequalities. CMP 83, 123-150. Brydges, D., J. Fröhlich, and A.D. Sokal 1983 The random walk representation of classical spin systems and correlation inequalities II. CMP 91, 117-139.

References

501

Burkov, S.E., and Ya.G. Sinai 1983 Phase diagrams of one-dimensional lattice models with long-range antiferromagnetic interaction. Russian Math. Surveys 38, 4, 235-257. Calheiros, F.J.L.C. 1983 The existence of a phase transition in classical antiferromagnetic models. JSP 31, 141-154. Campanino, M., D. Capocaccia, and E. Olivieri 1983 Analyticity for one-dimensional systems with long range superstable interactions. JSP 33, 437-476. Campanino, M., D. Capocaccia, and B. Tirozzi 1979 The local central limit theorem for a Gibbs random field. CMP 70, 125132. Campanino, M., G. Del Grosso, and B. Tirozzi 1979 Local limit theorem for Gibbs random field of particles and unbounded spins. JMP 20, 1752-1758. Campanino, M., and L. Russo 1985 An upper bound on the critical percolation probability for the threedimensional cubic lattice. Ann. Prob. 13, 478-491. Capel, H.W. 1966 On the probability of first-order phase transitions in I sing systems of triplet ions with zero-field splitting. Physica 32, 966-988. Cappocaccia, D., M. Cassandro, and G. Ciccotti 1973 Equilibrium states of an I sing ferromagnet in the low temperature region. CMP 29, 31-42. Cassandro, M., A. Da Fano, and E. Olivieri 1975 Existence of phase transition for a lattice model with a repulsive hard core and an attractive short range interaction. CMP 44, 45-51. Cassandro, M., G. Gallavotti, J.L. Lebowitz, and J.L. Monroe 1973 Existence and uniqueness of equilibrium states for some spin and continuum systems. CMP 32, 153-165. Cassandro, M., and E. Olivieri 1981 Renormalization group and analyticity in one dimension: a proof of Dobrushin's theorem. CMP 80, 255-269. Cassandro, M., E. Olivieri, A. Pellegrinotti, and E. Presutti 1978 Existence and uniqueness of DLR measures for unbounded spin systems. ZWvG 41, 313-334. Chay, S.C. 1972 On quasi-Markov random fields. J. Multivar. Anal. 2, 14-76. Chayes, J.T., and L. Chayes 1986 Critical points and intermediate phases on wedges of Zd. J. Phys. A 19, 3033-3048. Chayes, J.T., L. Chayes, and R.H. Schonmann 1987 Exponential decay of connectivities in the two-dimensional Ising model. JSP 49, 433-445.

502

References

Chung, K.L. 1980 Lectures from Markov Processes to Brownian Motion. Berlin etc.: Springer. Chung, K.L., and W.H.J. Fuchs 1951 On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. no. 6. Cohn, D.L. 1980 Measure theory. Boston etc.: Birkhäuser. Coniglio, A., C.R. Nappi, F. Peruggi, and L. Russo 1976 Percolation and phase transition in the Ising model. CMP 51, 315-323. 1977 Percolation points and critical points in the Ising model. J. Phys. A 10, 205-218. Conze, J.P. 1972 Entropie d'un group abélien de transformations. ZWvG 25, 11-30. Cox, T. 1977a 1977b 1979

An example of phase transition in countable one-dimensional Markov random fields. J. Appl. Prob. 14, 205-211. Entrance laws for Markov chains. Ann. Prob. 5, 533-549. An alternative proof of a theorem of Kesten concerning Markov random fields. Ann. Prob. 7, 377-378.

