Phase Transitions and Critical P h e n o m e n a
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Phase Transitions and Critical P h e n o m e n a Volume 18
Edited by C. Domb
Department of Physics, Bar-I/an University, Ramat-Gan, Israel
and
J. L. Lebowitz
Department of Mathematics and Physics, Rutgers University, New Brunswick, New Jersey, USA
ACADEMIC PRESS A HarcourtScienceand TechnologyCompany San Diego San Francisco London Sydney Tokyo
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Boston
This book is printed on acid-free paper. Copyright (~) 2001 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press A Harcourt Science and Technology Company Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com ISBN 0-12-220318-6
A catalogue record for this book is available from the British Library
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Contributors M. J. ALAVA, Laboratory of Physics, Helsinki University of Technology, PO Box 1100, HUT 02015, Finland E M. DUXBURY, Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, M148824, USA
H.-O. GEORGII, Mathematisches Institut, Universitiit, D-80333 Miinchen, Germany
Ludwig-Maximilians-
O. HAGGSTROM, Department of Mathematics, Chalmers University of Technology, S-412 96 Goteborg, Sweden C. MAES, Instituut voor Theoretische Fysica, K. U. Leuven, B-3001 Leuyen, Belgium C. E MOUKARZEL, Instituto de Fisica, Fluminense, 24210-340 Niteroi, R J, Brazil
Universidade
Federal
H. RIEGER, Institut fiir Theoretische Physik, Universitiit des Saarlandes, 66041 Saarbrucken, Germany
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General Preface This series of publications was first planned by Domb and Green in 1970. During the previous decade the research literature on phase transitions and critical phenomena had grown rapidly and, because of the interdisciplinary nature of the field, it was scattered among physical, chemical, mathematical and other journals. Much of this literature was of ephemeral value, and was rapidly rendered obsolete. However, a body of established results had accumulated, and the aim was to produce articles that would present a coherent account of all that was definitely known about phase transitions and critical phenomena, and that could serve as a standard reference, particularly for graduate students. During the early 1970s the renormalization group burst dramatically into the field, accompanied by an unprecedented growth in the research literature. Volume 6 of the series, published in 1976, attempted to deal with this new literature, maintaining the same principles as had guided the publication of previous volumes. The number of research publications has continued to grow steadily, and because of the great progress in explaining the properties of simple models, it has been possible to tackle more sophisticated models which would previously have been considered intractable. The ideas and techniques of critical phenomena have found new areas of application. After a break of a few years following the death of Mel Green, the series continued under the editorship of Domb and Lebowitz, Volumes 7 and 8 appearing in 1983, Volume 9 in 1984, Volume 10 in 1986, Volume 11 in 1987, Volume 12 in 1988, Volume 13 in 1989 and Volume 14 in 1991. The new volumes differed from the old in two new features. The average number of articles per volume was smaller, and articles were published as they were received without worrying too much about the uniformity of content of a particular volume. Both of these steps were designed to reduce the time lag between the receipt of the author's manuscript and its appearance in print. The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It is not longer an area of specialist interest, but has moved into a central place in
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General Preface
condensed matter studies. The editors feel that there is ample scope for the series to continue, but the major aim will remain to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments. CYRIL DOMB JOEL L. LEBOWITZ
Preface to Volume 18 The two review articles in this volume complement each other in a remarkable way. They both deal with what might be called the modern geometric approach to the properties of macroscopic systems. The first one by Georgii et al., is primarily analytical. It describes in a rigorous, yet generally accessible, mathematical way recent advances in the application of geometric ideas, such as percolation, to visualizing the structure present in a typical configuration of the spins or atoms making up the microscopic constituents of a macroscopic system. This leads to a better understanding of pure phases and phase transitions in equilibrium systems. The authors illustrate these ideas by carrying out an in depth analysis of some of the basic models in statistical mechanics. These include the Ising model (both ferromagnetic and antiferromagnetic), the Potts model, the Widom-Rowlinson model, etc. Typical of the geometric ideas discussed here are those underlying the behavior of the two-dimensional Ising ferromagnet with nearest neighbor interactions at zero magnetic field. For temperatures above the Onsager critical temperature neither the pluses nor the minuses percolate while below it there is percolation of pluses (minuses) and only pluses (minuses) in the plus (minus) phase. The review goes, however, much beyond such classical results to bring the reader right up to date on this exciting topic. The second article in this volume by Alava et al., also focuses on geometrical aspects of many-body systems. It does so in a hands-on way going beyond abstract theory to obtain practical answers. This requires the use of computers, but not just their blind use. Computing power alone is simply not enough. One also needs a deep understanding of the physics and cleverness of programming. This is, in fact, what this article is all about. This article, the first one in this series in which the computer is what might be called the star of the show, focuses on geometrical aspects of the use of computers in statistical mechanics. It provides, to quote the authors, "an introduction to combinatorial optimization algorithms and reviews their applications to ground-state problems in disordered systems." This covers an astonishingly large class of problems of current interest ranging from
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the random field Ising model to elastic media and rigidity percolation. It is fair to say that the review brings together the most recent advances in computer science which are useful for solving problems of interest to statistical mechanicians and to material scientists. To end this preface on a historical note we mention that computers have played a role in statistical mechanics for more than 50 years now. One of the earliest works was that of Fermi, Pasta and Ulam who used the computer to solve Newton's equations of motion for an anharmonic chain consisting of 32 particles and discovered an apparent lack of (or at least an extremely slow) equipartition of energy in that system. That led to what is now known as molecular dynamics. Another landmark work was the introduction of Monte Carlo sampling techniques by Metropolis, Rosenbluth, Rosenbluth, Teller and Teller for evaluating equilibrium properties of large systems. Other firsts included the use of computers by the King's College group of Domb and associates to help evaluate coefficients in the high- and low-temperature expansions of different spin systems which led to the first evaluation of critical exponents and the notions of universality. It is our expectation that the combination of computers and geometrical ideas described in this volume will play a major role in the development of statistical mechanics in the twenty-first century. CYRIL DOMB JOEL L. LEBOWITZ
Contents Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G e n e r a l Preface
v
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Preface to Volume 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1 The Random Geometry of Equilibrium Phases H . - O . GEORGII, O. HAGGSTR()M AND C. MAES 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 E q u i l i b r i u m phases
6
3 Some models
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 C o u p l i n g and stochastic d o m i n a t i o n 5 Percolation
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 R a n d o m - c l u s t e r representations . . . . . . . . . . . . . . . . . . . . . .
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7 U n i q u e n e s s and exponential mixing from n o n - p e r c o l a t i o n . . . . . . . .
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8 Phase transition and percolation
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. . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 R a n d o m interactions 10 C o n t i n u u m m o d e l s
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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Exact Combinatorial Algorithms: Ground States of Disordered Systems M. J.
ALAVA,P.
1 Overview
M.
DUXBURY,C.
F. MOUKARZEL AND H. RIEGER
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Basics o f graphs and algorithms
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. . . . . . . . . . . . . . . . . . . . .
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3 Flow algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 5 6 7 8 9 10
Contents
M a t c h i n g algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical programming . . . . . . . . . . . . . . . . . . . . . . . . Percolation and m i n i m a l path . . . . . . . . . . . . . . . . . . . . . . . R a n d o m Ising m a g n e t s . . . . . . . . . . . . . . . . . . . . . . . . . . Line, vortex and elastic glasses . . . . . . . . . . . . . . . . . . . . . . Rigidity theory and applications . . . . . . . . . . . . . . . . . . . . . C l o s i n g remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Subject index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175 181 186 196 234 264 295 297 297 318
Contents of Volumes 1-17 Contents of Volume 1 (Exact Results) t Introductory Note on Phase Transitions and critical phenomena. C. N. YANG. Rigorous Results and Theorems. R. B. GRIFFITHS. Dilute Quantum Systems. J. GINIBRE. C*-Algebra Approach to Phase Transitions. G. Emch. One Dimensional Models--Short Range Forces. C. J. THOMPSON. Two Dimensional Ising Models. H. N. V. TEMPERLEY. Transformation of Ising Models. I. SYOZI. Two Dimensional Ferroelectric Models. E. H. LIEB and E Y. Wu. Contents of Volume 2 t Thermodynamics. M. J. BUCKINGHAM. Equilibrium Scaling in Fluids and Magnets. M. VICENTINI-MISSONI. Surface Tension of Fluids. B. WIDOM. Surface and Size Effects in Lattice Models. P. G. WATSON. Exact Calculations on a Random Ising System. B. McCoY. Percolation and Cluster Size. J. W. ESSAM. Melting and Statistical Geometry of Simple Liquids. R. COLLINS. Lattice Gas Theories of Melting L. K. RUNNELS. Closed Form Approximations for Lattice Systems. D. M. BURLEY. Critical Properties of the Spherical Model. G. S. JOYCE. Kinetics of Ising Models. K. KAWASAKI. Contents of Volume 3 (Series Expansions for Lattice Models) t Graph Theory and Embeddings. C. DOMB. Computer Enumerations. J. L. MARTIN. Linked Cluster Expansions. M. WORTIS. ~ Out of print.
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Contents of Volumes 1-17
Asymptotic Analysis of Coefficients. D. S. GAUNT and A. J. GUTTMAN. Ising Model. C. DOMB Heisenberg Model. G. A. BAKER, G. S. RUSHBROOKE and R W. WOOD. Classical Vector Models. H. E. STANLEY. Ferroelectric Models. J. E EAGLE. X - Y Model. D. D. BETTS. Contents of Volume 4 t
Theory of Correlations in the Critical Region. M. E. FISHER and D. JASNOW. Contents of Volume 5at
Scaling, Universality and Operator Algebras. Leo R KADANOFF. Generalized Landau Theories. Marshall Luban. Neutron Scattering and Spatial Correlation near the Critical Point. JENS ALS-NIELSEN. Mode Coupling and Critical Dynamics. KYOZI KAWASAKI. Contents of Volume 5b t
Monte Carlo Investigations of Phase Transitions and Critical Phenomena. K. BINDER. Systems with Weak Long-Range Potentials. R C. HEMMER and J. L. LEBOWITZ. Correlation Functions and Their Generating Functionals: General Relations with Applications to the Theory of Fluids. G. STELE. Heisenberg Ferromagnet in the Green's Function Approximation. R. A. TAHIR-KHELI. Thermal Measurements and Critical Phenomena in Liquids. A. V. VORONEL. Contents of Volume 6 (The Renormalization Group and its Applications) t
Introduction. K. G. WILSON. The Critical State, General Aspects. F. J. WEGNER. Field Theoretical Approach. E. BREZIN, J. C. LE GUILLOU and J. ZINN-JUSTIN. The 1/n Expansion. S. MA. The e-Expansion and Equation of State in Isotropic Systems. D. J. WALLACE. Universal Critical Behaviour. A AHARONY. Renormalization: Ising-like Spin Systems. TH. NIEMEUER and J. M. J. VAN LEEUWEN Renormalization Group Approach. C. DI CASTRO and G. JONA-LASINIO.
Contents of Volumes 1-17
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Contents of Volume 7 t
Defect-Mediated Phase Transitions. D. R. NELSON. Conformational Phase Transitions in a Macromolecule: Exactly Solvable Models. E W. WIEGEL. Dilute Magnetism. R. B. STINCHCOMBE. Contents of Volume 8
Critical Behaviour at Surfaces. K. BINDER. Finite-Size Scaling. M. N. BARBER. The Dynamics of First Order Phase Transitions. J. D. GUNTON, M. SAN MIGUEL and P. S. SAHNI. Contents of Volume 9 t
Theory of Tricritical Points. I. D. LAWRIE and S. SARBACH. Multicritical Points in Fluid Mixtures: Experimental Studies. C. M. KNOBLER and R. L. SCOTT. Critical Point Statistical Mechanics and Quantum Field Theory. G. A. BAKER, JR. Contents of Volume 10
Surface Structures and Phase Transitions--Exact Results. D. B. ABRAHAM. Field-Theoretic Approach to Critical Behaviour at Surfaces. H. W. DIEHL. Renormalization Group Theory of Interfaces. D. JASNOW. Contents of Volume 11
Coulomb Gas Formulation of Two-Dimensional Phase Transitions. B. NIENHUIS. Conformal Invariance. J. L. C ARDY. Low-Temperature Properties of Classical Lattice Systems: Phase Transitions and Phase Diagrams. J. SLAWNY. Contents of Volume 12 t
Wetting Phenomena. S. DIETRICH. The Domain Wall Theory of Two-Dimensional Commensurate-Incommensurate Phase Transitions. M. DEN NUS. The Growth of Fractal Aggregates and their Fractal Measures. P. MEAKIN.
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Contents of Volumes 1-17
Contents of Volume 13
Asymptotic Analysis of Power-Series Expansions. A. J. GUTTMANN. Dimer Models on Anisotropic Lattices. J. E NAGLE,. S. O. YOKOI and S. M. BHATTACHARJEE. Contents of Volume 14
Universal Critical-Point Amplitude Relations. V. PRIVMAN,P. C. HOHENBERG and A. AHARONY. The Behaviour of Interfaces in Ordered and Disordered Systems. G. FORGACS, R. LIPOWSKY and TH. M. NIEUWENHUIZEN Contents of Volume 15
Spatially Modulated Structures in Systems with Competing Interactions. W. SELKE. The Large-n Limit in Statistical Mechanics and the Spectral Theory of Disordered Systems. A. M. KHORUNZHY, B. A. KHORUZHENKO, L. A. PASTUR and M. V. SHCHERBINA. Contents of Volume 16
Self-Assembling Amphiphilic Systems. G. GOMPPER and M. SCHICK. Contents of Volume 17
Statistical Mechanics of Driven Diffusive Systems. B. SCHMITTMANN and R. K. P. ZIA.
The Random Geometry of Equilibrium Phases Hans-Otto Georgii Mathematisches Institut, Ludwig-Maximilians-Universit#t, D-80333 MSnchen, Germany
Email" g e o r g i i @ r z , m a t h e m a t i k , u n i - m u e n c h e n , de
Olle H~iggstr6m Department of Mathematics, Chalmers University of Technology, S-412 96 G6teborg, Sweden
Email: o l l e h @ m a t h , c h a l m e r s , se
Christian Maes Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
Email: C h r i s t i a n . M a e s @ f y s . k u l e u v e n , ac .be
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Equilibrium phases
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T h e lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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R a n d o m fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The Hamiltonian
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Gibbs measures
PHASE TRANSITIONS
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Copyright ('6) 2001 Academic Press Limited
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Phase transition and phases . . . . . . . . . . . . . . . . . . . . . . . . .
3 Some models
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3.1
The ferromagnetic Ising model . . . . . . . . . . . . . . . . . . . . . . .
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The antiferromagnetic Ising model . . . . . . . . . . . . . . . . . . . . .
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The Potts model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The hard-core lattice gas model . . . . . . . . . . . . . . . . . . . . . . .
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The W i d o m - R o w l i n s o n lattice model
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4 Coupling and stochastic domination
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The coupling inequality . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Stochastic domination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Applications to the Ising model . . . . . . . . . . . . . . . . . . . . . . .
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Application to other models . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Percolation 5.1
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Bernoulli percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Dependent percolation: the role of the density . . . . . . . . . . . . . . .
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Examples of dependent percolation . . . . . . . . . . . . . . . . . . . . .
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The number of infinite clusters . . . . . . . . . . . . . . . . . . . . . . .
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6 Random-cluster representations . . . . . . . . . . . . . . . . . . . . . . . . . .
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Random-cluster and Potts models
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Infinite-volume limits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Phase transition in the Potts model . . . . . . . . . . . . . . . . . . . . .
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Infinite volume random-cluster measures . . . . . . . . . . . . . . . . . .
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An application to percolation in the Ising model . . . . . . . . . . . . . .
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Cluster algorithms for computer simulation
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Random-cluster representation of the W i d o m - R o w l i n s o n model
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7 Uniqueness and exponential mixing from non-percolation . . . . . . . . . . . . 7.1
Disagreement paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Stochastic domination by random-cluster measures
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Exponential mixing at low temperatures
8 Phase transition and percolation
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Agreement percolation from phase coexistence
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Plus-clusters for the Ising ferromagnet . . . . . . . . . . . . . . . . . . .
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Constant-spin clusters in the Potts model . . . . . . . . . . . . . . . . . .
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Further examples of agreement percolation
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Percolation of ground-energy bonds
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10 Continuum models
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1 Random geometry of equilibrium phases
10.1 Continuum percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The continuum Widom-Rowlinson model . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
Equilibrium statistical mechanics describes and explains the macroscopic behavior of systems in thermal equilibrium in terms of the microscopic interaction between their very many constituents. As a typical example, let us take some ferromagnetic material like iron: the constituents are then the spins of elementary magnets at the sites of some crystal lattice. Or we may think of a lattice approximation to a real gas, in which case the constituents are the particle numbers in the elementary cells of any partition of space. The central object is the Hamiltonian describing the interaction between these constituents. This interaction determines the relative energies between configurations that differ only microscopically. The equilibrium states with respect to the given interaction are described by the associated Gibbs measures. These are probability measures on the space of configurations which have prescribed conditional probabilities with respect to fixed configurations outside of finite regions. These conditional probabilities are given by the Boltzmann factor, the exponential of the inverse temperature times the relative energy. This allows one to compute, at least in principle, equilibrium expectations and spatial correlation functions following the standard Gibbs formalism. The so-called extremal Gibbs measures are very important since they describe the possible macrostates of our physical system. In such a state, macroscopic observables do not fluctuate while the correlation between local observations made far apart from each other decays to zero. Since the early days of statistical mechanics, geometric notions have played a role in elucidating certain aspects of the theory. This has taken many different forms. Arguably, the thermodynamic formalism, as first developed by Gibbs, already admits some geometric interpretations primarily related to convexity. For example, entropy is a concave function of the specific energy, the pressure is convex as a function of the interaction potential, the Legendre-Fenchel transformation relates various fundamental thermodynamic quantities to each other, and the set of Gibbs measures for an interaction is a simplex with vertices corresponding to the physically realized macrostates, the equilibrium phases. Here, however, we will not be concerned with this kind of convex geometry which is described in detail e.g. in the books by Israel (1979) and Georgii (1988). Rather, the geometry considered here is a way of visualizing the structure in the typical realizations of the system's constituents. To be more specific, let
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us consider for a moment the case of the standard ferromagnetic Ising model on the square lattice. At each site we have a spin variable taking only two possible values, +1 and - 1 . The interaction is nearest-neighbor and tends to align neighboring spins in the same direction. By the ingenious arguments first formulated by Peierls (1936) (see also Dobrushin, 1965; Sinai, 1982; Georgii, 1988), the phase transition in this model can be understood from looking at the typical configurations of contours, i.e., the broken lines separating the domains with plus resp. minus spins. The plus phase (the positively magnetized phase) is realized by an infinite ocean of plus spins with finite islands of minus spins (which in turn may contain lakes of plus spins, and so on). On the other hand, above the Curie temperature (first computed by Onsager) there is no infinite path joining nearest neighbors with the same spin value. So, for this model the geometric picture is rather complete (as we will show later). In general, however, much less is known, and much less is true. Still, certain aspects of this geometric analysis have wide applications, at least in certain regimes of the phase diagram. These "certain regimes" are, on the one hand, the high-temperature (or, in a lattice gas setting, low-density) regime and, to the other extreme, the low-temperature behavior. At high temperatures, all thermodynamic considerations are based on the fact that entropy dominates over energy. That is, the interaction between the constituents is not effective enough to enforce a macroscopic ordering of the system. As a result, every constituent is more or less free to behave at random, not much influenced by other constituents which are far apart. So, the system's behavior is almost like that of a free system with independent components. This means, in particular, that in the center of a large box we will typically encounter more or less the same configurations no matter what boundary conditions outside this box are imposed. That is, if we compare two independent realizations of the system in the box with different boundary conditions outside then, still at high temperatures, the difference between the boundary conditions cannot be felt by the spins in the center of the box; specifically, there should not exist any path from the boundary to the central part of the box along which the spins of the two realizations disagree. This picture is rather robust and can also be applied when the interaction is random (see Sections 7 and 9). At low temperatures, or large densities (when the interaction is sufficiently strong), the above picture no longer holds. Rather, the specific characteristics of the interaction will come into play and determine the specific features of the low-temperature phase. In many cases, the low-temperature behavior can be described as a random perturbation of a ground state, i.e., of a fixed configuration of minimal energy. Then we can expect that at low temperatures, and sometimes even up to the critical temperature, the equilibrium phases are realized as a deterministic ground state configuration, perturbed by finite random islands on which the configuration disagrees with the ground state. This means that the ground
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Random geometry of equilibrium phases
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state pattern can percolate through the space to infinity. One prominent way of confirming this picture is provided by the so-called Pirogov-Sinai theory which has been described in detail by Zahradnfk (1996). In Section 8 we will discuss some other techniques of establishing the same geometric picture. It is evident from the above that percolation theory will play an important role in this text. In fact, we will mainly be concerned with dependent percolation, but one can say that independent percolation stands as a prototype for the study of statistical equilibrium properties in geometric terms. In independent percolation, the model is extremely simple: the components are binary-valued and independent from each other. What is hard is the type of question one asks, namely the question of existence of infinite paths of l's and their geometry. We will introduce percolation below but refer to other publications (e.g. Grimmett, 1999) for a systematic account of the theory. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc. will be useful for describing certain geometric features of equilibrium phases, allowing characterizations of phases in percolation terms. Examples will be presented where the (thermal) phase transition goes hand in hand with a phase transition in an associated percolation process. Next, percolation techniques can be used to obtain specific information about the phase diagram of the system. For example, equilibrium correlation functions are sometimes dominated by connectivity functions in an associated percolation problem which is easier to investigate. Finally, representations in terms of percolation models will yield explicit relations between certain observables in equilibrium models and some corresponding percolation quantities. In fact, the resulting percolation models, like the random-cluster model, have some interest in their own right and will also be studied in some detail. This text is supposed to be self-contained. Therefore we need to introduce various concepts and techniques which are well known to some readers. On the other hand, important related issues will not be discussed when they are not explicitly needed. For more complete discussions on the introductory material we will refer the interested reader to other sources. More seriously, we will not include here a discussion of some important geometric concepts developed in the 1980s for the study of critical behavior in statistical mechanical systems, namely random walk expansions or random current representations. Fortunately, we can refer to an excellent book (Fernandez et al., 1992) where the interested reader will find all the relevant results and references. Important steps in this context include (Aizenman, 1982; Aizenman and Barsky, 1987; Aizenman et al., 1987; Aizenman and Fernandez, 1986; Brydges et al., 1982, 1983) and the references therein. Finally, to avoid misunderstanding, the random geometry in the title of this work should not be confused with stochastic geometry (or geometric probability) which, as a branch of integral geometry, provides a very interesting tool-box
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al.
for the discussion of morphological characteristics of random fields appearing in statistical physics and beyond, see Klain and Rota (1997); Adler (2000) and Mecke (1998).