Csiszâr, I. 1967 Information-type measures of difference of probability distributions and indirect observations. Studia Sei. Math. Hungar. 2, 299-318. 1975 I-divergence geometry of probability distributions and minimization problems. Ann. Prob. 3, 146-158. 1984 Sanov property, generalized I-projection and a conditional limit theorem. Ann. Prob. 12, 768-793. Csiszâr, I., and J. Körner 1981 Information theory, coding theorems for discrete memoryless channels. Budapest: Akadémiai Kiadô. Dang-Ngoc, N., and G. Royer 1978 Markov property of extremal local fields. Proc. Am. Math. Soc. 70, 185188. Dang-Ngoc, N., and M. Yor 1978 Champs markovien et mesures de Gibbs sur R. Ann. Sei. École Normale Sup. (4) 11, 29-69. Daniels, H.A.M., and A.C.D. van Enter 1980 Differentiability properties of the pressure in lattice systems. CMP 71, 65-76. Darroch, J.N., S.L. Lauritzen, and T.P. Speed 1980 Markov fields and log-linear interaction models for contingency tables. Ann. Stat. 8, 522-539. Dashyan, Yu.R., and Yu.M. Sukhov 1984 Gibbsian description of a class of random processes with discrete time. Soviet J. Contemporary Math. Anal. 19, 4, 30-46. See also Soviet Math. Dokl. 19, 1122-1126.

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De Coninck, J. 1987a On limit theorems for the bivariate {magnetization, energy) variable at the critical point. CMP 109, 191-205. 1987b Gaussian fluctuations for the magnetization of Lee-Yang ferromagnets at zero external field. JSP 47, 397-407. Del Grosso, G. 1974 On the local central limit theorem for Gibbs processes. CMP 37, 141 — 160. Dellacherie, C , and P.A. Meyer 1978 Probabilities and potential. Amsterdam etc.: North-Holland. Derrienic, Y. 1983 Un théorème ergodique presque sous-additif Ann. Prob. 11, 669-677. Deuschel, J.D. 1985 Lissage de diffusions a dimension infinie et leurs propriétés en tant que mesure de Gibbs. Thèse EPF Zürich, no. 7823. 1986 Non-linear smoothing of infinite-dimensional diffusion processes. Stochastics 19, 237-261. 1987 Infinite-dimensional diffusion processes as Gibbs measures on C[0, l ] z " . PTRF 76, 325-340. Dinaburg, E.I. 1970 The relation between topological entropy and metric entropy. Soviet Math. Dokl. 11, 13-16. Dinaburg, E.I. and Ya.G. Sinai 1984 Interacting contour models and their applications. Tr. Conf. Mat. Fiz., Dubna (in Russian). 1985 An analysis of ANNNI model by Peierls contour method. CMP 98, 119— 144. Dobrushin, R.L. 1965 Existence of a phase transition in two and three dimensional Ising models. Th. Prob. Appl. 10, 193-213. 1967 Existence of phase transitions in models of a lattice gas. Prov. V. Berkeley Symposium on Prob. Th. and Statistics 1965, Vol. Ill, Berkeley: Univ. California Press, pp. 73-87. 1968a The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Prob. Appl. 13, 197-224. 1968b Gibbsian random fields for lattice systems with pairwise interactions. Funct. Anal. Appl. 2, 292-301. 1968c The problem of uniqueness of a Gibbs random field and the problem of phase transition. Funct. Anal. Appl. 2, 302-312. 1969 Gibbsian random fields. The general case. Funct. Anal. Appl. 3, 22-28. 1970 Prescribing a system of random variables by conditional distributions. Th. Prob. Appl. 15, 458-486. 1972a Gibbs states describing coexistence of phases for a three-dimensional Ising model. Th. Prob. Appl. 17, 582-600. 1972b Asymptotic behavior of Gibbsian distributions for lattice systems and its dependence on the form of the volume. Theor. Math. Phys. 12, 699-711. 1973a Investigation of Gibbsian states for three-dimensional lattice systems. Th. Prob. Appl. 18, 253-271.