2 2.1
Equilibrium phases The lattice
Our object of study are physical systems with many constituents, spins or particles, which will be located at the sites of a crystal lattice s The standard case is when s = Z d, the d-dimensional hypercubic lattice. In general, we shall assume that s is the vertex set of a countable graph. That is, s is an at most countable set and comes equipped with a (symmetric) adjacency relation. Namely, we write x -~ y if the vertices x, y E s are adjacent, and this is visualized by drawing an edge between x and y. In this case, x and y are also called neighbors, and the edge (or bond) between x and y is denoted by (xy). We write B for the set of all edges (bonds) in s A complete description of the graph is thus given by the pairs (s ~ ) or (s B). In the case s = Z d, the edges will usually be drawn between lattice sites of distance one; hence x -~ y whenever Ix - Yl = 1. Here, I 9I stands for the sum-norm, i.e. Ixl - ~/d_l Ixil whenever x - (xl . . . . . xd) C Z d. This choice is natural because then Ix - Y l coincides with the graph-theoretical (or lattice) distance, viz. the length of the shortest path (consisting of consecutive edges) connecting x and y. For convenience, we sometimes use the same notation in the case of a general graph. On Z d we will occasionally distinguish between the standard metrics d l ( x , y) - - Z i Ixi - Yil, d2(x, y) = [zid=l ( x i - yi)2)] 1/2 and de~(x, y) = max/Ixi - Y i l . Given any metric d on s we write d ( A , A) = infxeA,y~zX d(x, y) for the distance of two subsets A, A C s We will always assume that the graph (s ~ ) is locally finite, which means that each x 6 s has only a finite number Nx of nearest neighbors. In other words, Nx is the number of edges emanating from x. Nx is also called the degree of the graph at x. In many cases we will even assume that (s ~ ) is of bounded degree, which means that N -- SUPx~z; Nx < e~. Common examples of such graphs, besides Z d, are the triangular lattice in two dimensions, and the regular tree Td (also known as the Cayley tree or the Bethe lattice), which is defined as the (unique) infinite connected graph containing no circuits in which every vertex has exactly d + 1 nearest neighbors. A region of the lattice, that is a subset A C s is called finite if its cardinality IAl is finite. We write E for the collection of all finite regions. The complement of a region A will be denoted by A c = s \ A. The boundary 0 A of A is the set of all sites (vertices) in A c which are adjacent to some site of A.
1 Random geometry of equilibrium phases
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On some occasions we will need the notion of thermodynamic (or infinite volume) limit, and we need to describe in what sense a region A ~ ,5' grows to the full lattice s For our purposes, it will, in general, be sufficient to take an arbitrary increasing sequence (An) with [.-Jn>__l An = / ~ . In the case/2 = Z a, we will often make the standard choice An -- [ - n , n] a N Z a, the lattice cubes centered around the origin. As ,5' is a directed set ordered by inclusion, we will occasionally also consider the limit along E. In each of these cases we will use the notation A 1" s
2.2 Configurations The constituents of our systems are the spins or particles at the lattice sites. So, at each site x 6 s we have a variable cr (x) taking values in a non-empty set S, the state space or single-spin space. In a magnetic set-up (to which we mostly adhere for simplicity), cr (x) is interpreted as the spin of an elementary magnet at x. In a lattice gas interpretation, there is a distinguished vacuum state 0 6 S representing the absence of any particle, and the remaining elements correspond to the types and/or the number of the particles at x. Unless stated otherwise, we will always assume that S is finite. Elements of S will typically be denoted by a, b . . . . . A configuration is a function cr : s --+ S which assigns to each vertex x 6 s a spin value cr(x) 6 S. In other words, a configuration cr is an element of the product space S2 = S s f2 is called the configuration space and its elements are usually written as cr, r/, ~ . . . . . (It is sometimes useful to visualize a, b . . . as colors. A configuration is then a coloring of the lattice.) A configuration cr is constant if for some a ~ S, cr (x) -- a for all x ~ / 2 . Two configurations cr and r/ are said to agree on a region A C s written as "or _= q on A", if or(x) = q(x) for all x c A. Similarly, we write "cr - 77 off A" if cr(x) = O(x) for all x r A. We also consider configurations in regions A C s These are elements of S A, again denoted by letters like cr, 77, ~ . . . . . Given o-, ~ 6 f2, we write crA r/Ac for the configuration ~ 6 f2 with ~(x) = or(x) for x 6 A and ~(x) = O(x) for x 6 A c. Then, obviously, ~ ---- cr on A. The cylinder sets A/'A (cr) = {~ ~ S2 :~ -- cr on A}, A 6 E, form a countable neighborhood basis of cr 6 f2; they generate the product topology on f2. Hence, two configurations are close to each other if they agree on some large finite region, and a diagonal-sequence argument shows that f2 is compact in this topology. We will often change a configuration cr ~ f2 at just one site x ~ / 2 . Changing or(x) into a prescribed value a E S we obtain a new configuration written crx,a. In particular, for S -- { - 1 , + 1} we write cry (y) _ [ cr (y) ! -or(x)
for y r x for y - x
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for the configuration resulting from flipping the spin at x. We will also deal with automorphisms of the underlying lattice (s .-~). Each such automorphism defines a measurable transformation of the configuration space f2. The most interesting automorphisms are the translations of the integer lattice/2 = zd; the associated translation group acting on f2 is given by Ox~r(y) = cr(x + y), y ~ Z d. In particular, any constant configuration is translation invariant. Similarly, we can speak about periodic configurations which are invariant under Ox with x in some sublattice of Z a. Later on, we will also consider configurations which refer to the lattice bonds rather than the vertices. These are elements 7/of the product space {0, 1}t3, and a bond b 6 13 will be called open if q(b) = 1, and otherwise closed. The above notations apply to this situation as well.
2.3
Observables
An observable is a real function on the configuration space which may be thought of as the numerical outcome of some physical measurement. Mathematically, it is a measurable real function on ~. Here, the natural underlying o--field of measurable events in f2 is the product ~r-algebra Y -- (70) c, where .To is the set of all subsets of S..T" is defined as the smallest a-algebra on f2 for which all projections X ( x ) : f2 -+ S, X ( x ) ( o ) = or(x) with cr ~ f2 and x c s are measurable. It coincides with the Borel a-algebra for the product topology on f2. We also consider events and observables depending only on some region A C /2. We let .T'A denote the smallest sub-a-field of Y containing the events A/'A(or) for o- e S A and A c ,f with A C A. Equivalently, YA is the o--algebra generated by the projections X (x) with x 6 A. YA is the a-algebra of events occurring in A. An event A is called local if it occurs in some finite region, which means that A 6 .T'A for some A 6,5'. Similarly, an observable f :S2 --+ R is called local if it depends on only finitely many spins, meaning that f is measurable with respect to YA for some A 6 ,5'. More generally, an observable f is called quasilocal if it is (uniformly) continuous, i.e., if for all ~ > 0 there is some A 6 E such that I f ( a ) - f ( $ ) l < ~ whenever ~ = cr on A. The set C ( ~ ) of continuous observables is a Banach space for the supremum norm Ilfll = sup~ If(~r)l, and the local observables are dense in it. The local events and observables should be viewed as microscopic quantities. On the other side we have the macroscopic quantities which only depend on the collective behavior of all spins, but not on the values of any finite set of spins. They are defined in terms of the tail or-algebra T = (-]A~E YA c, which is also called the o'-algebra of all events at infinity. Any tail event A 6 T and any T measurable observable is called macroscopic.
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As a final piece of notation we introduce the indicator function IA of an event A; it takes the value 1 if the event occurs (IA (or) = 1 if cr 6 A) and is zero otherwise.
2.4
Random fields
As the spins of the system are supposed to be random, we will consider suitable probability measures/z on (f2, .T'). Each such # is called a random field. Equivalently, the family X = (X(x), x ~ E) of random variables on the probability space (S2, 9t-, #) which describe the spins at all sites is called a random field. Here are some standard notations concerning probability measures. The expectation of an observable f with respect to # is written as # ( f ) = f f d # . The probability of an event A is # ( A ) -- lZ(IA) = fA dlz, and we omit the set braces when A is given explicitly. For example, given any x 6 /2 and a c S we write # (X (x) = a) for # (A) with A = {or 6 f2 : o (x) = a }. Covariances are abbreviated as # ( f ; g) = # ( f g ) - # ( f ) # ( g ) . Whenever we need a topology on probability measures on f2, we shall take the weak topology. In this (metrizable) topology, a sequence of probability measures #n converges to #, denoted by #n ~ #, if #n(A) -+ #(A) for all local events A E U A~s f A . This holds if and only if lZn(f) -+ # ( f ) for all local, or equivalently, all quasilocal functions f . In applications, #n will often be an equilibrium state in a finite box An tending to s as n -+ c~, and we are interested in whether the probabilities of events occurring in some fixed finite volume have a well-defined thermodynamic (or bulk) limit. That is, we observe what happens around the origin (via the local function f ) while the boundary of the box in which we realize the equilibrium state receeds to infinity. As there are only countably many local events, one can easily see by a diagonal-sequence argument that the set of all probability measures on f2 is compact in the weak topology.
2.5
The Hamiltonian
We will be concerned with systems of interacting spins. As usual, the interaction is described by a Hamiltonian. As the spins are located at the sites of a graph (s ~ ) , it is natural to consider the case of homogeneous neighbor potentials. (We will deviate from homogeneity in Section 9 when considering random interactions.) The Hamiltonian H then takes the form H(o-) = ~ x~y
U(o-(x), ~r(y)) + ~
V(cr(x))
(2.5.1)
x
with a symmetric function U : S • S --+ R U {~}, the neighbor-interaction, and a self-energy V : S --+ R. The infinite sums are formal; the summation index
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x ~ y means that the sum extends over all bonds (xy) ~ 13 of the lattice. U thus describes the interaction between spins at neighboring sites, while V might come from the action of an external magnetic field. In a lattice gas interpretation when S = {0, 1 } (the value 1 being assigned to sites which are occupied by a particle), V corresponds to a chemical potential. To make sense of the formal sums in (2.5.1) we compare the Hamiltonian for two different configurations ~r, r/ 6 g2 which differ only locally (or are "local perturbations" or "excitations" of each other), in that cr = r/off some A 6 g. For such configurations we can define the relative Hamiltonian H(cr It/) -- ' ~ [ U ( c r ( x ) ,
cr (y)) -
U(rl(x), r/(y))] + Z [ V ( c r ( x ) )
x"~y
- V(r/(x))]
x
(2.5.2) in which the sums now contain only finitely many non-zero terms: the first part is over those neighbor pairs (xy) for which at least one of the sites belongs to A, and the second part is over all x E A.
2.6
Gibbs measures
Gibbs measures are random fields which describe our physical spin system when it is in macroscopic equilibrium with respect to the given microscopic interaction at a fixed temperature. Here, macroscopic equilibrium means that all parts of the system are in equilibrium with their exterior relative to the prescribed interaction and temperature. So it is natural to define Gibbs measures in terms of conditional probabilities. Definition 2.1 A probability measure lz on the configuration space f2 is called a Gibbs measure for the Hamiltonian H in (2.5.1) or (2.5.2) at inverse temperature ~ 1 / T iffor all A E g and all cr E f2, # ( X = o" on AIX ~ r/off A) --/Z~,A(Cr)
(2.6.1)
for #-almost all q c f2. In the above, #~,A (or) is the Boltzmann-Gibbs distribution in A f o r fi and H, which is given by
U~,A (or) -
I{~_~ off A} exp[-fiH(alq)]. ZA (fi, ~)
(2.6.2)
Here, ZA (13, q) is a normalization constant making lz~, A a probability measure, and the constraint that cr has to coincide with rl outside A is added because we want to realize these probability measures immediately on the infinite lattice. Note that/zfi,A~ only depends on the restriction of rl to A c.
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So, # is a Gibbs measure if it has prescribed conditional distributions inside some finite set of vertices, given that the configuration is held fixed outside, and these conditional distributions are given by the usual Boltzmann-Gibbs formalism. This definition goes back to the work of Dobrushin (1968a) and Lanford and Ruelle (1969), whence Gibbs measures are often called DLR-states. By this work, equilibrium statistical physics and the study of phase transitions made firm contact with probability theory and the study of random fields. A thermodynamic justification of this definition can be given by the variational principle, which states that (in the case s = Z d) the translation invariant Gibbs measures are precisely those translation invariant random fields which minimize the free energy density (Lanford and Ruelle, 1969; Ellis, 1985; Georgii, 1988). For better distinction, the Gibbs distributions/z~,A are often called finite volume Gibbs distributions, whereas the Gibbs measures are sometimes specified as infinite volume Gibbs measures. We write G(fiH) for the set of all Gibbs measures with given Hamiltonian H and inverse temperature ft. In the special case U - 0 of no interaction, there is only one Gibbs measure, namely the product measure with one-site marginals /z(X(x) - a) - e-flV(a)/ Y~bES e-~V(b). In general, several Gibbs measures for the same interaction and temperature can coexist. This is the fundamental phenomenon of nonuniqueness of phases which is one of our main subjects. We return to this point in Section 2.7 below. First we want to emphasize an important consequence of our assumption that the underlying interaction U involves only neighbor spins. Due to this assumption, the Gibbs distribution/zt~,A~ only depends on the restriction r/~A of r/to the boundary OA of A, and this implies that each Gibbs measure/z E G(fiH) is a Markov random field. By definition, this means that for each A 6 ~c and a c S A /z(X = a on AI~A~) : / z ( X ~ a on AI~0A)
(2.6.3)
/z-almost surely. This Markov property will be an essential tool in the geometric arguments to be discussed in this review. There is, in fact, an equivalence between Markov random fields and Gibbs measures for nearest neighbor potentials, see Grimmett (1973); Averintsev (1975) or Georgii (1988). As an aside, let us comment on the case when the interaction of spins is not nearest-neighbor but only decays sufficiently fast with their distance. The Boltzmann-Gibbs distributions in (2.6.2), and therefore also Gibbs measures, can then still be defined, but the Gibbs measures fail to possess the Markov property (2.6.3). Rather their local conditional distributions/z~,A~ satisfy a weakening of the Markov property called quasilocality or almost-Markov property: for every A c ~c and A c .TA,/z~,A (A) is a continuous function of r/. So, in this case, Gibbs measures have prescribed continuous versions of their local conditional probabilities. To obtain a sufficiently general definition of Gibbs measures including this
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and other cases, one introduces the concept of a specification G = (GA, A 6 s This is a family of probability kernels GA from (~, 9rAc) to (f2, .T). GA(-, 0) stands for any distribution of spins with fixed configuration r/Ac 6 S s outside A; the standard case is the Gibbs specification GA(., r/) -- #~,A- A Gibbs measure is then a probability measure # on f2 satisfying #(AlgAe) - GA (A,-) /z-almost surely for all A c ,5" and A 6 ~ ; this property can be expressed in a condensed form by the invariance equation/zGA = / z . In order for this definition to make sense the specification G needs to satisfy a natural compatibility condition for pairs of volumes A C A' expressing the fact that if the system in A' is in equilibrium with its exterior, then the subsystem in A is also in equilibrium with its own exterior. It is easy to see that the Gibbs distributions in (2.6.2) are compatible in this sense. Details and further discussion can be found in many books and articles dealing with mathematical results in equilibrium statistical mechanics, (Preston, 1974a; Ruelle, 1978; Israel, 1979; Sinai, 1982; Georgii, 1988; van Enter et al., 1993). Liggett (1985) explains the relation between Gibbs measures and the condition of detailed balance (reversibility) in certain stochastic dynamics. The state of Statistical Mechanics just before the introduction of Gibbs measures is described in Ruelle (1969). Finally, we mention an alternative and constructive approach to the concept of Gibbs measures. Starting from the finite-volume Gibbs distributions/z~,A, one might ask what kind of limits could be obtained if r/is randomly chosen and A increases to the whole lattice s (This slightly older but still important approach was suggested by Minlos (1967)). To make this precise we consider the measures /z~,A -- f/z~,AP(do) , where p is any probability measure on f2 describing a "stochastic boundary condition". Any such/z/~,AP is called a (finite volume) Gibbs distribution with respect to H at inverse temperature/3, and their collection is denoted by (TA(fill). The set of all (infinite volume) Gibbs measures is then equal to
G(fiH) -- ("] ~A(flH). Equivalently, a probability measure # on f2 is a Gibbs measure for the Hamiltonian/3 H if it belongs to the closed convex hull of the set of limit points of #/~,A asA ?s One important consequence is that G(fiH) (= 0. This is because each GA (fl H) is obviously non-empty and compact. Equivalently, to obtain an infinite volume Gibbs measure one can fix a particular configuration r/ and take it as boundary condition. By compactness, we obtain an infinite volume Gibbs measure #~ by taking the limit of (2.6.2) as A 1' s at least along suitable subsequences; for details see Preston (1974a) or Georgii (1988). We remark that, in general, there is no unique limiting measure #~; rather there may be several such limiting measures obtained as limits along different subsequences.
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Fortunately, however, this is not the case for a wide class of models, either at low temperatures (/~ large) when ~ is a ground state configuration (in the realm of the Pirogov-Sinai theory) (Pirogov and Sinai, 1976), or at high temperatures when g is small. We conclude this subsection with a general remark. As all systems in nature are finite, one may wonder why we consider here systems with infinitely many constituents. The answer is that sharp results for bulk quantities can only be obtained when we make the idealization to an infinite system. The thermodynamic limit eliminates finite size effects (which are always present but which are not always relevant for certain phenomena) and it is only in the thermodynamic limit of infinite volume that we can get a clean and precise picture of realistic phenomena such as phase transitions or phase coexistence. This is a consequence of the general probabilistic principle of large numbers. In this sense, infinite systems serve as an idealized approximation to very large finite systems.