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References

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Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. CMP 32, 269-289. Conditions for the absence of phase transitions in one-dimensional classical systems. Math. USSR Sbornik 22, 28-48. Analyticity of the correlation functions for one-dimensional classical systems with power law decay of the potential. Math. USSR Sbornik 23,13-44. Gaussian random fields - Gibbsian point of view. In: R.L. Dobrushin and Ya.G. Sinai (eds.), Multicomponent random systems, Adv. in Prob, and Related Topics Vol. 6, New York: Dekker. Induction on volume and no cluster expansion. In: M. Mebkhout and R. Sénéor (eds.), VIII. Internat. Congress on Mathematical Physics, Marseille 1986, Singapore: World Scientific, pp. 73-91.

Dobrushin, R.L., J. Kolafa, and S.B. Shlosman 1985 Phase diagram of the two-dimensional I sing antiferromagnet (computer assisted proof). CMP 102, 89-103. Dobrushin, R.L., and B.S. Nahapetian 1974 Strong convexity of the pressure for lattice systems of classical statistical physics. Theor. Math. Phys. 20, 782-790. Dobrushin, R.L., and E.A. Pecherski 1981 Uniqueness conditions for finitely dependent random fields. In: J. Fritz, J.L. Lebowitz, and D. Szâsz (eds.), Random fields. Esztergom (Hungary) 1979, Vol. I, Amsterdam: North Holland, pp. 223-261. 1983 A criterion for the uniqueness of Gibbsian fields in the non-compact case. In: K. Ito and J.V. Prokhorov (eds.), Prob. Th. and Math. Statistics, LNM 1021, pp. 97-110. Dobrushin, R.L., and S.B. Shlosman 1975 Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics. CMP 42, 31-40. 1980 Nonexistence of one- and two-dimensional Gibbs fields with noncompact group of continuous symmetries. In: R.L. Dobrushin and Ya.G. Sinai (eds.), Multicomponent random systems, Adv. in Prob, and Related Topics Vol. 6, New York: Dekker. 1981 Phases corresponding to minima of the local energy. Selecta Math. So v. 1, 317-338. 1985a Constructive criterion for the uniqueness of a Gibbs field. In: J. Fritz, A. Jaffe, and D. Szâsz (eds.), Statistical mechanics and dynamical systems, Boston etc.: Birkhäuser, pp. 347-370. 1985b Completely analytical Gibbs fields. In: J. Fritz, A. Jaffe, and D. Szâsz (eds.), Statistical mechanics and dynamical systems, Boston etc.: Birkhäuser, pp. 371-403. 1985c The problem of translation invariance of Gibbs states at low temperatures. In: S.P. Novikov (ed.), Soviet Scientific Reviews C, Math. Phys., Vol. 5, Chur etc.: Harwood Ac. Publ., pp. 53-196. 1987 Completely analytical interactions: constructive description. JSP 46, 983— 1014. Dobrushin, R.L., and B. Tirozzi 1977 The central limit theorem and the problem of equivalence of ensembles. CMP 54, 173-192.

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Ekeland, I., and R. Teman 1976 Convex analysis and variational problems. Amsterdam: North Holland. Ellis, R.S. 1985 Entropy, large deviations and statistical mechanics. New York etc.: Springer. Ellis, R.S., and J.L. Monroe 1975 A simple proof of the GHS and further inequalities. CMP 41, 33-38. Ellis, R.S., J.L. Monroe, and C M . Newman 1976 The GHS and other correlation inequalities for a class of even ferromagnets. CMP 46, 167-182. Ellis, R.S., and C M . Newman 1978 Necessary and sufficient conditions for the GHS inequality with applications to analysis and probability. Trans. Am. Math. Soc. 237, 83-99. Ellis, R.S, C M . Newman, and M.R. O'Conell 1981 The GHS inequality for a large external field. JSP 26, 37-52. van Enter, A.CD. 1981 A note on the stability of phase diagrams in lattice systems. CMP 79, 2 5 32. 1982 Instability of phase diagrams for a class of "irrelevant" perturbations. Phys. Rev. B(3)26, 1336-1339. Evstigneev, I.V. 1978 A note on Gibbs representation. In: R.L. Dobrushin, V.l. Kryukov, and A.L. Toom (eds.), Locally interacting systems and their application in biology. LNM 653, pp. 194-202. Falk, H. 1975

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Fannes, M , P. Vanheuverzwijn, and A. Verbeure 1982 Energy-entropy inequalities for classical lattice systems. JSP 29, 547560. Farrell, R.H. 1962 Representation of invariant measures. Illinois J. Math. 6, 447-467. Fenchel, W. 1949 On conjugate convex functions. Canad. J. Math. 1, 73-77. de Finetti, B. 1931 Funzione caratteristica di un'fenomeno aleatorio. Atti della R. Accad. Naz. dei Lincei, Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali, 4, 251-300. Fisher, M.E, and V. Privman 1986 First order transitions in spherical model: Finite size scaling. CMP 103, 527-548. Flood, R.G, and W.G. Sullivan 1980 Consistency of random field specifications. ZWvG 53, 147-156. Föllmer, H. 1973 On entropy and information gain in random fields. ZWvG 26, 207-217.

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Ioffe, D., and R.H. Schonmann 1998 Dobrushin-Kotecky-Shlosman theorem up to the critical temperature. CMP 199, 117-167. Ioffe, D., S. Shlosman, and Y. Velenik 2002 2D models of statistical physics with continuous symmetry: the case of singular interactions. CMP 226, 4 3 3 ^ 5 4 . Johnson, O. 2003 Entropy and a generalisation of "Poincaré's obser\>ation". Math. Proc. Cambridge Philos. Soc. 135, 375-384.

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Jonasson, J., and J. Steif 1999 Amenability and phase transition in the Ising model. J. Theor. Probab. 12, 549-559 Keller, G. 1998 Equilibrium states in ergodic theory. Cambridge: Cambridge University Press. Kiilske, C , A. Le Ny and F. Redig 2004 Relative entropy and variational properties of generalized Gibbsian measures. Ann. Probab. 32, 1691-1726. Lebowitz, J.L., and A.E. Mazel 1997 On the uniqueness ofGibbs states in the Pirogov-Sinai theory. CMP 189, 311-321. Lefevere, R. 1999 Variational principle for some renormalized measures. JSP 96, 109-133. Lewis, J.T., C.-E. Pfister, and W.G. Sullivan 1994 The equivalence of ensembles for lattice systems: some examples and a counterexample. JSP 77, 3 9 7 ^ 1 9 1995 Entropy, concentration of probability and conditional limit theorems. Markov Process. Related Fields 1, 319-386' Lörinczi, J., F. Hiroshima and V. Betz 2011 Feynman-Kac-type theorems and Gibbs measures on path space. De Gruyter Studies in Mathematics 34. Berlin, New York: de Gruyter. Lyons, R. 2000 Phase transitions on nonamenable graphs. J. Math. Phys. 41, 1099-1126. Maes, C , F. Redig, and A. Van Moffaert 1999a Almost Gibbsian versus weakly Gibbsian measures. Stochastic Process. Appl. 79, 1-15. 1999b The restriction of the Ising model to a layer. JSP 96, 69-107. Maes, C , F. Redig, S. Shlosman, and A. Van Moffaert 2000 Percolation, path large deviations and weakly Gibbs states. CMP 209,517— 545. Malyshev, VA., and R.A. Minlos 1991 Gibbs random fields. Cluster expansions. Dordrecht: Kluwer. Merkl, F , and H. Wagner 1994 Recurrent random walks and the absence of continuous symmetry breaking on graphs. JSP 75, 153-165. Mossel, E., and Y. Peres 2003 Information flow on trees. Ann. Appl. Probab. 13, 817-844. Newman, C M . 1997 Topics in disordered systems, Lectures in Mathematics ETH Zürich, Basel: Birkhäuser. Olivieri, E., and M.E. Vares 2005 Large deviations andmetastability. Cambridge: Cambridge University Press. Osada, H., and H. Spohn 1999 Gibbs measures relative to Brownian motion. Ann. Probab. 27,1183-1207.