2.7
Phase transition and phases
As pointed out above, in general there may exist several solutions # to the DLRequation (2.6.1) for given U, V and g, which means that multiple Gibbs measures exist. The system can then choose between several equilibrium states. (In a dynamical theory this choice would depend on the past; but here we are in a pure equilibrium setting.) The phenomenon of non-uniqueness therefore corresponds to a phase transition. In fact, it is then possible to construct different Gibbs measures as infinite volume limits of Gibbs distributions with different choices of boundary conditions (Georgii, 1973, 1988). Since any two Gibbs measures can be distinguished by a suitable local observable, a phase transition can be detected by looking at such a local observable which is then called an order parameter. Varying the external parameters such as temperature or an external magnetic field (which can be tuned by the experimenter) one will observe different scenarios; these are collected in the so-called phase diagram of the considered system. As we have indicated in the introduction, the phase transition phenomenon is of central interest in equilibrium statistical mechanics. When phase transitions occur and when they do not is also one of the primary questions (although we will encounter many others) that we will try to answer with the geometric methods to be developed in subsequent sections. If multiple Gibbs measures for a given interaction exist, the structure of the set ~(g H) of all Gibbs measures becomes relevant. We only state here the most basic results; a detailed exposition can be found in Georgii (1988), for example. The basic observation is that G ( g H ) is a convex set. Its extremal elements, the extremal Gibbs measures, have a trivial tail o--field 7- (which means that all events in 7" have probability 0 or 1). Equivalently, all macroscopic observables
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are almost surely constant. In addition, the tail triviality can be characterized by an asymptotic independence (or mixing) property. On the other hand, any Gibbs measure # can be decomposed into extremal Gibbs measures; therefore every configuration which is typical for # is in fact typical for some extremal Gibbs measure. This shows that the extremal Gibbs measures correspond to what one can really see in nature as far as large systems in equilibrium are concerned. The extremal Gibbs measures therefore correspond to the physical macrostates, whereas non-extremal Gibbs measures only provide a limited description when the system's precise state is unknown. For all these reasons, the extremal Gibbs measures are called (equilibrium) phases. The central subject of this review is the geometric analysis of their typical configurations, and thereby the analysis of the phase diagram giving the variation in the number and the nature of the phases as one changes various control parameters (coupling, temperature, external fields, etc.). Often it is natural to consider automorphisms of the graph (/2, ~). For example, if s -- Z d we consider the translation group (Ox)xcZd . A homogeneous phase is then an extremal Gibbs measure which is also translation invariant. On the other hand, we can regard the extremal points of the convex set of all translation invariant Gibbs measures. These are ergodic, which means that they cannot be decomposed into distinct translation invariant probability measures, and are trivial on the a-algebra of all translation invariant events. However, these extremal translation invariant Gibbs measures need not be homogeneous phases; they are only ergodic. Yet, ergodic measures # satisfy a law of large numbers: for any observable f and any sequence of increasing cubes A, 1
lim ~-" f o Ox -- # ( f ) al, za IAI
/z-almost surely.
Hence, ergodic Gibbs measures are suitable for modelling macrostates in equilibrium if one limits oneself to measuring certain bulk observables or macroscopic quantities with additivity properties. Notice, however, that there exists a certain non-uniformity in the literature concerning the nomenclature. Sometimes these ergodic Gibbs measures are called (pure) phases. It is then argued that it might happen that two phases (as defined above) for a system can by no means be macroscopically distinguished (for example if one is a translation of the other). We do not wish to enter into a detailed discussion of these points.
3
S o m e models
In this section we discuss briefly the phase transition behaviour of some prototypical examples of Gibbs systems. Although these examples are fairly standard and
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well-known to most of our readers, we need to include them here to set the stage. They will be studied in detail in the later sections. An account of phase transition phenomena in more general lattice models can be found in many other sources, including Sinai (1982); Georgii (1988); Simon (1993); Miracle-Sole (1994) and Koteck2~ (1995).
3.1 The ferromagnetic Ising model The Ising model was introduced by Wilhelm Lenz (1920) and his student Ernst Ising (1925) as a simple model for magnetism and, in particular, ferromagnetic phase transitions. Each site x 6 /2 can take either of two spin values, +1 ("spin up") and - 1 ("spin down"), so that the state space is equal to S -- {-1, +l}. The Hamiltonian is given by (2.5.1) with U ( ~ ( x ) , ~ ( y ) ) -- - ~ ( x ) ~ ( y ) and V ( ~ ( x ) ) - - h c ~ ( x ) . The parameter h 6 R describes an external field. The finite volume Gibbs distribution in a box A with external field h at inverse temperature/3 > 0 with boundary condition r/ is thus the probability measure #~,r on ~2 - {-1, + 1}s which to each ~ 6 ~2 assigns probability proportional to
x~AoryEA
For /3 - 0 ("infinite temperature") the spin variables are independent under lZh,~, A , ~ but as soon as/3 > 0 the probability distribution starts to favour configurations with many neighbor pairs of aligned spins. This tendency becomes stronger and stronger as/3 increases. In the case h -- 0 of no external field, the model is symmetric under interchange of the spin values - 1 and + 1, so that there is an equal chance of having many pairs of plus spins or having many pairs of minus spins. This dichotomy gives rise to the following interesting behavior. Suppose that s - Z a, d > 2. If /3 is sufficiently small (i.e., in the high temperature regime), the interaction is not strong enough to produce any long range order, so that the boundary conditions become irrelevant in the infinite volume limit and the Gibbs measure is uniquely determined. By ergodicity and the -t- symmetry, the limiting fraction of plus spins will almost surely be 1/2 under this unique Gibbs measure. In contrast, when/3 is sufficiently large (in the low-temperature regime), the interaction becomes so strong that a long-range order appears: the bias towards neighbor pairs of equal spin then implies that Gibbs measures prefer configurations with either a vast majority of plus spins or a vast majority of minus spins, and this preference even survives in the infinite volume limit. The system thus undergoes a phase transition which manifests itself in a non-uniqueness of Gibbs measures. Specifically, there
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exist two particular Gibbs measures # + and tz-, obtained as infinite volume limits with respective boundary conditions 77 =_ § 1 and r/ = - 1 , which can be distinguished by their overall density of + 1s: the density is greater than 1/2 under # + and (by symmetry) less than 1/2 under # - . This is the spontaneous magnetization phenomenon that Lenz and Ising were looking for but were discouraged by not finding it in one dimension. In higher dimensions, the uniqueness regime and the phase transition regime are separated by a sharp critical value/3c, as is summarized in the following classical theorem (Peierls, 1936; Dobrushin, 1965, 1968b): Theorem 3.1 For the ferromagnetic Ising model on the integer lattice Z a of dimension d > 2 at zero external field, there exists a critical inverse temperature ~c E (0, cx~) (depending on d) such that for ~ < ~c the model has a unique Gibbs measure while for ~ > ~c there are multiple Gibbs measures. A stochastic-geometric proof of this result will be given in Section 6. The result (as well as its proof) holds for any graph (s ,~) in place of Z a, except that/~c may then take the values 0 or cx~. For instance, on the one-dimensional lattice Z 1 we have/~c - cx~, which means that there is a unique Gibbs measure for all /~. For Z 2, the critical value has been found to be/~c - 89log(1 + ~/2). This calculation is a remarkable achievement which began with Onsager (1944). An account of various (algebraic and/or combinatorial) methods can be found in Thompson (1972) and McCoy and Wu (1973). Let us also mention the work done by Abraham and Martin-L6f (1973) relating these exact computations to the real magnetization in the appropriate Gibbs measures; it also gives the result that there is a unique Gibbs measure at the critical value/~ =/~c. A rigorous calculation of the critical value in higher dimensions is beyond current knowledge. It is believed that uniqueness holds at criticality in all dimensions d > 2, but so far this is only known for d -- 2 and d > 4 (Aizenman and Fernandez, 1986). The case of a non-zero external field h ~ 0 is less interesting, in that one finds a unique Gibbs measure for all/~ and d. The intuitive explanation is that for h ~: 0 there is no 4- symmetry which could be broken; depending on the sign of h, the system is forced to prefer either § l s or - l s . This comes from the fact that the magnetic field acts on the whole volume, whereas the influence of a boundary condition is of smaller order as the volume increases. In contrast, a phase transition for h ~ 0 does occur when Z d is replaced by certain nonamenable graph structures for which the boundary of a volume is of the same order of magnitude as the volume itself (which makes them physically perhaps less realistic) - an example is the regular tree Td with d >_ 2; we refer to Spitzer (1975); Georgii (1988) and Jonasson and Steif (1999). A phase transition can also occur for a non-zero external field for the Ising model on a half-space where it is due to the so-called Basuev phenomenon (Basuev, 1984, 1987).
1 Random geometry of equilibrium phases
17
Because of the simplicity of its model assumptions, the standard Ising model has inspired a variety of techniques for analyzing interacting random fields. Its ferromagnetic structure suggests various monotonicity properties which can be checked by the coupling methods to be described in Section 4, and the assumption of neigbor interaction implies the spatial Markov property (2.6.3) which plays a fundamental role in the geometric analysis of typical configurations. Many techniques which were developed on this testing ground also turned out to be fruitful in more general cases.
3.2 The antiferromagnetic Ising model The Ising antiferromagnet is defined quite similarly to the ferromagnetic case, except that U(a(x), a(y)) is taken to be +a(x)a(y) rather than - a ( x ) a ( y ) . This means that neighboring sites now prefer to take opposite spins. Suppose that h = 0 and that the underlying graph is bipartite. This means that/2 can be partitioned into two sets l~even and 12odd such that sites in ~2even only have edges to sites in ~2odd, and vice versa. Clearly, Z a is an example of a bipartite graph. In this situation, we can reduce the antiferromagnetic Ising model to the ferromagnetic case by a simple spin-flipping trick: The bijection a ~ 6 of f2 defined by
6(x)-
[ a(x) I - a (x)
if x E ~,even, if x ~ s
(3.2.1)
maps any Gibbs measure for the antiferromagnetic Ising model to a Gibbs measure for the ferromagnetic Ising model with the same parameters, and vice versa. As a consequence, a phase transition in the antiferromagnetic model is equivalent to a phase transition in the ferromagnetic model with the same parameters. Hence, Theorem 3.1 immediately carries over to the antiferromagnetic case. The model becomes more interesting (or, at least, more genuinely antiferromagnetic) if either h ~: 0 or the graph is taken to be non-bipartite. Suppose first that h r 0 but still/2 = Z d. If Ihl is small and fl sufficiently large, we have the same picture as in the case h -- 0: there exist two distinct phases, one having a majority of plus spins on the even sublattice and a majority of minus spins on the odd sublattice, the other one having a majority of plus spins on the odd sublattice and a majority of minus spins on the even sublattice. We will show this in Section 8.5, Example 8.17; see also Dobrushin (1968b) and Georgii (1988). Note that this phase transition is somewhat different in flavor compared to that in the ferromagnetic Ising model: whereas in the Ising ferromagnet the phase transition produces a breaking of a state-space symmetry, the phase transition in the Ising antiferromagnet instead breaks the translation symmetry between the sublattices ~even and 12odd.
18
H.-O. Georgii et al.
To see what happens in the case of a non-bipartite graph we consider the triangular lattice which can be obtained by taking the usual square lattice Z 2 and adding an edge between each vertex x and its north-east neighbor x + (1, 1). In this case, one expects uniqueness when h --- 0, and existence of three distinct phases when Ihl # 0 is small and fl is large. Phase transitions in these models were studied by Dobrushin (1968b) and Heilmann (1974), for example.
3.3
The Potts model
A natural generalization of the ferromagnetic Ising model is the (ferromagnetic) Potts model (Potts, 1952), in which each spin may take q >_ 2 (rather than only two) different values. The state space is then S = {1, 2 . . . . . q}, and the pair interaction is given by U(cr(x),
~(y))
= 1 - 2I{,~(x)=cr(y)}.
We confine ourselves to the case of zero external field, so that V(cr (x)) -- 0. Taking q = 2 and identifying the state space {1, 2} with {-1, + 1} we reobtain the ferromagnetic Ising model with zero external field. Just as in the latter case, the Potts interaction favours configurations where many neighbor pairs agree, and Theorem 3.1 can be extended to the Potts model as follows, as we will show in Section 6.3. Theorem 3.2 For the q-state Potts model on Z d, d > 2, there exists a critical inverse temperature fie ~ (0, e~) (depending on d and q) such that f o r fl < fie the model has a unique Gibbs measure while f o r fl > fie there exist q mutually singular Gibbs measures. In the same way as Theorem 3.1, this theorem also holds on general graphs provided we allow fie to be 0 or ec. The Potts model differs from the Ising model in that, for q large enough, there are also multiple Gibbs measures at the critical value fi - fie, as demonstrated by Koteck3) and Shlosman (1982); an outline of a proof will be given in Example 8.21. The Onsager critical value for the twodimensional Ising model is believed to extend to the Potts model on Z 2 through the formula fie(q) - 89log(1 + x/if); see Welsh (1993), for example. This has so far only been established when q is sufficiently large (Lannait et al., 1991).
3.4
The hard-core lattice gas model
The hard-core lattice gas model (or hard-core model for short) describes a gas of particles which can only sit on the lattice sites but are so large that adjacent
1
Random geometry of equilibrium phases
19
sites cannot be occupied simultaneously. The state space is S - {0, 1}, the pair interaction oo i f a ( x ) = a ( y ) = 1 U ( a ( x ) , a ( y ) ) -- 0 otherwise, describes the hard core of the particles, and the chemical potential is V ( a ( x ) ) = -(log)~) a ( x ) . Here Z > 0 is the so-called activity parameter. The hard-core model shows some similarities to the Ising antiferromagnet in an external field and can be obtained from it by a limiting procedure (/3 --+ c~, h --+ 2d, fl(h - 2d) = const., (Dobrushin et al., 1985)). Since U is either 0 or c~, the inverse temperature/3 is irrelevant and will thus be fixed as 1, and we can vary only the parameter ~.. Finite volume Gibbs distributions can then be thought of as first letting all spins be independent, taking values 0 and 1 with respective 1 )~ and then conditioning on the event that no two l s sit probabilities 1-~ and 1--~, next to each other anywhere on the lattice. The phase transition behavior of the hard-core model on Z a, d > 2, is as follows. For ~. sufficiently close to 0, the particles are spread out rather sparsely on the lattice, and we get a unique Gibbs measure, just as in the Ising antiferromagnet at high temperatures. When ~. increases, the particle density also increases, and the system finally starts looking for optimal packings of particles. There are two such optimal packings, one where all sites in l~even a r e occupied and those in s are empty, and one vice versa. We denote these configurations by r]even and florid, respectively. (These chessboard configurations look similar to those favoured in the Ising antiferromagnet.) For sufficiently large ,k, the infinite volume construction of Gibbs measures with these two choices of boundary condition produces different Gibbs measures, so we get a phase transition (Dobrushin, 1968b). T h e o r e m 3.3 For the hard-core model on Z d, d >_ 2, there exist two constants 0 < )~c < Uc < cx~ (depending on d) such that f o r )~ < )~c the model has a unique Gibbs measure while f o r )~ > )~'c there are multiple Gibbs measures.
This result will be proved in Section 6.7. From a computer-assisted proof (Radulescu, 1997) we know that ~.c > 1.507 62. It is widely believed that one should be able to take ~.c - Uc in this result, which would mean that the occurrence of phase transition is increasing in ~. Such a result, however, would (unlike Theorems 3.1 and 3.2) not extend to arbitrary graph structures; some counterexamples were recently provided by Brightwell et al. (1999). The hard-core model analogue of introducing an external field in the Ising model on Z d is obtained by replacing the single activity parameter )~ by two different activities ~.even and )~odd, one for sites in s and the other for sites By analogy with the Ising model, one would expect to have a unique in s Gibbs measure as soon a s ~,even ~ ~.odd; this was conjectured by van den Berg and Steif (1994) and proved for the case d - 2 by H~iggstr6m (1997a).
20
3.5
H.-O. Georgii et aL
The Widom-Rowlinson lattice model
The Widom-Rowlinson model is another lattice gas model, where this time there are two types of particles, and two particles are allowed to sit on neighboring sites only if they are of the same type. Actually, Widom and Rowlinson (1970) originally introduced it as a continuum model of particles living in Rd; see Section 10.2 below. The lattice variant described here was first studied by Lebowitz and Gallavotti (1971). The state space is S = { - l, 0, + 1}, where - 1 and + 1 are the two particle types, and 0s correspond to empty sites. The pair interaction is given by zx3 if ~ ( x ) ~ ( y ) - - 1 , U(~(x), ~(y)) 0 otherwise, and the chemical potential by
-logT._ V (t7 (x)) =
0 log)~+
if o ( x ) - - 1 , if if
~ (x) - 0, ~(x)-+l.
Here k_, k+ > 0 are the activity parameters for the two particle types - 1 and + 1. As in the hard-core model, we fix the inverse tempreature/3 = 1 and only vary the activity parameters. Gibbs measures can then be thought of as first picking all spins independently, taking values - l, 0 or + 1 with probabilities proportional to X_, 1, and )~+, and then conditioning on the event that no two particles of different type sit next to each other in the lattice. We are mainly interested in the symmetric case 7._ -- )~+ -- k, where the phase transition behavior on Z d, d >_ 2 is similar to the Ising model. For )~ small, there is a unique Gibbs measure in which the overall density of plus particles is almost surely equal to that of the minus particles. For 7. sufficiently large, the system wants to pack the particles so densely that the -t-l symmetry is broken. As for the Ising model, one can construct two particular Gibbs measures # + and # - using boundary conditions 7/-- + 1 and r/-- - l ' f o r small )~ we get # + = # _ whereas for large 7. the two measures are different (and distinguishable through the densities of the two particle types), producing a phase transition. T h e o r e m 3.4 For the Widorn-Rowlinson model on Z d, d >_ 2, with activities )~_ - )~+ = 7, there exist 0 < )~c <_ )~c < ~ (depending on d) such that f o r )~ < )~c the model has a unique Gibbs measure while f o r 7. > )~tc there are multiple Gibbs measures. As in the hard-core model, we expect that one should be able to take ~.c - ~/c, but such a monotonicity is not known. Examples of graph structures where the desired monotonicity fails can be found in Brightwell et al. (1999). We furthermore expect that the asymmetric Widom-Rowlinson model on Z d with 7._ ~= )~+ always has a unique Gibbs measure (similarly to the Ising model with a non-zero external field), but this also is not rigorously known.
1 Random geometry of equilibrium phases
4
21
Coupling and stochastic domination
Geometry alone will not be sufficient for our analysis of equilibrium phases. We also need some probabilistic tools which allow us to compare different configurations and different probability measures. So we need to include another preparatory section describing these tools and their basic applications to our setting. Coupling is a probabilistic technique which has turned out to be immensely useful in virtually all areas of probability theory, and especially in its applications to statistical mechanics. The basic idea is to define two (or more) stochastic processes jointly on the same probability space so that they can be compared realizationwise. This direct comparison often leads to conclusions which would not be easily available by considering the processes separately. Although an independent coupling is sometimes quite useful (as we will see in Section 7.3, for example), it is usually more efficient to introduce a dependence which relates the two processes in an efficient way. One such particularly good relationship is that one process is pointwise smaller than the other in some partial order. This case is related to the central concept of stochastic domination, via Strassen's theorem (Theorem 4.6) below. We will confine ourselves to those parts of coupling theory that are needed for our applications; a more general account can be found in the monograph by Lindvall (1992).
4.1 The coupling inequality In this and the next subsection of general character, 12 will be an arbitrary finite or countably infinite set. As the notation indicates, we think of the standard case that 12 is the lattice introduced in Section 2.1, but the following results will also be applied to the case when 12 is replaced by its set B of bonds. We consider again the product space S2 = S z;, where for the moment S is an arbitrary measurable space. Suppose X and X' are random elements of f2, and let # and #I be their respective distributions. We define the (half) total variation distance lilt - #'11 between # and #I by
I1~- ~'11 =
sup I#(A) - #'(A)I
(4.1.1)
ACf2
where A ranges over all measurable subsets of f2. The coupling inequality below provides us with a convenient upper bound on this distance. To state it we first need to define what we mean by a coupling of X and X'. Definition 4.1 A coupling P of two Q-valued random variables X and X', or of their distributions tt and lz t, is a probability measure on f2 x f2 having marginals
22
H.-O. Georgii et al.
# and #~, in that for every event A C f2 P((~, ~ ' ) ' ~ E A) -- #(A)
(4.1.2)
P((~, ~') 9~' E A) = #t(A).
(4.1.3)
and
We think of a coupling as a redefinition of the random variables X and X' on a new common probability space such that their distributions are preserved. Sometimes it will be convenient to keep the underlying probability space implicit, but in general, as in (4.1.2) and (4.1.3), we make the canonical choice, which is the product space t2 • t2; X and X' are then simply the projections on the two coordinate spaces. With this in mind, we write P ( X E A) and P(X' E A) for the left-hand sides of (4.1.2) and (4.1.3), respectively. In the same spirit, P (X = X') is a shorthand for P((~, ~') : ~ = ~').
Proposition 4.2 (The coupling inequality) Let P be a coupling of two f2-valued random variables X and X ~, with distributions # and #'. Then
II~- ~'ll ~ P(X #
(4.1.4)
X').
Proof: For any A E ~2, we have #(A) - #'(A)
--
P(X E A)-
=
P(XEA,
P ( X ' E A)
X'r
tEA)
<_ P ( X E A , X' r <_ P (X ~: X'), whence (4.1.4) follows by symmetry. The next result states that there always exists some coupling which achieves equality in (4.1.4). We call such a coupling optimal.
Definition 4.3 A coupling P of two f2-valued random variables X and X', with distributions # and #~, is said to be an optimal coupling if
I1~- ~'11- P(X ~ x'). Proposition 4.4 For any two f2-valued random variables X and X t, there exists an optimal coupling of X and X'.