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List of Symbols

» 0&% C

29 84 the complex numbers

C (in Part IV)

372 374 115 129 (j-algebra product 320 12 384 16 28, 212 86 92, 386 27 314 322 312 395 291 31 383 13 the natural numbers 322 13 86 the rational numbers 368 373 the real numbers 12 386 82 31,15 384

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28 329, 375 67 7" 7? ? Dirac measure àm ( = point mass) at co 141, 142 à(f) dA 45, 156 340 ôP(O) 82 epA@ product measure A 18,19 ^A' ^. A-inf, A-sup 383, 386 62 KS^ 13 M/) 13 /i-a.a. /i-almost all /i-a.s. /i-almost surely 59 Ç(G,co), eP(G,co) 385 27,28 p >p° ff ff 12 i> A x((p), x(y) 82,83 84 *(/*)
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540

List of Symbols

1Mb 29 [ •, • ] closed interval ] •, • [ open interval ['>'[>]')'] half-open intervals {/ = g}, {/ < gf} the set of all elements x of the common domain off and g with f(x) = g(x) resp. f(x) < g(x)

\ A x

v, A =

321 set difference symmetric difference Cartesian product of sets, product of a-algebras, product of measures maximum, minimum equality by definition

Index

admissible potential 27 approximation of — ergodic Gibbs measures 300 — extreme Gibbs measures 121, 122, 154, 157-159 Bernoulli scheme 471 Bethe lattice 245 Blume-Capel model 412-413, 486 Boltzmann factor 27 bond 30 boundary condition 15, 28 - , free 69, 91, 463, 474, 481 - , periodic 69-70, 92, 374-381, 386, 435, 440, 475 - , ( ± ) 462-463 boundary law 212, 242 breakdown of - shift invariance 101,108, 224, 230, 235, 274-275, 358-359, 462, 488,490 - SO(N) symmetry 355, 444, 474 - spin reversal symmetry 96, 100, 108, 176, 250, 281-282, 355, 398, 400, 402, 404-405,412,444, 460 Brownian bridge 462, 493 central limit theorems 462, 469-470, 474 chessboard estimate 367-374, 387, 391, 428, 436, 439, 441, 459, 483, 492 Choquet theory 129, 466 circuit 101 cluster 385 - expansions 468, 469, 481, 491 cofinal set, net 17, 62 compactness results 60 — 64, 72, 315, 326-327, 458-459 competing potential wells 421-425, 488 configuration 12 conformai invariance 490 continuous dependence of Gibbs measures on potentials 67-69, 72, 326-327 contour 95, 99. 102, 109, 382, 460, 468 - estimates 95, 103, 110, 387, 484

correlation inequalities 100, 455 — 457, 459, 467, 469, 472, 486, see also "FKGHolley", "Griffiths", "Lebowitz" correlations - , decay of 28, 159-162,457,468-469, 472, 489 - , differentiability of 162, 468, 480-481 countable core 78, 459 C-potential 374 Cramer's theorem 330 cube 292 cylinder event 12 cylinder function 31 decomposition - , ergodic 293-294, 298, 318, 465, 476 - , extreme 129-136, 293-294, 298, 465-466 denseness of — ergodic Gibbs measures 359 — 364 — ergodic random fields 295, 330 discrete Gaussian model 7, 94, 106—113, 177-178, 182, 464 discriminating system 349 disordered system 494 DLR state 28 Dobrushin's uniqueness condition 142 — 149 Dynkin system 65, 121 empirical field 328 energy 3, 27 —, specific 320 ensemble 4, 336 — 337 entrance law 213, 473 entropy 308, 312-313 — , conditional 316 — 318 — function 478 - , Kolmogorov-Sinai 477, 478 - , relative 56, 309-312, 472, 477 - , specific 312-319, 477-478 — , topological 478 entropy-energy conflict 420 — 431