1
Random geometry of equilibrium phases
23
To construct an optimal coupling, one simply puts the common mass Ix/x Ix' of # and IX' on the diagonal of f2 • f2 and adds any measure with marginals IX- Ix/x #' and IX' - IX/x Ix'; the simplest choice of such a measure is the appropriately scaled product measure. For details we refer to Lindvall (1992), where such a coupling is called the ),-coupling of # and #'. This construction shows, in particular, that the optimal coupling is, in general, not unique. Applications of optimal couplings to interacting particle systems can be found in Liggett (1985) and Maes (1993), for example.
4.2
Stochastic domination
Suppose now that S is a closed subset of R, so that S is linearly ordered. The product space S2 is then equipped with a natural partial order • which is defined coordinatewise: For ~, ~' 6 f2, we write ~ 5 ~' (or ~' ~_ ~) if ~(x) < ~'(x) for every x 6 s A function f 9f2 ~ R is said to be increasing (or, non-decreasing) if f ( ~ ) < f ( ~ ' ) whenever ~ • ~'. An event A is said to be increasing if its indicator function Ia is increasing. The following standard definition of stochastic domination expresses the fact that Ix' prefers larger elements of f2 than #. Definition 4.5 Let # and IX' be two probability measures on if2. We say that IX is stochastically dominated by IX', or IX' is stochastically larger than IX, writing IX -
(i) # ___D #'. (ii) For all continuous bounded increasing functions f #'(f).
9 f2 --+ R, # ( f )
<
(iii) There exists a coupling P of IX and IX' such that P ( X ~ X') - 1. Sketch of proof: While the implications (i) =~ (ii) and (iii) ~ (i) are trivial, the assertion (ii) =~ (iii) is too deep to be explained here in detail. To start one should note that # and # ' may be considered as measures on the compact space [ - c r co] L;, and one can then follow the arguments outlined by Liggett (1985,
24
H.-O. Georgii et al.
pp. 72 ft.) or Lindvall (1992). Further discussion of Strassen's theorem can be found in Kamae et al. (1977) and Lindvall (1992). [] The equivalence (i) r
(ii) in Theorem 4.6 readily implies the following corollary.
Corollary 4.7 The relation -<7) of stochastic domination is preserved under weak limits. Next we recall a famous sufficient condition for stochastic domination. This condition is (essentially) due to Holley (1974) and refers to the finite-dimensional case when 1/21 < 0o. We also assume for simplicity that S C R is finite. Hence f2 is finite. In this case, a probability measure # on S2 is called irreducible if the set {rl 6 f2 : #(7) > 0} is connected in the sense that any element of ~ with positive /x-probability can be reached from any other via successive coordinate changes without passing through elements with zero/x-probability.
Theorem 4.8 (Holley) Let/2 be finite, and let S be a finite subset of R. Let # and lz ! be probability measures on f2. Assume that #! is irreducible and assigns positive probability to the maximal element of f2 (with respect to <_). Suppose further that l z ( X ( x ) >_ a l X - ~ offx) < / z ' ( X ( x ) > a l X - rl offx)
(4.2.1)
whenever x ~ /2, a ~ S, and ~, ~ ~ S s are such that ~ __%_r/, # ( X = ~ off x) > 0 and # ! ( X - ri off x) > 0. Then # ~D #!.
Proof: Consider a Markov chain (Xk)~=O with state space b2 and transition probabilities defined as follows. At each integer time k > 1, pick a random site x 6 / 2 according to the uniform distribution. Let Xk = Xk-1 on/2 \ {x}, and select Xk (x) according to the single-site conditional distribution prescribed by #. This is a socalled Gibbs sampler for #, and it is immediate that if the initial configuration X0 is chosen according to #, then Xk has distribution # for each k. Define a similar ! oO Markov chain (Xk)k= o with # replaced by #!. Next, define a coupling of (Xk)~_O and (X~)~_ 0 as follows. First pick the initial values (X0, X~) according to the product measure # x / z !. Then, for each k, pick a site x c /; at random and let Uk be an independent random variable, uniformly distributed on the interval [0, 1]. Let Xk(y) -- Xk-1 (y) and X'~(y) -X k-1 ! (Y) for each site y 7~ x, and update the values at site x by letting Xk(x) ----max{a ~ S" # ( X ( x ) > a l X -- ~ offx) > Uk} and ' _ alX' - ~ o f f x ) >_ U~} Xk(X) = max{a ~ S 9# ' ( X ' ( x ) >
1 Random geometry of equilibrium phases
25
!
where ~ -- Xk-1 (/2 \ {x}) and r/ = Xk_ 1(/2 \ {x}). It is clear that this cont co struction gives the correct marginal behaviors of (Xk)~=O and (Xk)k= o. The t t assumption (4.2.1) implies that Xk -< X k whenever Xk-1 -< Xk_ 1. By the t oo irreducibility of #i, the chain (Xk)k= 0 will almost surely hit the maximal state of f2 at some finite (random) time, and from then onwards we will thus have t t Xk <_ X k. Since the coupled chain (Xk, X k ) ~ O is a finite state aperiodic Markov chain, (Xk, X'k) has a limiting distribution as k --+ c~. Picking (X, X I) according to this limiting distribution gives a coupling of X and X' such that X <_ X I almost surely, whence # ___79#I by Theorem 4.6. [] A non-dynamical proof of Holley's inequality by induction on IEI, together with an extension to non-finite S, was given by Preston (1974c); the simplest induction proof of an even more general result can be found in Batty and Bollmann (1980). As a consequence of Holley's inequality we obtain the celebrated FortuinKasteleyn-Ginibre (FKG) inequality (Theorem 4.11 below) of Fortuin et al. (1971), who stated it under slightly different conditions. It concerns the correlation structure in a single probability measure rather than a comparison between two probability measures.
Definition 4.9 A probability measure # on f2 is called monotone if # ( X ( x ) >_ a l X - ~ off x) < # ( X ( x ) >_ a[X - rl off x) whenever x ~ 12, a c S, and ~, ~ c S s O and # ( X - rI off x) > 0 .
(4.2.2)
are such that ~ -< tl, # ( X - ~ off x) >
Intuitively, # is monotone if the spin at a site x prefers to take large values whenever its surrounding sites do.
Definition 4.10 A probability measure # on f2 is said to have positive correlations if f o r all bounded increasing functions f , g 9f2 --+ R we have # ( f g) > # ( f ) # ( g ) .
Since the preceding inequality is preserved under rescaling and addition of constants to f and g, # has positive correlations whenever # ___79 #f for any probability measure #f with bounded increasing Radon-Nikodym density relative to #. Theorem 4.6 thus shows that # has positive correlations whenever # ( f g ) >_ # ( f ) # ( g ) for all continuous bounded increasing functions f and g. Hence, the property of positive correlations is also preserved under weak limits.
Theorem 4.11 (The F K G inequality) Let 12 be finite, S a finite subset o f R, and # a probability measure on f2 which is irreducible and assigns positive probability to the maximal element o f ~ (relative to <_). I f # is monotone, it also has positive correlations.
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H.-O. Georgii et aL
Proof: Suppose #(r/)g(r/) for all a 6 Sand~ 6 lz(X(x) >_ alX -
#i is a second probability measure on f2 such that/d(7/) r/ 6 f2 and some positive increasing function g. For x 6 12, = Ss such t h a t / z ( X = ~ o f f x ) > 0 w e w r i t e q x ( a , ~ ) ~ off x) and define q~x(a, ~) similarly in terms of/z ~. Then
q~x(a, ~ ) / ( 1 - q~x(a, ~))
Z
--
lz(~x'S)g(~X'S)/s~
s>a
sea
qx(a,~)/(1 -qx(a,~)). Together with assumption (4.2.2), this implies that # and /z' satisfy (4.2.1). Theorem 4.8 thus implies that/z -<79/z I, and the corollary follows. [] Finally, we state a simple observation showing that, under the condition of stochastic domination, the equality of the single-site marginal distributions already implies the equality of the whole probability measures.
Proposition 4.12 Let s be finite or countable, and let lz and lZ' be two probability measures on f2 - R s satisfying lZ -<79 Izl. If in addition, # ( X ( x ) < r) = Izt (X (x) < r ) f o r all x ~ ~ and r ~ R then lZ = lz ~. Proof: Let P be a coupling of/z and/z t such that P (X _ X t) -- 1 which exists by Theorem 4.6. Writing Q for the set of rational numbers, we have for each x~/~
P ( X ( x ) # X'(x))
= <
P ( X ( x ) < X1(x)) ZP(X(x)
< r, X'(x) > r)
r6Q
--
r~o(P(X(x)<_r)-P(Xt(x)
Summing over all x 6 s we obtain P ( X by (4.1.4).
4.3
~= X I) -
0, whence /z =
/z' []
Applications to the Ising model
We will now apply the results of the previous subsection to the ferromagnetic Ising model. Let (s be any infinite locally finite graph. For definiteness, one
1 Random geometry of equilibrium phases
27
may think of the case/2 - zd; the arguments are, however, independent of the particular graph structure. As in Section 3.1, we write I~h,fl,A for the Gibbs distribution in a finite region A with boundary condition r/6 f2 and external field h 6 R at inverse temperature fl > 0. Our first application of Holley's theorem asserts that if one boundary condition dominates another, then we also have stochastic domination between the corresponding finite volume Gibbs distributions. L e m m a 4.13 If the boundary conditions ~, 77 ~ ~2 satisfy ~ -< ~, then q [J
, fi, A
-~ ~D [~ h , fi, A "
Also, each #~h,/~,A has positive correlations. The conditional probability of having a plus spin at a given site x given the configuration ~ everywhere else is equal to (1 + exp[-2fl(h + Y~y:y~x ~(y))])-l, which is an increasing function of ~. Theorems 4.8 and 4.1 1 thus imply stochastic domination between the projections of the Gibbs distributions to S A and the positive correlations property. As their behavior outside A is deterministic, the lemma follows. []
Proof:
We write #h,t~,A+ and #h,t~,A for the finite volume Gibbs distributions obtained with respective boundary conditions 17 = +1 and r/ ~ - 1 . Lemma 4.13 then shows that ' ___z) #h,t~,A + #h,~,A ___D#h,/~,A
( 4 . 3 1)
for any q 6 ~. This sandwich inequality reveals the special role played by the "all plus" and "all minus" boundary conditions. Next we establish the existence of the limiting "plus measure" discussed in Section 3.1. We will say that a measure # on f2 is homogeneous if it is invariant under all graph automorphisms of (/2, "~). In the case s -- Z d, a homogeneous measure is thus invariant under translations, lattice rotations, and reflections in the axes. We write A 1" s for the limit along an arbitrary increasing sequence of finite regions which exhaust the full graph/2. Proposition 4.14 The limiting probability measure + - A~s lim #h,~
+
exists. #h,~+ is a homogeneous Gibbs measure for the Ising model on (12, "~) with external field h and inverse temperature fl and has positive correlations.
28
H.-O. Georgii e t al.
P r o o f : By the general theory in Section 2.6, the limit is a Gibbs measure whenever it exists. Also, by Lemma 4.13 the limit must have positive correlations. To show the existence of the limit we note that + ~79 /Zh,t~,zX + wheneverA /Zh,/~,A
C A
(4.3.2)
This follows from Lemma 4.13 because ~h,/3,A + by con+ is obtained from ~h,fl,A ditioning on the increasing event that X _-- + 1 on A \ A . Now, for any finite A C A, if A increases, then, by (4.3.2), #h,~,A + (X -+1 on A) decreases and therefore converges to infA #h,~,A + (X _= + l o n A ) . Note that this limit is obviously invariant under any automorphism of (/2, ~). By inclusion-exclusion it follows that, for any local observable f , #h,C~,A+( f ) converges to an automorphism invariant limit as A "1" /2. These limits determine a unique homogeneous probability measure #h,r + which, as a weak limit of finite volume Gibbs distributions, is a Gibbs measure. The lemma is thus proved. [] Obviously, replacing the "all plus" boundary condition by the "all minus" boundary condition, we obtain in the same way an automorphism invariant infinite volume Gibbs measure #h--,t~with positive correlations. In the same way, Lemma 4.13 shows that any extremal Gibbs measure has positive correlations (since it is a weak limit of #~,,,A for suitable r/). However, positive correlations may fail for suitable (/2, -~) and some particular non-extremal Gibbs measures. For instance, when s = Z 3 and/~ is sufficiently large, one can take a convex combination of two different so-called Dobrushin states, see Dobrushin (1972) and van Beijeren (1975). We now take the limit in the sandwich inequality (4.3.1). Let # be any Gibbs measure for the Ising model with parameters h and/~. Taking the mean f lz(do) in (4.3.1), we obtain that ~h2fl,A -'<79# ~79 #h,fl,A'+ and since stochastic domination is preserved under weak limits, we end up with #h-~ ___7)# ___7)#h,~+
(4.3.3)
when # is any Ising-model Gibbs measure for/3, h. On the one hand, this shows that #h,~ and #h,~ + are extremal, and thus equilibrium phases in the sense of Section 2.7. On the other hand, we obtain an efficient criterion for the existence of a phase transition which was first observed by Lebowitz and Martin-L6f (1972) and Ruelle (1972). T h e o r e m 4.15 For the Ising model on an infinite locally finite graph (/~, ~) with external field h ~ R and inverse temperature 1~, the following statements are equivalent.
1 Random geometry of equilibrium phases
29
(i) There is a unique infinite volume Gibbs measure
(ii)
/Lh,/5
+
--
#h,fl +
(iii) /Zh,/~(X (x) -- + 1) =/Lh,/5 (X (x) -- + 1) f o r all x ~ s Proof: The implications (i) ==> (ii) =~ (iii) are immediate. (iii) =~ (ii) follows directly from (4.3.3) and Proposition 4.12, and (ii) =r (i) from (4.3.3). [3 Remarks: (a) In the case h - 0 of no external field, assertion (iii) is equivalent + ( X ( x ) = +1) = 1/2 for all x E 12, by the 4- symmetry of the model. An to #h,fl extension of Theorem 4.15 in this case to the q-state Potts model will be given in Theorem 6.10. (b) If the graph automorphisms act transitively on (s ~) then, by homogeneity, assertion (iii) is equivalent to having the equation only for some x ~ 12. Using for example the random-cluster methods of Section 6 one can also obtain the same equivalence in the general case, assuming only that 12 is connected. (c) If L; - Z d, (iii) is equivalent to the condition that the free energy density is differentiable with respect to h at the given values of h and t5 (Lebowitz and Martin-L6f, 1972). By the celebrated Lee-Yang circle theorem (Ruelle, 1971), this is the case whenever h #- 0. Alternatively, one can use the Griffiths-HurstSherman (GHS) inequality to establish (iii) for h # 0 (Preston, 1974b). Hence, for a non-zero external field, the Ising model on Z d does not exhibit a phase transition. We conclude this subsection comparing the Ising-model "plus" measures for different values of the parameters. (For similar results in a lattice gas setting see Lebowitz and Penrose, 1977). Proposition 4.16 Consider the Ising model on an arbitrary graph (/2, ~) at two inverse temperatures/51,152 and two external fields hi, h2. Suppose that either [31 = ~2 and hi <_ h2, or (/2, ~ ) is of bounded degree N = SUPx~s Nx and /52h2 >/51hl + N]fll -/52]. Then #h+l,fll "<'D #h+2,f12 9 r/
Proof: The stated conditions imply that #~hl,~1 ,{x}(+ 1) _< ].Lh2,fl2,{x } (-3I- 1) whenever ~ __ rl. Hence, by Theorem 4.8, #+~,~,A ___z~ #+2,~,A for all A, and the proposition follows by letting A 1" s [] If 12 has bounded degree N, we obtain in particular a comparison of Ising and Bernoulli measures. Let ~p denote the Bernoulli measure on {-1, 1}s with
30
H.-O. Georgii et aL
density p. Then #h,~ + --+ 7tp as fl --+ 0 and flh -+ 89log l@p- The preceding proposition and Corollary 4.7 thus show that +
>_
~
1
p
#h,fl _7) lp'p for h > 2fl log 1 - p and + #h,r
4.4
1
_
+N
p
___z~7*p for h -< ~2fl log ~1 - - p
No
Application to other models
Do the arguments of Section 4.3 extend to the other models of Section 3? The answer to this question is different for the different models. For the Potts model with q >_ 3, Lemma 4.13 fails because the conditional probability that the spin at a site x takes a large value is not increasing in the surrounding spin configuration. Nevertheless, the Potts model admits some analogues of Proposition 4.14 and Theorem 4.15. These results are deeper than their Ising counterparts, and will be demonstrated by random-cluster arguments in Section 6. The Widom-Rowlinson model exhibits the same monotonicity properties as the Ising model. We thus obtain Widom-Rowlinson analogues of Lemma 4.13 and Proposition 4.14. Fixing any two activity parameters ,k+, ~._ > 0 and writing #+ (resp. # - ) for the associated limiting Gibbs measures with "all plus" (resp. "all minus") boundary conditions, we find that any other Gibbs measure for the same parameters is sandwiched (in the sense of (4.3.3)) between # + and # - The analogue of Theorem 4.15 reads as follows. Theorem 4.17 For the Widom-Rowlinson model on an infinite locally finite graph (s ~ ) with activity parameters ~.+, )~_ > O, the following statements are equivalent. (i) There is a unique infinite volume Gibbs measure (ii)
# + --
#-
(iii) / , + ( X ( x ) -- +1) = # - ( X ( x )
-- + l ) f o r a l l x
~ E.
The Ising antiferromagnet and the hard-core lattice gas model are far from satisfying the monotonicity properties needed for the arguments in Section 4.3. The conditional probability that a site x takes the value + 1 is decreasing, rather than increasing, in the surrounding configuration. However, when (/2, ~) is bipartite (as in the case s - Z d) we can use again the trick of (3.2.1) to flip all spins on the odd sublattice. The Ising antiferromagnet is then mapped onto the Ising ferromagnet with a staggered external field (having alternating signs on the even and the
1 Random geometry of equilibrium phases
31
odd sublattices). Similarly, the hard-core model is mapped into a model which also exhibits the necessary monotonicity properties. We thus obtain analogous results which we spell out only for the hard-core model with activity ,k > 0. There 9 e v e n and #~dd obtained as infinite volume exist two particular Gibbs measures ~z limits of finite volume Gibbs distributions with respective boundary conditions rleven and rlodd, defined in Section 3.4. In terms of these two Gibbs measures, the existence of a phase transition can be characterized as follows. Theorem 4.18 For the hard-core model on an infinite locally finite bipartite graph (~, ~ ) with activity parameter )~, the following are equivalent.
(i) There is a unique infinite volume Gibbs measure .even
(ii) tz z
=
lzodd
.even (iii) t~ x (X(x) = 1)-
5
~odd( X ( x )
- 1) f o r all x E s
Percolation
We will now introduce the ideas of random geometry we referred to in the title of this review. As these were developed first in the framework of percolation theory, we devote this section to a description of this subject. We will start with the classical case of independent, or Bernoulli, percolation, and will then proceed to the case of dependent percolation. In the subsequent sections we will see how these results and ideas can be used for the geometric analysis of equilibrium phases.
5.1
Bernoulli percolation
Bernoulli percolation was introduced in the 1950s in papers by Broadbent and Hammersley (Broadbent and Hammersley, 1957; Hammersley, 1957a, b) as a model for the passage of a fluid through a porous medium. In fact, the model has appeared first in the context of polymerization (Flory, 1941). We give here only a brief introduction; a thorough treatment can be found in the books and lectures by Grimmett (1997, 1999), Chayes and Chayes (1986), Kesten (1982) and Menshikov et al. (1986). The porous medium is modelled by a graph (s ~), and either the sites or the bonds of this graph are considered to be randomly open or closed (blocked). We begin with the case of site percolation. The alternative case of random bonds will be discussed at the end of this subsection. The basic question of percolation theory is: how can a fluid spread through the medium. This involves the connectivity properties of the set of open vertices.
32
H.-O. Georgii e t al.
To describe this we introduce some terminology. A finite path is a sequence ( V l , e l , v2, e2 . . . . . e k - 1 , Vk), where Vl . . . . . vk ~ s are pairwise distinct vertices and el . . . . . ek-1 ~ 13 are pairwise distinct edges such that, for each i c { 1 . . . . . k - 1}, the edge ei connects the vertices vi and Vi+l. Obviously, a path is equivalently described by its sequence (vl, v2 . . . . . Vk) of vertices or its sequence (el, e2 . . . . . ek-1) of edges. The number k is called the length of the path. In the same way, we can also speak of infinite paths (Vl, el, v2, e2 . . . . ) and doubly infinite paths ( . . . . v - l , e - l , v0, e0, vl . . . . ). A region C C s is called connected if for any x, y c C there exists a path which starts at x, ends at y, and which only contains vertices in C. An open path is a path on which all vertices are open. An open cluster is a maximal connected set C in which all vertices are open; here maximal means that there is no larger region C' D C which is connected and only contains open vertices. An infinite open cluster (or infinite cluster, for short) is an open cluster containing infinitely many vertices. Using these terms, we may say that the existence of an infinite open cluster is equivalent to the fact that a fluid can wet a macroscopic part of the medium. We now turn to the classical case of Bernoulli site percolation with retention parameter p 6 [0, 1]. In this case, each vertex of s independently o f all others, is declared to be open (and represented by the value 1) with probability p and closed (with value 0) with probability 1 - p. We write 7tp for the associated (Bernoulli) probability measure on the configuration space {0, 1 }s The first question to be asked is whether or not infinite clusters can exist. This depends, of course, on both the graph (s ~ ) and the parameter p. The basic observation is the following. Proposition 5.1 For Bernoulli site percolation on an infinite locally finite graph (s ~), there exists a critical value Pc c [0, 1] such that lpp (3 an infinite open cluster) = / 0 [ 1
if p < Pc ifp>pc.
At the critical value p - Pc, the ~pp-probability o f having an infinite open cluster is either 0 or 1. Proof: A moment's thought reveals that the existence of infinite clusters is invariant under a change of the status of finitely many vertices. By Kolmogorov's zero-one law, the 7tp-probability of having infinite clusters is therefore either 0 or 1. It remains to show that this probability is increasing in p. The existence of an infinite open cluster is obviously an increasing event, so we have proved our case if we can show that ~ppj ___~ 7tp2 whenever pl < p2.
(5.1.1)
1 Random geometry of equilibrium phases
33
This is intuitively obvious and can be proved by the following elementary coupling argument. Let Y = ( Y ( x ) ) x ~ s be a family of i.i.d. (independent identically distributed) random variables with uniform distribution on [0, 1], and for p e [0, 1] let Xp = ( X p ( x ) ) x c s be defined by X p ( x ) - I{y(x)<__p}. It is then clear that X p has distribution ~p and X p~ -< X p2 whenever Pl < P2. This implies (5.1.1) by (the trivial part of) Theorem 4.6. (Note that we have in fact constructed a simultaneous coupling of all ~pS, and that this construction would be used in Monte Carlo simulations of Bernoulli percolation.) [] We write {x +-~ ec} for the event that x e s belongs to an infinite cluster, and set Ox(p) - ~pp(X ~ c~). For homogenous graphs such as s - Z d, Ox(p) does not depend on x, and then we write simply O(p). Equation (5.1.1) shows that Ox (p) is increasing in p. We also make the following observation. P r o p o s i t i o n S . 2 For any infinite locally finite connected graph ( s any x ~ s and any p c (0,1), we have Ox(p) > 0 if and only if ~pp (3 an infinite open cluster) - 1. From Proposition 5.1 we know that an infinite cluster exists with probability 0 or 1. The implication "only if" is therefore immediate. For the "if" part, we note that if an infinite cluster exists with positive probability, then there is some N such that ~p(AN) > 0, where AN is the event that some vertex within distance N from x belongs to an infinite cluster. On the other hand we have ~p(BN) > O, where BN is the event that all vertices within distance N - 1 from x are open. The event AN f-) BN implies that x belongs to an infinite cluster. But AN and BN a r e increasing events (see Section 4.2), so that we can apply Theorem 4.11 to obtain Ox(p) >_ ~ p ( a N f"l BN) >_ ~ p ( a N ) ~ p ( B N ) > O. [] Proof:
Note that the proof above applies to the much broader class of all measures with positive correlations (recall Definition 4.10), rather than only the Bernoulli measures. Next we ask whether both possibilities in Proposition 5.2 really occur, that is, if 0 < Pc < 1. For, only in this case we really have a non-trivial critical phenomenon at Pc. The answer depends on the graph. For/2 - Z d with dimension d >_ 2, the threshold Pc is indeed non-trivial, as is stated in the theorem below. This non-triviality of Pc is a fundamental ingredient of many of the stochasticgeometric arguments employed later on. On the other hand, it is easy to see that pc - 1 for/2 - Z1. 5.3 The critical value Pc d > 2, satisfies the inequalities
p c ( d ) f o r site percolation on s -
Theorem
1
2d-
1
6 < Pc <_ -~.
-
Z d,
(5.1.2)
34
H.-O. Georgii et al.
Proof: We begin with the lower bound on pc. For k = 1, 2 . . . . . we write Nk for the random number of open paths of length k starting at 0. On the event {0 +-> c~} we have Nk _ 1 for each k, whence
O(p) < ~pp(Nk).
(5.1.3)
The number of all paths of length k starting at 0 is at most 2d(2d - 1)k-l, and each path is open with probability pk. Hence
7zp(Nk) < 2 d ( 2 d - 1)k-lp k which tends to 0 as k --+ cxz whenever p < 1 / ( 2 d - 1). In combination with (5.1.3) this implies that O(p) = 0 for p < 1/(2d - 1), and the first half of (5.1.2) is established. The second half of (5.1.2) only needs to be proved for d = 2; this is because Z 2 c a n be embedded into Z d for any d >_ 2, so that pc(d) <_ pc(2). So let d - 2. We first need some additional terminology. A ,-path in Z 2 is a sequence (Vl, v2 . . . . . vk) of distinct vertices such that doo(vj, vj+l) = 1 for j = 1 . . . . . k - 1. Note that two consecutive vertices in a ,-path need not be nearest neighbors; they may also be "diagonal neighbors". A ,-circuit is a sequence (l)1, V2 . . . . . t)k, Vl) such that (U1, l)2 . . . . . Vk) is a ,-path and d~(vk, U1) --" 1. Informally, a ,-circuit is a ,-path which ends where it starts. ,-circuits with the same set of sites are identified. A closed ,-circuit is a ,-circuit in which all vertices are closed. Now let M be the number of closed ,-circuits that surround the origin 0. As Z 2 is a planar graph, the "outer boundary" of a finite open cluster containing 0 defines a closed ,-circuit around 0. Hence, the event {0 +-> co} occurs if and only i f M = 0. The number of all (not necessarily closed) ,-circuits of a given length k surrounding 0 allows the following crude estimate. Consider the leftmost crossing of the x-axis of such a circuit; the location of such a crossing is at distance at most k from the origin, so there are at most k such locations to choose from. Starting at this location, we may trace the ,-circuit clockwise (say), and at each step we have at most 7 d~-neighbors to choose from. Hence, the number of ,-circuits of length k around 0 is at most k 7 k-1 . Each one is closed with probability (1 - p)k, so oo
~rp(m) < Z k 7 k - 1
(1 -
p)k.
k---1
The last sum is finite for p > 6/7. Hence, for such a p and n large enough, there is a positive probability for having no closed ,-circuit around 0 which contains a site of distance at least n from 0. By the argument in the proof of Proposition 5.2,
1 Random geometry of equilibrium phases
35
it follows that 7tp (M = 0) > 0 for such p. Hence 0 (p) > 0 for p > 6/7, and the second half of (5.1.2) follows. [] The exact value of the percolation threshold pc (d) of Z d is not known for any d > 2. The best rigorous bounds for d = 2 are presently 0.556 < pc(2) < 0.680
(5.1.4)
where the first inequality is due to van den Berg and Ermakov (1996) (inspired by Menshikov and Pelikh (1989)) and the second to Wierman (1995). For high dimensions, it is known that lim 2d pc(d) = 1,
(5.1.5)
d--+~
see Kesten (1990), Gordon (1991) and Hara and Slade (1995) for this and finer asymptotics. On some particular graphs, pc can be determined exactly. For example, for s - Td, the regular (Cayley or Bethe) tree, branching-process arguments immediately show that pc(Td) -- 1/d, and for the triangular lattice it follows from planar duality that Pc - 1/2 (Kesten, 1982). The preceding considerations do not tell us what happens at the critical value Pc. It is believed that, for the integer lattice Z d of any dimension d >_ 2, there is ~pc-a.s. no infinite cluster, which means that O(pc) - 0; so far this is only known for d -- 2 and d > 19, see Russo (1981) and Hara and Slade (1994). One can show that the relation O(pc) - 0 implies continuity of O(p) at p - pc, so in combination with the trivial continuity of O(p) in the subcritical regime and the following more interesting result (which can be found e.g. in Grimmett (1999)) we obtain continuity of 0 (p) throughout [0, 1] as soon as absence of infinite clusters at criticality is established. T h e o r e m 5.4 For Bernoulli site percolation on Z d, d >_ 2, the function O(p) is continuous throughout the supercritical regime (Pc, 1]. So far we have been interested in the existence of infinite clusters. In the subcritical regime p < pc when no infinite cluster exists, one may ask for the size of a typical cluster. Let ICol be the random number of vertices in the open cluster containing the origin; we set ICol - 0 if the origin is closed. By the definition of pc, ~Pp(IC0l - c~) - 0. For/2 = Z d, we even have the stronger statement that the expected value of IC01 is finite.
Theorem 5.5 For Bernoulli site percolation on Z d with retention parameter p < Pc, we have 7tp(IC01) < ~ . This was proved independently by Menshikov (1986) and by Aizenman and Barsky (1987). The proofs are rather involved, so we refer the reader to the
36
H.-O. Georgii e t al.
original articles and Grimmett (1999). It is worth noting that Theorem 5.5 fails in the setting of general graphs; a striking counterexample is the "three-one-tree" discussed by Lyons (1990, p. 936). Menshikov even showed that the distribution of the radius of the open cluster containing the origin decays exponentially. The following even stronger result states that the same is true for the distribution of IC01; see Grimmett (1999) for a proof (in the case of bond percolation).
Theorem 5.6 For Bernoulli site percolation on Z d with retention p a r a m e t e r p < Pc, there exists a constant c (depending on p) such that
~p(]C01 ~ n) < e -cn for all n.
Looking at a fixed path of length n starting at 0, we immediately obtain the lower bound ~Pp(IC01 > n) >__pn. So the preceding upper bound is best possible, except that the optimal constant c is unknown. We conclude this section with some remarks on Bernoulli bond percolation. The model is similar, except that now the edges rather than the vertices in (s ~) are independently open (described by the value 1) or closed (with value 0) with respective probabilities p and 1 - p. The associated configuration space is thus {0, 1}t3. We write 4~p for the associated (Bernoulli) probability measure on {0, 1}/3. In the present context of bond percolation, an open path is a path in which all edges are open, and an open cluster is a maximal region C c s which is connected, in that for any x, y E C there is an open path in C from x to y. All results for site percolation discussed so far extend to the bond percolation set-up. This is no surprise because bond percolation is equivalent to site percolation on the so-called covering graph for which 13 is taken as a set of vertices, and edges are drawn between any two coincident elements of/3. In particular, there exists again a critical value pc for the occurrence of infinite open clusters, and Propositions 5.1 and 5.2, Theorems 5.3-5.6 and the asymptotic formula (5.1.5) are still true in the case of bond percolation. What are generally different, are the critical values for site and bond Bernoulli percolation on a given graph. One remarkable case is that of bond percolation on Z 2, where (again by planar duality) pc -- 1/2; this is a famous result by Kesten (1980). In the specific case of trees, however, site and bond percolation are equivalent. In particular, for C = Ta we have Pc = 1 / d for both site and bond percolation; for more general trees a formula for pc has been given by Lyons (1990).
1
5.2
Random geometry of equilibrium phases
37
Dependent percolation: the role of the density
Our main subject is the analysis of equilibrium phases by means of percolation methods. In this case, a site will be considered as open if, for example, the configuration in a neighborhood of this site shows a specified pattern, and the events "site x is open" with x 6 12 are then far from being independent. This leads us to considering the case of dependent percolation. In this subsection we take some first steps in this direction. Our starting point is the following question. In the case of Bernoulli percolation, there is a unique parameter, the occupation probability or density p, which governs the phase diagram and allows us to distinguish between subcritical ("no infinite cluster") and supercritical ("at least one infinite cluster") behavior. Does this also hold in general? Specifically, is it true that for any translation invariant probability measure/z on {0, 1}Zd, the occurrence of an infinite cluster only depends on the density p(/z) = / z ( X ( x ) = 1) of open sites x c Zd? In general, the answer is obviously "no", as we will now show by two simple examples. There exist translation invariant measures/z on {0, 1}Zd with arbitrarily small densities such that infinite clusters exist almost surely, and also translation invariant measures with densities arbitrarily close to 1 for which no infinite clusters exist with probability 1. Example 5.7 For q c (0, 1), let (Y(x), x 6 Z) be i.i.d, random variables taking values 0 and 1 with probability 1 - q resp. q. We define a translation invariant random field ( X ( x ) , x 6 Zd), d > 2, by setting X ( x ) = X ( x l . . . . . xa) = Y(xl) for each x c Z a. Writing # for the distribution of (X(x), x c Za), we have p(/z) = q, but with/z-probability 1 there exist infinitely many infinite clusters, even if q is arbitrarily small. Example 5.8 Again let q 6 (0, 1), d > 2, and (Y(x, i), x 6 Z, i 6 {1 . . . . . d}) be i.i.d, random variables taking values 0 and 1 with probabilities 1 - q and q. We define a translation invariant random field (X (x), x 6 Z d) by setting d
X ( x ) -- X(Xl . . . . , X d ) -
1--I Y ( x i , i ) . i=1
A moment's thought reveals that #-a.s. there exist no infinite clusters, despite the fact that p ( # ) = qd may be arbitrarily close to 1. These examples suggest looking for additional assumptions under which high (resp. low) density guarantees existence (resp. nonexistence) of infinite clusters. Positive correlations (in the sense of Definition 4.10) do not suffice, because both examples above obviously have positive correlations.
38
H.-O.
Georgii e t
aL
An alternative might be to assume R-independence in the sense that X(A) and X (A) are independent for any two finite regions A, A C s such that min
xEA,yEA
Ix
-
Yl > R
for some given R. For Z d, d >_ 2, this gives non-trivial thresholds 0 < pl < p2 < 1 (depending on R) such that existence (resp. non-existence) of infinite clusters is guaranteed as long as p ( # ) > P2 (resp. p ( # ) < Pl); see e.g. Liggett et al. (1997). However, R-independence rarely holds in Gibbs models. For instance, for plus phase of the Ising model with vanishing external field and inverse temperature 15 > 0, the spins at any two vertices are always strictly positively correlated no matter how far apart they are (although the correlation does tend to 0 in the distance). However, in contrast to what we just saw in the case of the cubic lattices, the density does play a significant role for the regular trees Td. To show this we consider a probability measure # on {0, 1}s where now L; = Td with d >_ 2. The natural analogue of translation invariance in this setting is automorphism invariance of/z, which means that # inherits all the symmetries of Td. In particular, this implies that # ( X (x) = 1) is independent of x, so that the density p ( # ) is well defined. As opposed to the Z d case, having p ( # ) sufficiently close to 1 now does guarantee that an infinite cluster exists with positive probability. This is also true in the bond percolation case, where p(/z) is defined as the probability that a given edge is open. The following result is due to H~iggstr6m (1997b). T h e o r e m 5.9 For any automorphism invariant site percolation model # on Td with density p ( # ) > d2+----!1,we have # ( 3 an infinite open cluster) > 0. The same
2 is true f o r bond percolation on Td with density p ( # ) > d+l"
These bounds are in fact sharp, in that for any p < d2+--!lthere exists some automorphism invariant probability measure on {0, 1}Ta with density p, which does not allow an infinite cluster with probability 1, and similarly for the case of bond percolation; see H~iggstr6m (1997b). It follows from Example 5.8 that the corresponding threshold for Z d is trivial: only density 1 is enough to rule out the non-existence of infinite clusters. The intuitive reason is the following. On Z d, one can find finite regions A C /~ with arbitrarily small surface-to-volume ratio, which means that a vast majority of sites is not adjacent to a vertex outside A; we can simply take A = An = [ - n , n] d N Z d with large n; this property of Z d is known as amenability. Hence, a relatively small number of closed vertices may easily "surround" a large number of open sites. In contrast, every region in rid has a surface of the same order of magnitude as its volume; this makes it impossible for a small minority of closed vertices to surround a large number of open vertices. This intuition can be turned into a proof using the so-called masstransport method sketched below. Benjamini et al. (1999) have recently extended
1 Random geometry of equilibrium phases
39
this method to derive a similar dichotomy for a large class of graphs, including Cayley graphs of finitely generated groups. Sketch proof of Theorem 5.9: For simplicity we confine ourselves to the case of bond percolation on T2. We want to show that if p(/z) > 2/3, then an infinite cluster exists with positive /z-probability. Imagine the following allocation of mass to the edges of T2. Originally every edge receives mass 1. Then the mass is redistributed, or transported, as follows. If an edge e is open and is contained in a finite open cluster, then it distributes all its mass equally among those closed edges that are adjacent to the open cluster containing e. If e is open and contained in an infinite open cluster, then it keeps its mass. Closed edges, finally, keep their own mass and happily accept any mass that open edges decide to send them. The expected mass at each edge before transport is obviously 1, and one can show B this is an instance of the mass-transport principle (Benjamini et al., 1999) B that the expected mass at a given edge is also 1 after the transport. Suppose now, for contradiction, that p(/z) > 2/3 and that all open clusters are finite/z-a.s. Then all open edges have mass 0 after transport. Furthermore, since each open cluster containing exactly n edges has exactly n + 3 adjacent closed edges (as is easily shown by induction ~ it is here that the tree structure is used), the mass after transport at a closed edge adjacent to two open clusters of sizes n l and n2 has mass nl n2 1-~ ~ <3 n1+3 n2+3 Hence the expected mass after transport at a given edge e is strictly less than 3/z(X(e) = 0) = 3(1 - p(/z)) < 1, contradicting the mass-transport principle.
5.3 Examples of dependent percolation From the previous subsection the reader might get a rather pessimistic view of the possibilities of establishing existence (or non-existence) of infinite clusters for dependent percolation models on Z d. This is certainly not the case, and a lot can be done. One standard way of determining the percolation behavior of a dependent model is by stochastic comparison with a suitable Bernoulli percolation model. For the existence of infinite clusters, it is sufficient to show that the given dependent model is stochastically larger than the Bernoulli model for some parameter p > Pc, and the absence of infinite clusters will follow if the model at hand is stochastically dominated by the Bernoulli model for some p < Pc. Let us demonstrate this technique for the Ising model on Z d.
40
H.-O. Georgii et al.
Consider percolation of plus spins in the plus measure #h,t~' + defined in Section 4.3. If we keep I3 fixed then Proposition 4.16 tells us that #h,~+ is stochastically increasing in h. Consequently, both the probability of having an infinite cluster of plus spins, as well as the probability that a given vertex is in such an infinite cluster, are increasing in h. Furthermore, as Z d is of bounded degree N -2d, the remarks after the same proposition imply that, for any given p e (0, 1) + stochastically dominates the Bernoulli measure and fl, the Ising m e a s u r e ~h,fl lpp when h is large enough, and is dominated by Op for h below some bound. We may combine this observation with Proposition 5.1 to deduce the following critical phenomenon: Theorem 5.10 For the Ising model on Z d, d > 2, at a fixed temperature fi, there exists a critical value hc ~ R (depending on d and fl) f o r the external field, such that
#h,r + (3 an infinite cluster of plus spins) -- [ 01 ifif hh >< hc.hC As we shall see later in Theorem 8.2, we have hc = 0 when d > 2 and fl > tic. Higuchi (1993b) has shown that the percolation transition at hc is sharp, in that the connectivity function decays exponentially when h < hc, and that the percolation probability is continuous in (fl, h), except on the critical half-line h = O, fl > tic. In Section 6 below, we will make a similar use of stochastic comparison arguments for random-cluster measures, cf. Proposition 6.11. The stochastic domination approach works also in the framework of lattice gases with attractive potential; see Lebowitz and Penrose (1977). In the rest of this subsection we shall give some examples of strongly dependent systems where other approaches to the question of percolation are needed. Typically in these examples, the probability that all vertices in a finite region A are open (or closed) fails to decay exponentially in the volume of A, and as a consequence, the random field neither dominates nor is stochastically dominated by any non-trivial Bernoulli model. The geometry of level heights of a random field forms an important object of study both from the theoretical and the applied side. For example, it relates to the presence of hills and valleys on a rough surface, or to the random location of potential barriers in a doped semiconductor. To fix the ideas we consider a random field X - ( X ( x ) , x e Z d) with values X ( x ) e S C R which are not necessarily discrete. It is often interesting to divide S into two parts $1 and So and to define a new discrete random field Y via Y ( x ) - I{x(x)eS~}. For S - R one typically considers S1 = [g, cx~) for some level g e R. In this way we obtain a coarse-grained description of a system of continuous spins. One question is to which extent one can reconstruct the complete image from this information. We consider here a different question: what is the geometry of the random set {x ~ Z d 9Y ( x ) - 1}? This set is called the excursion or exceedance set when it
1
Random geometry of equilibrium phases
41
corresponds, as in the example above, to the set on which the original random field exceeds a given level. For a recent review of this subject we refer to Adler (2000). We now give four examples of equilibrium systems with continuous spins where one can show (the absence of) percolation of an excursion (exceedance) set. Here we only state the results. Some hints on the proofs will be given later in Section 8 via Theorem 8.1. Details can be found in the paper by Bricmont et al. (1987). E x a m p l e 5.11 Consider a general model of real-valued spins ( a ( x ) , x c Z d) with ferromagnetic nearest-neighbor interaction. The formal Hamiltonian is given by (5.3.1) H ( a ) -- - Z a ( x ) a ( y ) . x,~y
The reference (or single-spin) measure ~, ~: 30 on R is assumed to be even and to decay fast enough at q - ~ so that the model is well defined. Then, for any Gibbs measure # relative to (5.3.1) with # ( s g n ( a ( 0 ) ) ) > 0, there will be percolation of all sites x with a (x) >_ 0. Such Gibbs measures always exist at sufficiently low temperatures when d > 2. E x a m p l e 5.12 Consider again a spin system ( a ( x ) , x Hamiltonian has the 'massless' form H(a) - - ~
c z d ) , where now the
7r(a(x) - a ( y ) )
x,~y
with a (x) 6 R or Z and ~ an even convex function. The single-spin measure ~ is either a Lebesgue measure on R or a counting measure on Z. The case a (x) 6 Z and ~ ( t ) - It[ corresponds to the so-called solid-on-solid (SOS) model of a ddimensional surface in z d + l ; the choice a ( x ) 6 R and 7r(t) - t 2 gives the harmonic crystal. Let # be a Gibbs measure which is obtained as infinite volume limit of finite volume Gibbs distributions with zero boundary condition. (In the continuous-spin case, such Gibbs measures exist for any temperature when d > 3 and ~ ( t ) - ottz+4~(t), where ot > 0 and ~b is convex (Brascamp and Lieb, 1976).) Then, for any s < 0, there is percolation of the sites x 6 Z d with a (x) _> s E x a m p l e 5.13 Consider next a model of two-component spins a (x) 6 R 2, x E Z d, a (x) - (rx cos dpx, rx sin ~bx), with formal Hamiltonian H ( a ) -- - ~
a(x). a(y)
x~,y
and some rotation-invariant and suitably decaying reference measure )~ on R 2. Then, for any Gibbs measure # with #(cos 4~0) > 0, there is percolation of the sites x c Z d with cos qSx > 0. Such Gibbs measures exist at low temperatures if d>3.