542

Index

equilibrium state 1, 5, 454, 482-483, 489 equivalence —, macroscopic 136 — 139 — of ensembles 336-337 — of potentials 39 —43 — of transfer matrices 210-212 — , physical 455 ergodic decomposition 293 — 294, 298, 318, 465,476 ergodic random field 291 ergodic theorem — for Markov chains 49, 55-56, 125, 205-207, 458 — , multidimensional 302 — 307, 477 - , random 449, 450, 488 evaluation maps 59, 86, 129 evaluation a-algebra 129 everywhere dense measure 22 exchangeable distributions 35, 126, 136, 465 existence of — Gaussian Gibbs measures 269 — 275 — Gibbs measures 71, 72, 458-459 — invariant Gibbs meausures 85 — 93, 459 external field 3, 6, 30 extreme 115 — boundary 115 — decomposition 129-136, 293-294, 298,465-466 — Gibbs measures 118 — 124 , physical meaning of 119

GHS inequality 457, 482 Gibbs distribution 3, 28 - , averaged 91, 299-300 — with free boundary condition 69, 91, 463, 474, 481 — with periodic boundary condition 69 — 70, 92, 374-381, 386, 435, 440, 475 Gibbs measure 5, 16, 28 - , almost 492 — , averaged limiting 302 —, canonical 454, 466 —, fully invariant 459 - , gradient 493 —, I-invariant 86 — , limiting 135 —, maximal 456 - , shift-invariant 296-302, 325, 346, 481-482 - , tempered 2 3, 279, 459, 468, 475 - , weakly 492 Gibbs representation theorem 25, 35 — 36, 454-455, 479, 493 Gibbs state 28 Griffiths' inequalities 100, 176, 456-457, 460, 486, 488 ground random field 464 — 465 ground state 51, 98, 100, 107, 464 —, absolute 393 — degeneracy 395 - , local 384 , stable partition of 395 —, random 464

ferromagnet 1, see also "Heisenberg", "Ising", "potential" Fibonacci numbers 53 Finetti, theorem of de 35, 136, 465 FKG-Holley inequality 455-456, 461, 468, 470, 474, 482, 485, 489, 490 free energy 308 —, specific 320 —, specific Gibbs 323 —, specific Helmholtz 479 -symmetric partition 396

Hamiltonian 3, 26, 27 harmonic oscillator 182, 271, 281, 475, 493 Heisenberg models 180, 355, 376-381, 393, 433-451, 464, 471-472, 476, 488 - , anisotropic 402-403, 486 hierarchical model 488 — 489 homogeneous 44, 46, 84, 219, 273 - , completely 240-241, 246

Gaussian — domination 488 — fields 257-283, 469,475 — inequalities 457, 476 Gertsik-Dobrushin model 399-400, 413, 485

infinite volume limit 4, 67, 121, 122, 135, 157, 300 infrared bound 433, 440-443, 488 interface 101,119, 462-463, 464, 485, 488, 493 invariant — modifications, potentials, specifications 84 — random fields 84, 116-117

Index -

sets 115 — 117, see also "shift-invariant" irreducible 199, 200, 263, 443 Ising antiferromagnet - , multidimensional 106, 297, 398-399, 413, 464, 467, 481, 485 - on Cayley trees 253-255, 474 - , one-dimensional 49 - 54, 297, 464 - , stabilized critical 413, 486 Ising ferromagnet 6 — 7, 489 — 492 - on Cayley trees 247-253, 474, 491 - , one-dimensional 49-52, 95-99, 139, 167, 458, 464 - , three-dimensional 101, 119, 398, 460, 461, 462, 464, 470, 479, 481, 484, 490 - , two-dimensional 99-105, 398, 4 6 0 464, 470, 479, 481, 484-485, 4 9 0 494