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Example 5.14 Consider again the massless harmonic crystal of Example 5.12 above (with 7z(t) - t 2) in d -- 3 dimensions. There exists a value gc < cx~ so that for all g > gc there is no percolation of sites x 6 Z d with a (x) _ ~. Finally, we give an example of a strongly correlated system, sharing some properties with the harmonic crystal of Example 5.14, where at present there is no proof of a percolation transition. The model is one of the simplest examples of an interacting particle system. What makes the problem difficult is that the random field is not Markov (not even Gibbsian) and not explicitly described in terms of a family of local conditional distributions. Example 5.15 The voter model is a stochastic dynamics in which individuals (voters) sitting at the vertices of a graph update their position (yes/no) by randomly selecting a neighboring vertex and adopting its position, see Liggett (1985) for an introduction. Using spin language and putting ourselves on Z 3, the time evolution of this voter model is specified by giving the rate c(x, ~) for a spin flip at the site x when the spin configuration is cr 6 {+1, - 1 }z3, 1
Z ('y~x
There is a one-parameter family of extremal invariant measures #p each obtained asymptotically (in time) from taking the Bernoulli measure ~p with density p as the initial condition. These stationary states #p are strongly correlating. The spin-spin correlations decay as the inverse 1/r of the spin-distance r on Z 3. It is an open question whether for p sufficiently close to 1 the plus spins percolate, and whether for sufficiently small p there is no percolation. Simulations by Lebowitz and Saleur (1986) indicate that there is indeed a non-trival percolation transition with critical value Pc ~ O. 16. The same problem may be considered for d _> 4. For d - 1, 2, however, the problem is not interesting because in these cases #p is known to put mass p on the "all + 1" configuration and mass 1 - p on the "all - 1 " configuration. Alternatively, one may consider the same model with Z 3 replaced by Td" Theorem 5.9 can then be applied to show that the plus spins do percolate for p > d+l 2d" Other interesting examples of dependent percolation with remarkable properties can be found in Molchanov and Stepanov (1983). -
5.4
-
The number of infinite clusters
Once infinite clusters have been shown to exist with positive probability in some percolation model, the next natural question is: How many infinite clusters can exist simultaneously ? For Bernoulli site or bond percolation on Z d, Aizenman et
1 Random geometry of equilibrium phases
43
al. (1987) obtained the following, now classical, uniqueness result: with probability 1, there exists at most one infinite cluster. Simpler proofs were found later by Gandolfi et al. (1988) and by Burton and Keane (1989). The argument of Burton and Keane is not only the shortest (and, arguably, the most elegant) so far. It also requires much weaker assumptions on the percolation model, namely: translation invariance and the finite-energy condition below, which is a strong way of stating that all local configurations are really possible. Its significance for percolation theory had been discovered before by Newman and Schulman (1981).
Definition 5.16 A probability measure lz on {0, 1}s with s a countable set, is said to have finite energy if f o r every finite region A C ~, /z(X~7?onAlX~offA)>0 for all 7? ~ {0, 1}A and #-a.e. ~ ~ {0, 1}AC.
Theorem 5.17 (The Burton-Keane uniqueness theorem) Let # be a probability measure on {0, 1}zd which is translation invariant and has finite energy. Then, lz-a.s., there exists at most one infinite open cluster.
Sketch proof: Without loss of generality we can assume that # is ergodic with respect to translations. For, one can easily show that the measures in the ergodic decomposition of # admit the same conditional probabilities, and thus inherit the finite-energy property. Since the number N of infinite clusters is obviously invariant under translations, it then follows that N is almost surely equal to some constant k c {0, 1. . . . . ~ } . In fact, k ~ {0, 1, zx~}. Otherwise, with positive probability each of the k clusters would meet a sufficiently large cube A; by the finite-energy property, this would imply that with positive probability all these clusters are connected within A, so that in fact k = 1, in contradiction to the hypothesis. (This part of the argument goes back to Newman and Schulman (1981).) We thus only need to exclude the case k -- ~ . In this case, # ( N > 3) = 1, and the finite-energy property implies again that # ( A x ) = 3 > 0, where Ax is the event that x is a triple point, in that there exist three disjoint infinite open paths with starting point x and that these paths would fall in three different components if the vertex x were removed. By the (norm-) ergodic theorem, for any sufficiently large cubic box A we have ~(IA]-I Z
lAx >_ 3 / 2 ) >
1/2.
(5.4.1)
xEA
On the other hand, for geometrical reasons (which are intuitively obvious but need some work when made precise), there cannot be more triple points in A than points in the boundary 0 A of A. Indeed, each of the three paths leaving a triple
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H.-O. Georgii et aL
point meets 0 A, which gives three boundary points associated to each triple point in A. If one identifies these boundary points successively for one triple point after the other one sees that, at each step, at least one of the boundary points must be different from those obtained before. Hence,
IAI-~ ~ IAx ~ IAI-]IOAI < 3/2 xEA
when A is large enough. Inserting this into (5.4.1) we arrive at the contradiction # (0) > 1/2, and the theorem is proved. For more details we refer to the original paper by Burton and Keane (1989). [] We stress that the last argument relies essentially on the amenability property of Z d discussed in Section 5.2. The finite-energy condition is also indispensable: In another paper, Burton and Keane (1991) construct, for any k E {2, 3 . . . . . cx~}, translation invariant percolation models on Z 2 for which finite energy fails and which have exactly k infinite open clusters. For example, we have k = cx~ in Example 5.7. Fortunately, the finite-energy condition holds in most of the dependent percolation models which show up in stochastic-geometric studies of Gibbs measures. The situation becomes radically different when Z d is replaced by the nonamenable tree Td. Instead of having a unique infinite cluster, supercritical percolation models on Td tend to produce infinitely many infinite clusters. It is not hard to verify that this is indeed the case for supercritical Bernoulli site or bond percolation (except in the trivial case when the retention probability p is 1), and a corresponding result for automorphism invariant percolation on Td can be found in H~iggstr6m (1997b). On more general non-amenable graph structures, the uniqueness of the infinite cluster property can fail in more interesting ways than on trees; see e.g. Grimmett and Newman (1990) and H~iggstr6m and Peres (1999). Let us next consider the particular case of (possibly dependent) site percolation on Z 2. We know from Theorem 5.17 that under fairly general assumptions there is almost surely at most one infinite open cluster. Under the same asumptions there is almost surely at most one infinite closed cluster (i.e., at most one infinite connected component of closed vertices). In fact, the proof of Theorem 5.17 even shows that almost surely there is at most one infinite closed ,cluster. (Here, a closed ,-cluster is a maximal set C of closed sites which is ,-connected, in that any two x, y E C are connected by a ,-path in C; ,-paths were introduced in the proof of Theorem 5.3. Any closed cluster is part of some closed ,-cluster.) But perhaps an infinite open cluster and an infinite closed , cluster can coexist? Theorem 5.18 below asserts that under reasonably general circumstances this cannot happen. Under slightly different conditions (replacing the finite-energy assumption by separate ergodicity under translations in the two coordinate directions), it was proved by Gandolfi et al. (1988).
1 Random geometry of equilibrium phases
45
Theorem 5.18 Let lz be an automorphism invariant and ergodic probability measure on {0, 1}Z2 with finite energy and positive correlations. Then
#(3 infinite open cluster, 3 infinite closed ,-cluster) = 0. Note that automorphism invariance in the Z2-case means that, in addition to translation invariance, # is also invariant under reflection in and exchange of coordinate axes. Under the conditions of the theorem, we have in fact some information on the geometric shape of infinite clusters. If an infinite open cluster exists and thus all closed ,-clusters are finite, each finite box of Z 2 is surrounded by an open circuit, and all these circuits are part of the (necessarily unique) infinite open cluster. Hence the infinite open cluster is a sea, in the sense that all "islands" (i.e., the ,-clusters of its complement) are finite. Similarly, if a closed ,-cluster in Z 2 exists, it is necessarily a sea (and in particular unique). The corresponding result is false in higher dimensions. To see this, consider Bernoulli site percolation on Z 3. The critical value Pc for this model is strictly less than 1/2 (see Campanino and Russo (1985)), whence for p = 1/2 there exist almost surely both an infinite open cluster and an infinite closed cluster. The proof of Theorem 5.18 below is based on a geometric argument of Yu Zhang who gave a new proof of Harris' (1960) classical result that the critical value Pc for bond percolation on Z 2 is at least 1/2. (Recall that this bound is actually sharp.) Zhang's proof appeared first in (the first edition of) Grimmett (1999) and was exploited later in other contexts by H~iggstr6m (1997a) and H~iggstr6m and Jonasson (1999). Proof of Theorem 5.18: Let A be the event that there exists an infinite open cluster, let B be the event that there exists an infinite closed ,-cluster, and assume by contradiction that #(A n B) > 0. Then, by ergodicity, /z (A N B) = 1. (This is the only use of ergodicity we make, and ergodicity could clearly be replaced by tail triviality or some other mixing condition.) Next we pick n so large that lz(An) > 1 - 10 -3 and # ( B n ) > 1 - l0 -3,
where An (resp. Bn) is the event that some infinite open cluster (resp. some infinite closed ,-cluster)intersects An -- [ - n , n] 2 n Z 2. Let AnL (resp. Ann, An~ and AnB) be the event that some vertex in the left (resp. right, top and bottom) side of the square-shaped vertex set An \ An-1 belongs to some infinite open path which contains no other vertex of An, and define B~, Bnn, Bn:r and Bn8 analogously. Then An -- A L u ARn U A T u ABn .
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H.-O. Georgii et al.
Since all four events in the right-hand side are increasing and /z has positive correlations,
/zCAn)
--
/z(ALn U Aft U A T U Aft)
=
1 - / z ( - - , A L n--,Aft n--,A T n--,Aft)
<_ 1 -/z(-,AI~n)/z(-,Aff)/z(-,ATn)/z(-,Aff), where --, indicates the complement of a set (for typographical reasons). By the automorphism invariance of #, A~, AnR, An/"and An8 all have the same/z-probability, so that /z(--,AnL) < ( 1 - / z ( A n ) ) 1/4 and therefore #(A L) - / z ( A R) > 1 - (1 -/Z(An)) 1/4 -- 1 - 10 -3/4 > 0.82.
(5.4.2)
In the same way, we obtain # ( B T) - # ( B B) > 0.82.
(5.4.3)
Now define the event D - Anc N Aft n BnT N Bff. From (5.4.2) and (5.4.3) obtain
we
/z(D) > 1 - 4(1 - 0.82) = 0.28 > 0. When D occurs, both the left-hand side and the right-hand side of An are intersected by some infinite open cluster. By Theorem 5.17, these infinite open clusters are identical and separate their (common) complement into (at least) two pieces, preventing the infinite closed ,-clusters intersecting the top and bottom sides of An from reaching each other (see the picture on p. 290 of Grimmett (1999)). Consequently, there exist two infinite closed ,-clusters, in contradiction to Theorem 5.17. [] Theorem 5.18 admits some variants. First, the assumption of ergodicity can be avoided if the assumption of positive correlations is strenghtened to the condition that # is monotone in the sense of Definition 4.9. (This is because monotonicity is preserved under ergodic decomposition, so that Theorem 4.11 implies positive correlations for each ergodic component.) Moreover, as the preceding proof shows, ergodicity is only needed to show that infinite clusters exist with probability either 0 or 1, and translation invariance and finite energy are only used for the uniqueness of infinite clusters. We also need only the invariance under lattice rotations rather than all reflections, and closed ,-clusters can be replaced by closed clusters. We may thus state the following result.
1 Random geometry of equilibrium phases
47
Proposition 5.19 There exists no probability measure # on {0, 1 }z2 which has positive correlations, is invariant under lattice rotations and the interchange of the states 1 ("open") and 0 ("closed"), and satisfies /z(~ a unique infinite open cluster) = 1. This proposition will be applied to the ferromagnetic Ising model in Section 8.2.
6
Random-cluster representations
In Section 5 we saw a number of dependent percolation models. Here we shall focus on a particular class of such models, namely the (Fortuin-Kasteleyn) randomcluster model (and some of its relatives), which has turned out to be of great value in analyzing the phase transition behavior of Ising and Potts models. An alternative source for much of the material in the present section is Hfiggstr6m (1998). In Sections 6.1 and 6.2, we introduce the random-cluster model and discuss its relation to Ising and Potts models. This relation is then applied in Section 6.3 to prove Theorems 3.1 and 3.2. Despite the fact that Theorems 3.1 and 3.2 concern infinite systems, these applications only require defining finite volume random-cluster measures. However, it may be interesting in its own right to study infinite volume random-cluster measures on graphs such as zd; this is done in Section 6.4. In Section 6.6, we describe how the random-cluster representation of Ising and Potts models can be used to construct highly efficient Monte Carlo simulation algorithms. Finally, in Section 6.7, we discuss a variant of the random-cluster model which is applicable to the Widom-Rowlinson model rather than to Ising and Potts models.
6.1
Random-cluster and Potts models
The random-cluster model, also known as the Fortuin-Kasteleyn (FK) model after its inventors (Fortuin, 1972a, b; Fortuin and Kasteleyn, 1972), is a two-parameter family of dependent bond percolation models living on a finite graph. Let G = (s ~) be a finite graph with vertex set s and edge set/3. For a bond configuration 7/ 6 {0, 1}t3, we write k(q) for the number of connected components (including isolated vertices) in the subgraph of G containing all vertices but only the open edges (i.e. those e c 13 for which q(e) = 1).
Definition 6.1 The random-cluster measure ~pGq f o r G with parameters p 6 [0, 1] and q > 0 is the probability measure on {0, 1}t3 which to each rl c {0, 1}B
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H.-O. Georgii et al.
assigns probability
~o (77)_ P,q
1
17 Pr/(e)(1 -- p) 1--r/(e) } qk(~),
ZG, q e~13
where ZG,q is a normalizing constant. Note that taking q = 1 yields the Bernoulli bond percolation measure ~bp defined at the end of Section 5.1. All other choices of q give rise to dependencies between edges (as long as p is not 0 or 1, and G is not a tree). Taking q 6 {2, 3 . . . . } yields a model which is intimately related to the q-state Potts model, in a way which we will now explain. Let/Z~,q be the Gibbs measure for the q-state Potts model on G at inverse temperature fl, on {1. . . . .
q}s
which to each o- ~ {1. . . . . q}s assigns probability
c
_1(
#q,~((r) = Z~,q exp where
i.e./Z~,q is the measure
--2/?
)
x"~yZI{cr(x)~r(y)} ,
again Z~,q is a normalizing constant.
For q c {2, 3 . . . . } and p 6 [0, 1],
let pGq be the probability measure
on {1 . . . . . q}Z: • {0, 1}13 corresponding to picking a random element of {1. . . . . q }z:: • {0, 1}13 according to the following two-step procedure. (1) Assign each vertex a spin value chosen from {1 . . . . . q} according to uniform distribution, assign each edge value 1 or 0 with respective probabilities p and 1 - p, and do this independently for all vertices and edges. (2) Condition on the event that no two vertices with different spins have an edge with value 1 connecting them. In other
words, ppG,q is the measure which to each element (or, r/) of {1. . . . . q}s x
{0, 1}13 assigns probability proportional to
I-I ( pO(e)(1 -- P)l-o(e)l{(cr(x)-a(Y))O(e)=O}) " e=(xy)r Here, (xy) denotes the edge linking x and y. The measure eGq was introduced by Swendsen and Wang (1987) and made more explicit by Edwards and Sokal (1988), and is therefore called the Edwards-Sokal measure. The following theorem states that the edge marginal of eGq is a random-cluster measure, and the vertex marginal is a Gibbs measure for the Potts model, meaning coupling
of/Z~,q and ~b~q.
that eGq is a
1 Random geometry of equilibrium phases
T h e o r e m 6.2 Let, loG, p,qvertex and l~ . p,q projecting
eGq
on
{ 1 . . . . . q}s and
49
be the probability measures obtained by Then
{0, 1}B, respectively. ---~ tZ;, q
pG,vertex
P,q
(6.1.1)
with/~ -- 1 log(1 - p), and
eft,q,edge __ ~Gp,q"
(6.1.2)
Proof: The proof is just a matter of summing out the marginals. Letting Z be the normalizing constant in pGq, fixing a 6 { 1 . . . . . q }z3, and summing over all r/6 {0, 1 }B we find
(0") eft,vertex ,q
=
E ppGq(r r/~{0,1}B
:
Z
E VI P~(e)(1 - P)l-rl(e)I{(cr(x)-cr(Y))rl(e)=O} 0~{0,1}B e=(xy)~B
=
-~
1
1-I (1 - p)l{cr(x)#cr(Y)} e=(xy)El3
----
--zlexp( -
--
lZ~,q (tY),
1
2flEIl~(x)C:cr(Y)') x-.~y
since Z must be equal to ZflG'q by normalization. This proves (6.1.1). To verify (6.1.2) we proceed similarly, fixing 77 6 {0, 1} B and summing over a { 1 . . . . . q}s Note that, given 77, there are exactly qk(O) spin configurations a that are allowed, in that any two neighboring vertices x -~ y with rl((xy)) -- 1 have the same spin. We obtain
epG,edge ,q (T])
again by normalization.
--
G (0% ri) E PP,q a~{1 ..... q}s 1
:
qk(O)_~ I-I Pr/(e)(1 -- p)l-•(e) e613
=
u. ,cp,q (01 ,
O
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H.-O. Georgii et al.
The Edwards-Sokal coupling pGq o f JZfl,q G and ~)p,q G is the key to using the random-cluster model in analyzing the Potts model. The following two results are each other's dual, and are immediate consequences of Theorem 6.2 and the definition of ppG,q.
Corollary 6.3 Let p = 1 - e -2~, and suppose we pick a random spin configuration X ~ {1 . . . . . q}s as follows: (1) Pick a random edge configuration Y ~ {0, 1}13 according to the randomcluster measure ~)pGq.
(2) For each connected component C of Y, pick a spin at random (uniformly) from { 1. . . . . q }, assign this spin to every vertex of C, and do this independently f o r different connected components. Then X is distributed according to the Gibbs
measure lZ~,q.
Corollary 6.4 Let p = 1 - e -2~, and suppose we pick a random edge configuration Y ~ {0, 1}t3 as follows: (1) Pick a random spin configuration X c {1 . . . . . q }s according to the Gibbs measure lZ~,q.