Kalikow's model 228-233, 463 Kirkwood-Salsburg equations 454, 468 Kosterlitz-Thouless transition 28, 472

^.-admissible 27 large deviations 220, 327-337, 477, 4 7 9 480, 493 — surface order, 490 lattice gas 2, 3, 35, 42, 144-145, 354, 413 - , hard core 400-401, 414, 485, 491 Lebowitz inequalities 457, 461, 482 Lee-Yang circle theorem 469 — 470, 481 Legendre-Fenchel transform 329, 344 — 345, 478-479 limiting Gibbs measure 135 —, averaged 302 ^.-modification 19 —, Markovian 191 -.positive 19, 20 local convergence 59 local function 31 locally equicontinous 61 low temperature limit 51-54, 98, 100, 108, 147-148, 297, 393, 394-407, 411-414, 444-447, 464-465, 468, 487 A-periodic 368 ^.-specification 19 - , positive 19, 20 J^-topology 59

543

macroscopic observables 31, 119 Markov chain 45, 50, 125, 193, 195, 1 9 6 208, 213, 227, 230, 239-246, 2 6 3 264, 271, 325 Markov field 157, 190, 193, 239 Markov potential 30 Markov property - , global 157, 194,470 - , local 157, 194, 239 - , one-sided 190-203, 239 - , two-sided 190, 193-195 McMillan's theorem 333 - 334 mean field approximation 146, 357, 4 6 7 468, 492 metastability 494 microcanonical distribution 336 microscopic quantities 31 minimum free energy principle 335 mixing coefficients 159 non-existence of Gibbs measures 66, 176-178, 181-182, 218-221, 2 6 9 272, 277, 280 nonnegative definite 257, 286 — 287 - relative to rk 377-378 TV-vector model 433, see also "Heisenberg", "plane rotor" occupation number 2 ocean 385 Onsager-Yang formula 460 — 461 order parameter 348, 363-364, 483 parameter set 11 partition function 27 path 101, 239, 385 P-bounded functional 340, 346 Peierls argument 9 4 - 9 5 , 99, 101-105, 109-111, 382-383, 460, 484, 485 Peierls condition 487 percolation 99, 383-394, 475, 484, 485, 490 - disagreement 491 Perron-Frobenius theorem 46, 54 phase 119 - diagram 338, 399, 487 - , p u r e 119, 298, 359-364, 461 - rule 360 - separation line 462, 463, 490, 494 phase transition 6, 28, 454, 480-481 - at low temperatures 98, 100-101, 108, 173, 297, 394-407, 411-414, 4 4 4 447, 487

544

Index

- , first order 420, 427, 461, 491, 492 - in Gaussian models 268-273, 2 7 4 280, 281-283 - in one dimension 96, 166, 173, 176, 224, 230, 235, 271-272, 282, 297, 351, 355, 358, 446-447, 473, 488-489 - on Cayley trees 250, 254-255, 491 - without symmetry 250, 351, 354,408431, 487 - with prescribed order parameters 351, 354, 355 - with symmetry breaking: see "breakdown" Pirogov-Sinai theory 95, 360, 462, 464, 468, 471, 487, 491, 493 plane rotor models 6 - 7 , 179, 403, 464, 472, 489, see also "Heisenberg" - , Shlosman's 180, 403-404, 486 Poincaré recurrence theorem 390 Poisson boundary value problem 2 6 3 264 positive definite 257, 274, 286 positive ^-modification, ^-specification 19, 20 potential 26, 27 —, absolutely summable 28 — 29, 84 —, a-normalized 35 —, chemical 3, 30 —, completely analytical 467 - , double-well 404-405, 414, 421-425, 462, 486, 488, 493 —, equivalent 39 - , ferromagnetic 3, 50, 96, 99, 250, 354, 355, 398, 399, 402, 439, 446, 455-457, 471, 481, 482, 484 —, finite range 29 - , g a s 35 —, homogeneous 46, 84 -,Kac492 —, ^-admissible 27 —, Markov 30 —, nearest-neighbour 30 —, pair 29 —, random 494 - , self- 30 —, shift-invariant 84 —, spin 42 - , superstable 458, 471 —, T-invariant 84 —, uniformly convergent 29 Potts model 148-149,401-402,414, 425-431, 475, 485, 488, 489-492

Poulsen simplex 296, 476 pre-modification 22 pressure 322-323, 480 - , analyticity of 480-481 - , computation of 325, 460-461 - , convexity of 339, 480 - , differentiability of 339-341, 348, 456, 468, 480, 481-482 —, strict convexity of 347, 480 proper kernel 14 pure — phase 119, 298, 359-364, 461 — spin transformation 91, 174, 395 — state 291 quasilocal — function 31 — pre-modification 35 — specification 32, 492 random-cluster model 489 — 490 random field 11 random walk 219, 271, 444, 448-451 recurrent 212, 492 - , null 212 — , positive 206, 212 reflection - , lattice 89, 91, 107, 368, 373 — positivity 367-381, 387, 441, 483, 492 - , s p i n 82, 107 rotation - , lattice 89, 107, 395 —,spin 82 roughening temperature 462 ^-positive 368 ^-positive 373 r-symmetric 384 shift 82 shift-invariant — events 291 — Gibbs measures 89, 9 1 - 9 3 , 200-208, 275, 278, 296-302, 325, 346, 3 4 8 364, 411, 437 — potential, specification 84 — random fields 290-296 Shlosman's rotator 180, 403-404, 486 short-range correlations 120 simple graph 30 Slawny's D-model 405-407, 486 solid-on-solid model 464, 493

Index specification 5, 16 —, Gaussian 265 —, Gibbsian 28 —, ground 464 —,independent 18 -,X- 19 - , Markov 44, 156, 191, 239 —, positive 1-19 - , product 127-129, 139, 176, 400, 444 Sperner's lemma 415, 431-432 spherical model 476 spin 1, 12 - flip 82 - wave 433, 439 Spitzer-Cox model 221 —228 Spitzer's model 233 — 237 spontaneous magnetization 1, 100, 253, 357, 444, 461, 490 standard Borel space 61, 73 — 80 state space 11 surface tension 463, 490 surrounding circuit 101 symmetry 84 - breaking 93, see also "breakdown" - , absence of 168-188, 217, 463, 471-472, 492 - c o n t i n u o u s 168, 178-188 —, discrete 168 tailCT-field31, 118-119 - , one-sided 125, 139, 196, 198, 215, 465, 473 tangent functional 340, 346 temperature 4, 26 -,critical 100, 397, 456, 461, 467-468, 474, 485, 494 thermodynamic limit 4

545

transfer matrix 209, 242, 246, 458, 460 transformation 82, 395 —, 1-preserving 82 —, u-preserving 84 - , p u r e spin 91, 174, 395 transient 212, 270-271, 444 translation: see "shift" tree 238-239 -,Cayley 245 trivial a-algebra 116 unbounded spin systems 106—113, 190 — 237, 256-283, 404-405, 434-438, 456, 458-459, 464, 466, 467, 468, 473, 478, 479, 488, 493 uniqueness theorems — for ferromagnetic potentials 476, 481-482 — ofDobrushin 142-152, 458, 466-467 - , one-dimensional 45, 164-167, 207, 215, 218-219,471,473 - , other 279-280, 461, 468,491 variational principle 308, 323, 325, 459, 474,478-480,493 Widom-Rowlinson model 401, 459-460, 462, 485, 490 Wulff shape 490, 494 zero-one law — for shift-invariant events 291 — for tail events 118, 120 — of Blackwell and Freedman 125 — of Hewitt and Savage 125-127 — of Kolmogorov 125

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