(2) Given X, assign each edge e = (xy) independently value 1 with probability p 0
ifX(x)= X(y) ifX(x) r X(y),
and value 0 otherwise. Then Y is distributed according to the random-cluster measure cpp,q" G
As a warm-up for the phase transition considerations in Section 6.3, we give the following result as a typical application of the random-cluster representation.
Corollary 6.5 If we pick a random spin configuration X ~ {1 . . . . . q }s accordC , then f o r i ~ { 1 . . . . , q } and two vertices x , y ~ s ing to the Gibbs measure lZ~,q the two events {X(x) - i} and {X(y) = i} are positively correlated, i.e.
l ~ , q ( g ( x ) - i, X ( y ) - i) > lZ~,q(X(x) =--i)#~,q(X(y) -- i).
Proof: The
measure
/Z~,q is invariant under permutation of the spin set
{1 . . . . . q}, so that 1
q
1 Random geometry of equilibrium phases
51
We therefore need to show that lZ~,q(X(x) = i, X ( y ) -- i) >
1
q2"
We may now think of X as being obtained as in Corollary 6.3 by first picking an edge configuration Y E {0, 1}t3 according to the random-cluster measure ~bP6, q and then assigning i.i.d, uniform spins to the connected components. Given Y, the conditional probability that X ( x ) = X ( y ) = i is 1/q if x and y are in the same connected component of Y, and 1/q 2 if they are in different connected components. Hence, for some ~ E [0, 1], lzGq(X(x)e,
--
1 1 1 i, X ( y ) -- i) -- or- + (1 - oe) > q --q-2---~"
An easy modification of the above proof shows that if G is connected and/5 > 0, then the correlation between I{x(x)=i} and I{x(y)=i} is strictly positive. Note that the relation between the random-cluster model and the Potts model depends crucially on the fact that all spins in {1 . . . . . q } are a priori equivalent. This is no longer the case when a non-zero external field is present in the Ising model. Several attempts to find useful random-cluster representations of the Ising model with external field have been made, but progress has been limited. Perhaps the recent duplication idea of Chayes et al. (1998) represents a breakthrough on this problem.
6.2
Infinite-volume limits
In this subsection we will exploit some stochastic monotonicity properties of random-cluster distributions on finite subgraphs of Z d. This will give us the existence of certain limiting random-cluster distributions, and also the existence of certain limiting Gibbs measures for the Potts model. The basic observation is stated in the lemma below which follows directly from definitions. L e m m a 6.6 Consider the random-cluster model with parameters p and q on a finite graph G with edge set 13. For any edge e = (xy) E 13, and any configuration /7 E {0, 1}t~\{e}, we have if x and y are connected via open edges in q q5G (e is openlq) - { p P,q
P p+(1-p)q
otherwise. (6.2.1)
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H.-O. Georgii et al.
For q > l, Lemma 6.6 means in particular that the conditional probability in (6.2.1) is increasing in 7/ (and also in p). This allows us to use Holley's theorem and the FKG inequality to prove the following very useful result. We write qSp G for Bernoulli bond percolation on G with parameter p. Corollary 6.7 For a finite graph G and the random-cluster p 6 [0, 1]andq > 1, we have
measure
qbpGq with
(a) (~pGqis monotone, and therefore it has positive correlations, (b) ~ 6p,q _<79 qbG ,
(C) ~pG,q >._79 ~G
P p+(1--p)q
9
Furthermore, for 0 < Pl < P2 < 1 and q > 1, we have (d) (DpG,q "<7) G ~)P2,q" Proof: The monotonicity in (a) is just the observation that the conditional probability in (6.2.1) is increasing in p and in 77. Positive correlations then follows from Theorem 4.1 1. Next, note that (6.2.1) implies that P < G p + (1 -- p)q - (/)p'q (e iS open]r/) _< p
(6.2.2)
for all r/as in Lemma 6.6. Theorem 4.8 in conjunction with the second (resp. first) inequality in (6.2.2) implies (b) (resp. (c)). Finally, (d) is just another application of (6.2.1) and Theorem 4.8. [] Consider now the integer lattice Z a (for definiteness and simplicity) with its usual graph structure. We associate with any finite region A C Z d two specific random-cluster distributions which correspond to two different choices of boundary condition. The latter will we distinguished by a parameter b 6 {0, 1}. Let/3 be the set of all nearest-neighbor bonds in Z d, B ~ the set of all edges of B that are contained in A, and B~ the set of edges with at least one endpoint in A. (The difference B~x \ B ~ thus consists of all edges leading from a point of A to a point of Ac.) We then let q~pb,q,A be the probability measure on {0, 1}t3 in which each r/6 {0, 1}B is assigned probability proportional to
I{~-=boffB~}{ I-I Prl(e)(1--p)l-~(e)] qk(rl'A)' e~Bba
1 Random geometry of equilibrium phases
53
b the where k(r/, A) is the number of all 0-open clusters meeting A. We call ~p,q,A random-cluster distribution in A with parameters p and q and boundary condition b. In the case b = 0, k(-, A) is simply the number of all clusters that are contained in A; this corresponds to forgetting all sites in A c and is therefore referred to as the free boundary condition. On the other hand, suppose that A has no holes, in the sense that A c has no finite connected components; since we can always assume without loss of generality that A is connected, we call such a A simply connected. Then, in the case b = 1, all sites of A c may be thought of as being firmly wired together, whence this is called the wired boundary condition. Suppose now that A C A are two finite regions in Z a. Then 4)bp,q,A is obtained b A by conditioning on the event {0 - b on B b \ B b } which is increasing from ~ p,q, for b = 1 and decreasing for b = 0. Hence, if q > 1 then Corollary 6.7 (a) implies that q5~ q,A --7) ~plp,q,A when A C A p,q,A and 4)p,q,A
(6.2.3)
in complete analogy to (4.3.2). Moreover, we obtain the following counterpart of Proposition 4.14 on the existence of infinite-volume limits. We write A 1" Z d for the limit along some (any) increasing sequence of finite simply connected subsets of Z d, converging to Z d in the usual way.
L e m m a 6.8 For p ~ [0, 1] and q >_ 1, the limiting measures 4)b
,q
=
lim ~bp,q,A, A],Z d
b E {0, l}
exist and are translation invariant. This convergence result has consequences for the convergence of Gibbs distributions for the Potts model, as we will show next. Let q 6 {2, 3 . . . . }, and for i { 1 . . . . . q} and any finite region A in Z d let #},q,h denote the Gibbs distribution in A for the Potts model at inverse temperature/3 with boundary condition r/-- i on A c. For i = 0, let #~,q,A be the corresponding Gibbs distribution with free boundary condition, which is defined by letting/2 = A in (2.6.2), i.e., by ignoring all sites outside A; we think of #~,q,A as a probability measure on the full configuration space {1 . . . . . q }zd by using an arbitrary extension. Still for i = 0, Theorem 6.2 shows t h a t / ~ , q , a and q~0,q,h admit an EdwardsSokal coupling (on A) when p = 1 - e -2~. A similar Edwards-Sokal coupling is possible for i 6 {1,.. ., q } when A is simply connected. Indeed, let Pp,q,Ai be the probability measure on {1. . . . . q}s x {0, 1}B corresponding to picking a random site-and-bond configuration according to the following procedure. (1) Assign to each vertex of A c value i, and to all edges of B \ B1A value 1.
54
H.-O. Georgii et al.
(2) Assign to each vertex in A a spin value chosen from {1. . . . . q } according to uniform distribution, assign to each edge in/31A value 1 or 0 with respective probabilities p and 1 - p, and do this independently for all vertices and edges. (3) Condition on the event that no two vertices with different spins have an edge with value 1 connecting them. It is now a simple modification of the proof of Theorem 6.2 to check that the vertex i and d/)p,q,A' 1 respectively. (Note that by and edge marginals of pip,q, A are #~,q,A the simple connectedness of A there is always a unique component containing Ac.) Analogues of Corollaries 6.3 and 6.4 follow easily. This leads us to the following result extending Proposition 4.14 to the Potts model.
Proposition 6.9 For any i E {0, 1. . . . . q }, the limiting probability measure i
~fl,q
--
lim
i
A.~Zd~fl,q, A
on { 1. . . . . q }zd exists and is a translation invariant Gibbs measure for the q-state Potts model on Z d at inverse temperature ft.
Proof: In view of the general facts reported in Section 2.6, the limits are Gibbs measures whenever they exist. We thus need to show that #~,q,A i ( f ) converges as A 1" Z d, for any local observable f . For definiteness, we do this for i c { 1 . . . . . q }; the case i = 0 is completely similar. Fix an f as above, and let A C Z d be the finite region on which f depends. As shown above, for a simply connected A we may think of a {1. . . . . q} zdas arising by first picking an valued random element X with distribution #ri edge configuration Y 6 {0, 1}A according to q~pl,q,A (with p -- 1 - e -2/3) and then assigning random spins to the connected components, forcing spin i to the (unique) infinite cluster. Forx, y c A, we write {x ~ y} for the event that x and y are in the same connected component in Y, and {x ~ ~ } for the event that x is in an infinite cluster. Clearly, the conditional distribution of f given Y depends only on the indicator functions (I{x~y})x,y~/X and ( l { x ~ } ) x c / X , since the conditional distribution of X on A is uniform over all elements of { 1 , . . . , q}A such that firstly X ( x ) = X ( y ) whenever x +-~ y, and secondly X ( x ) : i whenever x ~ e~. Hence, the desired convergence of #~,q,A ( f ) follows if we can show that the joint distribution of (I{x~y})x,y~A and ( I { x ~ } ) x ~ A converges as n -+ cx~. This, however, follows from Lemma 6.8 upon noting that (I{x~y})x,yc/X and (l{x~e~})x~/X are increasing functions. E3
1
6.3
Random geometry of equilibrium phases
55
Phase transition in the Potts model
As promised, this subsection is devoted to proving Theorems 3.1 and 3.2, using random-cluster arguments. The original source for the material in this subsection is Aizenman et al. (1988); see also H~iggstr6m (1998) for a slightly different presentation. We consider the Potts model on Z a, d > 2. All the arguments to be used here, except those showing that the critical inverse temperature fie is strictly between 0 and c~, go through on arbitrary infinite graphs; we stick to the Z d case for definiteness and simplicity of notation. We consider the limiting Gibbs measures /X~,q obtained in Proposition 6.9. For i 6 {1 . . . . . q}, these play a role similar to that of the "plus" and "minus" measures #~- and # ~ for the Ising model. In fact, we have the following result which extends Theorem 4.15 to the Potts model and also gives a characterization of phase transition in terms of percolation in the random-cluster model.
T h e o r e m 6.10 Let fl > 0 and p -
1 - e -2~. For any x E Z d and any i { 1 . . . . . q }, the f o l l o w i n g statements are equivalent.
(i) There is a unique Gibbs measure f o r the q-state Potts model on Z d at inverse temperature ft.
(ii) I~fl,q i (X (x) -- i) -- 1/q. (iii) qS~,q(x +-~ ec) = 0. As we will see in a moment, it is the percolation criterion (iii) which is most convenient to apply. In this context we note that 4); (x +-~ o c ) = 'q
inf dl);,q,A(X ~
A,A
A c) -- lim ~1 AtZ d
p,q,A (X +-~ A c ),
(6.3.1)
where {x +-~ A c} stands for the event that there exists an open path from x to some site in A c. This follows from (6.2.3) and the fact that {x ~-+ A c } decreases to {x ~ ~ } as A 1" Z d. The usefulness of the percolation criterion is demonstrated by the next result which extends the scenario for Bernoulli percolation to the random-cluster model. Together with Theorem 6.10, this gives Theorem 3.2 with tic - 89log(1 - Pc).
Proposition 6.11 For the random-cluster model on Z d, d >_ 2, and any fixed q >_ 1, there exists a percolation threshold Pc E (0, 1) (depending on d and q) such that
C/);,q (X ~ (X3) { -- 0 for p < Pc, >0
f o r p > Pc.
56
H.-O. Georgii et al.
Proof: The statement of the proposition consists of the following three parts:
(i) 4~plq(x ~ cx~) - 0 for p sufficiently small, (ii) ~bl,q (X +-> ~ )
> 0 for p sufficiently close to 1, and
(iii) 4)P,q 1 (x +-~ cx~) is increasing in p. We first prove (i). Suppose p < pc(Z d, bond), the critical value for Bernoulli bond percolation on Z d. For e > 0, we can then pick A large enough so that 4~p(0 ~ A ~) _< e. By Corollary 6.7(b), we have that the projection of q~p on {0, 1}B~ stochastically dominates the projection of q51p,q,A on {0, 1}~x for any A D A, so that
4~pl,q,A (O ~
A
c) < 4~1p,q,A (O ~
A c) < 6
for any A D A. Since e was arbitrary, we find lim 4) 1 Ac A'~Zd p,q,A(O ~ ) --0 which in conjunction with (6.3.1) implies (i). Next, (ii) can be established by a similar argument. Let p be such that p* = p / [ p + (1 - p)q] > p c ( Z d, bond). Corollary 6.7(c) then shows that ~plp,q,A >-7) Cpp. for every A, so that lim qSl,q,A (0 +-~ ~ )
> 0,
A~'Z d
proving (ii). To check (iii) we note that Corollary 6.7(d) implies that, for any A, cPll ,q,A _<7) cPp2,q 1 ,A whenever Pl < P2.
(6.3.2)
This proves (iii) and thereby the proposition. Before the proof of Theorem 6.10 we need another definition and a couple of lemmas. For a finite box A in Z d and a spin configuration ~ 6 {1. . . . . q}OA, let A ti ___~{X E 0 A " ~ ( x ) -- i}
1 Random geometry of equilibrium phases
for i -- 1. . . . . q. We now define the random-cluster
distribution
57
~,q,Ar for A
with boundary condition ~, as the probability measure on {0, 1}t3~ which to each 1/6 {0, 1} ~ assigns probability proportional to
ID(~,~)[ I-I P~(e)(1--p)I-~(e)} qk~(~)
e6/31A where k~(rl) is the number of connected components in ~ that do not intersect 0A, and D(~, rl) is the event that there is no open path in rl connecting any two vertices in A~i and A~ for any i r j. Lemma 6.12 Let p = 1 - e -2~, let A be a finite region in Z d, and fix some boundary condition ~ ~ { 1 . . . . . q }aA. Suppose that we pick a random spin configuration X 6 { 1 . . . . . q }A as follows. (1) Pick Y r {0, 1}t31 according to ~p,q,A"
(2) For each i ~ { 1 . . . . . q} and each connected component C in Y intersecting i assign spin i to every vertex in C. A~, (3) For all other connected components C in Y, pick a spin at random (uniformly) from { 1 . . . . . q }, assign this spin to every vertex of C, and do this independently for different connected components. Then X is distributed according to the Gibbs distribution #%,q,A~ model on A with boundary condition ~.
for the Potts
Proof: This is a straightforward generalization of the proofs of Theorem 6.2 and Corollary 6.3. []
Lemlna 6.13 With notation as above, we have, for any ~ ~ {1 . . . . . q}aA, that the projection of ~lp,q,A on {0, 1}B~ stochastically dominates dp~p,q,A. Proof: Just write down single-edge conditional distributions for 4~,q,A~ and
4~l,q,A (as in Lemma 6.6) and invoke Theorem 4.8.
[]
We are finally ready for the proof of Theorem 6.10. Proof of Theorem 6.10: We begin with the implication (i) :=~ (ii). If there is a unique Gibbs measure for the q-state Potts model on Z d at inverse temperature/3, then we have in particular that
58
H.-O. Georgii e t al.
But since by symmetry lZfl,qi ( X ( x )
-- i) -
lzJfl,q(S(x)
-- j) for any i, j
{1 . . . . . . q} we must have #~,qi ( X ( x ) -- i) = 1/q, and (ii) is established. Next we turn to the implication (ii) =:~ (iii). By the Edwards-Sokal coupling of edge and site processes introduced before Proposition 6.9, we have lztf,q (X(O)
-- i)
--
lim /Z~,q,A(X(O ) - - i ) A~Zd
=
1 q-1 - + q q
1 lim dpp,q,A (0 +-~ AC).
(6.3.3)
A~'Zd
Together with (6.3.1), the result follows. Most of the work is needed for the implication (iii) =:~ (i). Roughly speaking, the absence of percolation in the random-cluster model implies that every finite region is cut off from infinity by a set of closed edges. Thus, independently of what happens macroscopically, the local spins feel as if they are in a system with free boundary condition. This makes a phase transition impossible. To make this intuition precise we let # be an arbitrary Gibbs measure for the Potts model at inverse temperature/3. We will show that/z - #~,q, the limiting measure with free boundary condition obtained in Proposition 6.9. We fix any local observable f and some e > 0. We then can find a finite box A C Z d such that , ~- q , r ( f ) - / z ~ ~- q ( f ) l , < e for all finite F D A. ]#0 (6.3.4) I
!
_
In view of (6.3.1) we can also choose a finite box A 3 A satisfying t~l,q,A (X +-~ A c) < e/IA[ for all x 6 A, and thus
qSl,q,A(A ~ A c) < e. Here, {A +-~ A c} is the event that there exists an open path from A to AC. Consider the complementary event C -- {A +-~ Ar c. For any edge configuration r/ 6 C, A is cut off from A c by a set of closed edges. Indeed, let F be the union of all 0-open clusters meeting A. Then (a) A C F C A ,
and
(b) r/(e) - 0 for all edges from F to F c, i.e., r/-- 0 on 131 \ / 3 ~ Since these properties are stable under finite unions, there exists a largest set F (r/) satisfying (a) and (b). The maximality implies that, for each fixed region F, the event {r/ 9 F (r/) -- F} only depends on the status of the edges off/3 ~ Coming to the core of the argument, we fix any boundary spin configuration 6 {1 . . . . . q}OA and consider the Edwards-Sokal coupling P -- P~p,q,A of #~o q,h~ and 4~q,At, introduced in Lemma 6.12. We write X for the random spin
1 Random geometry of equilibrium phases
59
configuration in {1 . . . . . q} zd and Y for the random edge configuration in {0, 1}B Then P ( f o X)
=/z%,q,A( f ) ' P(Y ~ C)-
and
dp~p,q,A(C) >-- flplp , q , A ( C )
> 1- e
by Lemma 6.13 and the choice of A. Assuming without loss of generality that IIf II _< 1, we can therefore conclude that
]P(foXiY
E
C)-/z$,q,A(f)l
(6.3.5)
< 28.
However, the conditional expectation on the left is an average of the conditional = F) with A C F C A, and these in turn are averages expectations P ( f o X I F ( Y ) of conditional expectations of the form P ( f o X IY = r/off BilL, Y = 0 on B~ \ / 3 ~ which, by construction of P, are equal to /z~,q,F(f)" and (6.3.5), we conclude that
Together with (6.3.4)
]/z~,q,A (f) -- /zOl3,q(f)l <3e. Taking the/z-average over ~ and letting e -+ 0 we finally g e t / z ( f ) and the proof is complete,
6.4
=
/z~,q (f), n
Infinite volume random-cluster measures
The random-cluster arguments used in the previous subsections for studying infinite volume Ising and Potts models only required defining finite volume randomcluster distributions, although we have seen already the limiting random-cluster measures q5p,q" b (Their existence was convenient in the formulation of Theorem 6.10, but not really needed for the arguments.) Recent years have nevertheless witnessed a rapid development of a theory for infinite volume random-cluster measures, defined in the DLR spirit. Here we shall discuss the basics of such a theory. A similar theory of infinite volume Edwards-Sokal measures for joint spin and edge distributions was recently developed by Biskup et al. (2000). Let G be an infinite (locally finite) graph with vertex set/2 and edge set B. Fix p c [0, 1] and q > 0, and let B C B be a finite simply connected region; since the random-cluster model lives on edges rather than on vertices, we let "region" refer to edge sets rather than vertex sets in this subsection. Let V(B) = {x /2 : 3e ~ B incident to x}. For an edge configuration ~ ~ {0, 1}Be, define the
60
H.-O. Georgii et al.
random-cluster distribution CkpS~ in B as the probability measure on {0, 1}B in which each r/6 {0, 1}B is assigned probability proportional to
where k (7, B) is the number of connected components of rl which intersect V (B). b defined in This is a generalization of the random-cluster distributions ~)p,q,A Section 6.2, which are recovered by taking B = B~ and ~: = b, b ~ {0, 1}. It is easy to see that the random-cluster distributions are consistent in the sense that conditioning on a configuration in some B' C B yields the corresponding random-cluster distribution in B \ B'. Definition 6.14 A probability measure ~ on {0, 1}/3 is said to be a random-cluster measure with parameters p and q if its conditional probabilities satisfy q~(r/l~) = q~(~ in B[~ off B) -- ~bpB,~(r/)
for all finite simply connected B C B, ~-almost all ~ and all 77 such that rl = offB. This is the direct analogue of the definition of a Gibbs measure in the randomcluster setting. There is, however, also another possibility which differs from the preceding one for graphs in which the complements of finite regions are not connected. The idea is to connect all infinite clusters at infinity. This corresponds to a one-point compactification of G. Accordingly, we shall use a prefix "C" which stands for "compactified". Thus, we define a C-random-cluster distribution "B~ 4~p'~ as in (6.4.1), except that k(r/, B) is replaced by ~:(r/, B), defined as the number of all finite connected components of 77intersecting V (B). Definition 6.15 A probability measure dp on {0, 1}/3 is said to be a C-randomcluster measure for p and q if its conditional probabilities satisfy
for all finite simply connected B C B, ~-almost all ~ and all ~ such that rl = offB. The study of random-cluster measures in the case G = Z d was initiated by Grimmett (1995), and about simultaneously by Pfister and Vande Velde (1995) and Borgs and Chayes (1996); see also Seppal~iinen (1998) for some even more recent developments. The C-variant (Definition 6.15) was introduced in the regular tree case by H~iggstr6m (1996a, b), and further studied in a general graph context by
1 Random geometry of equilibrium phases
61
Jonasson (1999). In the following, we will try to convince the reader that randomcluster measures of both types are of interest, and also discuss their relation to each other. We shall concentrate mainly on the case q > 1. The reason for this is that it is only for q > 1 that the conditional probability in (6.2.1) is increasing in 7, which allows the use of the stochastic domination and correlation results in Section 4 (Theorems 4.8 and 4.11). For q < 1, these tools are not available, and for this reason the random-cluster model with q < 1 is much less understood than the q >_ 1 case, although in Grimmett (1995), H~iggstr6m (1995) and Sepp~il~iinen (1998) one can find at least some results in the q < 1 regime of the parameter space. ^8 1 ^S We write q~pS,,q0for 4~p8,~ with ~ ~ 0, and 4~p,'~ for 4~p,'~ with ~ - 1. We can omit the circumflex when G - Z d, because there is always exactly one infinite cluster regardless of the configuration on B (this is related to Proposition 6.19 below). On the other hand, the two different ways of counting clusters with wired boundary condition are not equivalent for all graph structures; a simple counterexample is G -- Td in which the wired boundary condition gives rise to several infinite clusters. Still in the context of general infinite graphs, we write B 1" /3 for the limit along some (any) sequence of finite simply connected regions increasing to/3 in the usual way. In complete analogy to (6.2.3) and Lemma 6.8 we then obtain the following monotonicity and convergence result. Lemma
6.16 For p ~ [0, 1], q > 1, a n d a n y two f i n i t e b o n d sets B1 C B2, we
have
(i) ,6Bj,0 .rp,q -<79 '6B2'0 v.p,q , so that the limit d?0
P'q
(ii) "rp,q 3,BJ'I ___Dr~B2'l v'p,q SO that the limit ~1 '
P'q
-- lira q~pB,q0exists; a n d B?~
"B,1 exists. -- lira qbp,q B?13
Now let 4~ be any random-cluster measure of either type, compactified or not, with the given parameters p and q. Further application of Theorem 4.8 (or Corollary 6.7) implies that 4~~ ^ p,q <_z~4~ <_z~ ~p,q" This is analogous to the sandwiching relation (4.3.3) for the Ising model. Furthermore, the arguments of Section 6.3 show that the q-state Potts model on G at inverse temperature fl has a unique Gibbs measure if and only if the q~tP,q -probability of having an infinite cluster is 0, where as usual p - 1 - e -2/~. For Definitions 6.14 and 6.15 to be of interest, we have to establish at least the existence of random-cluster measures of the two types. The following theorem
62
H.-O. Georgii et aL
tells us that at least for q > 1, such measures do exist. (The existence problem for q < 1 remains open in the setting of general graphs, although existence has been established for Z d and Td; see Grimmett (1995) and H~iggstr6m (1996a), respectively.) T h e o r e m 6.17 For p E [ 0 , 1 ] a n d q
> ~ is a random_ 1 we have that (i) dpp,q cluster measure, and (ii) ~ lp,q is a C-random-cluster measure. ,
For the proof we use the following lemma which characterizes random-cluster measures in terms of single-edge conditional probabilities. For e -- (xy) E 13 and E {0, 1}t3\{e}, we write as usual {x +-~ y} for the event that there exists an open C
path in ~ from x to y. We also write {x < r y} for the event that there either exists an open path in ~ from x to y, or x and y are both in infinite clusters of ~. C
We think of this C-connectivity notion x .~ ~ y as allowing paths between x and y to go "via infinity". L e m m a 6.18 Fix p E [0, 1] and q > O, and let ck be a probability measure on {0, 1}t3. Then dp is a random-cluster measure f o r p and q if and only if f o r each e - (xy) E 13 and dp-a.e. ~ E {0, 1}t3\{e} we have
~(e is open
I~) -- / p P / p+(1--p)q
if x +-~ y in otherwise,
(6.4.2)
Similarly, ~ is a C-random-cluster measure f o r p and q if and only if(6.4.2) holds C with x < ~ y instead o f x <-+ y.
Proof: We consider only the first statement, as the C-case is completely similar. For the "only if" part we only need to note that the right-hand side of (6.4.2) is equal to "r a,p,q le}'~ (e is open ) Passing to the "if" part, we may restrict ourselves to the case of p E (0, 1). Assume that 4~ satisfies (6.4.2) for each e = (xy) E 13 and 4)-a.e. ~ E {0, 1}t3\{e}. Let B C 13 be some finite edge set. We need to show that the conditional distribution q~('l~) of 4) given the configuration ~ on B c equals q~ff,~ for 4)-a.e. ~. For this, it suffices to check that for any two configurations r/, r/' E {0, 1}t~ which agree with ~ on B c we have ~b(/~l~)
{I-IeEB Pr/(e)( 1 -- p)l-rl(e)}qk(rI,B)
~(~'1~)
{ I l e E B PO'(e)( 1 -- p)l-rl'(e)}qk(O',B)
(6.4.3)
with k defined as in (6.4.1). If 0 and r/t differ only at a single edge e, then k(~7, B ) - k(r/', B) = k(r/, e ) - k(r/', e), whence (6.4.3) is immediate from (6.4.2). In the general case, we interpolate 77and r/~ by a sequence of configurations which successively differ in at most one edge, and use a telescoping argument. []
1 Random geometry of equilibrium phases
63
Proof of T h e o r e m 6.17: We prove (ii) only, as (i) follows from a similar argument and is also better known, see Borgs and Chayes (1996). Fix e = (xy) ~ B. By L e m m a 6.18, it is sufficient to establish (6.4.2) with the C-connectivity relation c
> in place of the standard connectivity relation +->. Let B1, B2 . . . . be an increasing sequence of finite edge sets containing e and converging to B in the usual sense. We write, with slight abuse of notation, ~ (Bi) for the restriction of ~ to Bi \ {e}. We recall from the Martingale convergence theorem that ^1 (e is open I~(Bj)) q~;,q (e is open I~) -- lim 4)p,q
(6.4.4)
j---> oo
^l for dpp,q-a.e. ~
E {0, 1 }t3\{e} .
c We suppose first that x < > y fails in ~. Then at least one of the vertices x and y is in a finite cluster of ~, and consequently there is some m (depending c
on ~) such that --,(x ~ > y) can be verified by just looking at ~(Bm). (This is a consequence of the special concept of C-connectivity.) For any n > j > m we then have P ~B,,,1 (e is open I~(Bj)) P,q p+(1-p)q so
that by the definition of -ea~l,q we obtain
4);,q(e is open I~(Bj)) -by letting n --+ cx). Then we let j ~ of (6.4.2) in the case ~: r {x ."
c
p + (1 -
p)q
oo and use (6.4.4) to deduce the C-version
> y}. c
We go on to the case ~ 6 {x ~. ; y}. In analogy to (6.4.4), we have ^1 l i m dpp,q(X ~ j--+ cx~
yI~(Bj))-- 1
C for q~l,q-a.e. ~ 6 {x < ; y}. For such ~ and any e > 0, we can thus find an m (depending on ~) such that
4;,q(X
<
C
> yI~(Bj))
for any j >_ m. Next we use the definition
of
>_ 1 - e
(6.4.5)
~l,q. For any n = 1, 2 . . . . . let Y
and Yn be {0, 1 }t3-valued random edge configurations with distributions ~lp,q and ^Bn, 1 (])p,q satisfying Yn >- Y; this is possible by L e m m a 6.16. We write Pn for the
64
H.-O. Georgii e t al.
probability measure underlying this coupling. By the same lemma and the order relation Yn >- Y, we have lim Pn(Yn(Bj) ~A Y ( B j ) ) = 0.
/'/---+o~
SinceYn c { x <
~p,q
I "Bn,1
C
> y}wheneverY6{x
(X y , ~ ( B j ) ) -
<
C
> y},wecanwrite
"1 (x y , ~ ( O j ) ) ~)p,q
<_ Pn {x < > y in Yn, Yn(Bj) -- ~ ( B j ) } A { x "Bn 1 (x < C > y ) - - ~ p"1 _< Y)
(6.4.6)
I < > y in Y, Y ( B j ) -- ~ ( B j )
+ Pn(Yn(Bj) ~= Y ( B j ) ) c
where A denotes symmetric difference. Since {x < > y} is the decreasing limit of the local events {x +-> y in A} U {x ++ A c, y ++ A c} as A 1" s an analogue of (6.3.1) together with (6.4.6) shows that the last expression tends to zero as n --+ cx). It follows that "Bn,1
lim ~p,q (X <
C
n---~oG
> yl~(Bj))-
"1 C qbp,q(X < > y ] ~ ( B j ) )
"Bn,1 which is at least 1 - e by (6.4.5). But since qbp,q (e is open I~!) = P for each n
and a l l ~ ' ~ { x
<
C
> y},weget p-
"Bn,1 (e is open [~(Bj)) _< p. e < lim Cp,q n----+oo
Hence,
p-
"1 g <_ dpp,q(e is open [~(Bj)) _< p,
and since e was arbitrary we can use (6.4.4) to deduce the C-version of (6.4.2) in the c a s e ~ ~ { x <
C
> y}.
[]
Let us now briefly address the issue of whether the two types of random-cluster measures are any different. The following result says that very often they are the same.
Proposition 6.19 Let r be a probability measure on {0, 1 }/3 with r (3 at most one infinite open cluster) - 1.
Then, f o r any p c [0, l] and q > O, r is a random-cluster measure f o r p and q if and only if it is a C-random-cluster measure f o r p and q.
1 Random geometry of equilibrium phases
65
This means that whenever "uniqueness of the infinite cluster" can be verified, the two types of random-cluster measures coincide. An example is obtained if we consider translation invariant random-cluster measures for Z d, since the BurtonKeane uniqueness theorem (Theorem 5.17) applies in this situation. For Z a, the "1 measures q~0,q a n d ~p,q are translation invariant, by Lemma 6.8. On the other hand, uniqueness of the infinite cluster typically fails on trees, leading to very different behavior for the two types of random-cluster measures; see H~iggstr6m (1996a, b) for a discussion. Proof of Proposition 6.19: For p = 0 or 1 the result is trivial, so we may assume that p 6 (0, 1). The conditional probabilities in (6.4.2) and its C-counterpart differ only on the event C
Axy : {X < > y} \ {x +-~ y}.
Hence if 4~ is a random-cluster measure but not a C-random-cluster measure (or vice versa), then Axy has to have positive 4~-probability for some edge e - (xy) E /3. But then the event Axy O {e is closed} has positive 4~-probability, and since this event implies the existence of at least two infinite clusters, we are done. n Much of the study of infinite volume random-cluster measures that has been done so far concerns the issue of uniqueness (or non-uniqueness) of randomcluster measures. A discussion of this issue would, however, lead us too far, so instead we advise the reader to consult Grimmett (1995), H~iggstr6m (1996a) and Jonasson (1999) to find out what is known and what is conjectured in this field.
6.5
An application to percolation in the Ising model
In Theorem 5.10 we have seen that the probability of percolation of plus spins in the Ising model is an increasing function of the external field. A much harder question is to determine monotonicity properties of percolation probabilities as fl (rather than h) is varied. An interesting open problem is to decide whether for G = Z d, d > 2, the probability a~-(x <
+
>ec)
is increasing in/3. Here we write #~- for the plus phase in the Ising model at +
inverse temperature i3 with external field h - 0, and {x ~. , ec} is the event that there exists an infinite path of plus spins starting at x. At first sight, one might be seduced into thinking that this would be a consequence of the connection between Ising and random-cluster models, and the stochastic monotonicity of random-cluster measures as p varies; see (6.3.2). However, such a conclusion
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is unwarranted. For example, in the coupling of Theorem 6.2 the existence of an open path between x and y in the random-cluster representation is a sufficient but not necessary condition for x and y to be in the same spin cluster. In fact, H~iggstr6m (1996c) showed, by means of a simple counterexample and in response to a question of Cammarota (1993), that the probability that x and y are in the same spin cluster need not be increasing in fl, and similarly for the expected size of the spin cluster containing x. However, when the underlying graph is a tree, monotonicity in fl of the probability of plus percolation can be established: Theorem 6.20 For the lsing model on the regular tree Td, d >__ 2, with a dis+ tinguished vertex x, the percolation probability # ~ (x < > (x~) is increasing in
An interesting aspect of this result is that its proof, unlike those of the monotonicity results mentioned earlier in this section, is not based on stochastic domination between the probability measures in question. In fact, stochastic domination fails, i.e. it is not always the case (in the setting of Theorem 6.20) that #~-1%Z~ # ~
(6.5.1)
when/31 < f12. An easy way to see this is as follows. Just as in Theorem 3.1, let/?c be the critical inverse temperature for non-uniqueness of the Gibbs measure. (It is straightforward to show, using either the random-cluster approach or the methods in Section 7, that fie > 0 for L; = Ta.) Pick fll < f12 in (0,/?c). By Theorem 4.15, we then have #~- (~ 9~ (y) - 4-1) - - ]-/~fl-2(~ " ~ (y) = + 1) -- 1/2 for every vertex y. If now (6.5.1) was true we would have #~-1 = #~-2 by Proposition 4.12. This, however, is impossible because the two measures have different conditional distributions on finite regions. Theorem 6.20 can be proved using the exact calculations for the Ising model on Td, which can be found in e.g. Spitzer (1975) and Georgii (1988). Here we present a simpler proof which does not require any exact calculation, but which exploits random-cluster methods. Proof of Theorem 6.20: As usual we write L; and B for the vertex and edge sets of Ta. Since lim
+
+ (x < > c x z ) - - #
A1,s ~fl, A
+
(x < > ~ )
for any fl in analogy to (6.3.1), it suffices to show that for/~l < f12 and any A, we have +
+
,A (x < > oo).
(6.5.2)
1 Random geometry of equilibrium phases
67
This we will do by constructing a coupling P of two {-1, + 1}C-valued random objects X1 and X2 with respective distributions/z~l ,A and/z~-2, A and the property + that if x < > c~ in X1, then the same thing happens in X2. Recall the Edwards-Sokal coupling of spin and edge configurations described ahead of Proposition 6.9. In the present case of the tree Td, this construction requires the C-version of counting clusters, which corresponds to making the complement of A connected. Therefore we will work with the C-random-cluster distributions. Let Pl - 1 - e -2/3~ and P2 - 1 - e -2/32, and let B --/31 C / 3 be the set of edges incident to at least one vertex in A. We first let Y1 and Y2 be two {0, 1}t3-valued random edge configurations distributed according to the random^B,1 and ~bp2,2, ^B,1 and such that P(Y1 -< Y2) - 1" this is possible cluster measures 4~p~,2 ^B,1 by the ~bp,z-analOgue of (6.3.2). X1 and X2 can now be obtained by assigning spins to the connected components of Y1 and Y2 in the usual way; these spin assignments are coupled as follows. First we must assign spin + 1 to all infinite clusters in Y1 and Y2. Then we let (Z(y))ycA be i.i.d, random variables taking values + 1 and - 1 with probability 1/2 each, and assign to each finite cluster C of Y1 and Y2 the value Z(y), where y is the (unique) vertex of C that minimizes the distance to x. This defined X 1 and X2. A moment's thought reveals that the set of vertices that can be reached from x via spins in X1 is almost surely contained in the corresponding set for X2. Hence (6.5.2) is established, and we are done. [] Note that this proof did not use any property of Td except for the tree structure, so Theorem 6.20 can immediately be extended to the setting of arbitrary trees.
6.6
Cluster algorithms for computer simulation
An issue of great importance in statistical mechanics which we have not touched upon so far is the ability to perform computer simulations of large Gibbs systems. Many (most?) questions about phase transition behavior etc. can, with current knowledge, only be answered partially (or not at all) using rigorous mathematical arguments. Computer simulations are then important for supporting (or rejecting) heuristic arguments, or (in case not even a good heuristic can be found) to provide ideas for what a good conjecture might be. This topic is somewhat beside the main issue of our survey, but since random-cluster representations have played a key role in simulation algorithms for more than a decade we feel that it is appropriate to describe some of these algorithms. In fact, it was the need of efficient simulation which, in the late 1980s, sparked the revival of the randomcluster model (Swendsen and Wang, 1987) which up to then had raised only little interest since its introduction by Fortuin and Kasteleyn in the early 1970s. Consider for instance the Ising model with free boundary condition on a large cubic region A C Z d. Direct sampling from the Gibbs distribution ~h,/3,A with
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free boundary condition is not feasible, due to the huge cardinality of the state space S2, and the (related) intractability of computing the normalizing constant for the Gibbs measure. The most widely used way to handle this problem is the Markov chain Monte Carlo method, which dates back to the paper by Metropolis et aL (1953). The idea is to define an ergodic Markov chain having as unique stationary distribution the target distribution #h,~,A. Starting the chain in an arbitrary state and running the chain for long enough will then produce an output with a distribution close to the target distribution. An example of such a chain is the single-site heat bath algorithm, whose evolution is as follows. At each integer time, a vertex x ~ A is chosen at random, and the spin at x is replaced by a new value according to the conditional distribution (under #h,~,A) of the spin at x given the spins at its neighbors. It is immediate that #h,~,A is stationary for this chain, and ergodicity of the chain follows from elementary Markov chain theory upon checking that it is aperiodic and irreducible. The problem with this approach is that the time taken to come close to equilibrium may be very long. For example, let h - 0. Then, for fi < fie (with fie defined as in Theorem 3.1), the time taken to come within a fixed small variational distance from the target distribution grows only like n log n in the size of the system (here n is the number of vertices in A) whereas in contrast the time grows (stretched) exponentially in the size of the system for/3 > fie; see Martinelli and Olivieri (1994) and Martinelli (1999). This means that simulation using this heat bath algorithm is computationally feasible even for fairly large systems provided that/3 < fi~, but not for fi > /~c. What happens for/~ > /3c is that if the chain starts in a configuration dominated by plus spins, then the plus spins continue to dominate for an astronomical amount of time, and similarly for starting configurations dominated by minus spins. The set of configurations where the fraction of plus spins is around 1/2 (rather than around the fractions predicted by the magnetization in the infinite-volume Gibbs measures # ; and # ~ ) has small probability and thus can be seen as a "bottleneck" in the state space, slowing down the convergence rate. A way to tackle the exceedingly slow convergence rate in the phase coexistence regime is to use the heat bath algorithm for the corresponding randomcluster model rather than for the Ising model itself, and only in the end go over to the Ising model by the random mapping described in Corollary 6.3. This has the disadvantage that the calculation of single-site (or, rather, single-edge) conditional probabilities become computationally more complicated due to the possible dependence on edges arbitrarily far away (see Lemma 6.6). This disadvantage, however, seems to be by far outweighed by the fact that the convergence rate of the Markov chain (for fi > tic) appears to be very much faster than for the heat bath applied directly to the spin variables. The reason for this phenomenon is that the random-cluster representation "doesn't see any difference" between the plus state and the minus state. This approach can, of course, be used also for the q > 3
1 Random geometry of equilibrium phases
69
Potts model, and is due to Sweeny (1983). Later, Propp and Wilson (1996) built on this approach by coupling several such Markov chains (i.e. running them in parallel) in an ingenious way, producing an algorithm which runs for a random amount of time (determined by the algorithm itself) and then outputs a state which has exactly the target distribution. The running time of this algorithm turns out (from experiments) to be moderate except for the case of large q and fi close to the critical value. The Propp-Wilson approach, known as exact or perfect simulation, has received a vast amount of attention among statisticians during the last few years (see the annotated bibliography (Wilson, 1998)) and we believe that it also has interesting potential in physics. There is, however, another Markov chain which appears to converge even faster than those of Sweeny, Propp and Wilson. We are talking about the Swendsen-Wang (1987) algorithm, which runs as follows for Ising and Potts models on a graph with vertex set /2 and edge set /3: starting with a spin configuration X0 6 {1 . . . . . q}s a bond configuration Y0 6 {0, 1}t3 is chosen according to the random mapping defined in Corollary 6.4. Then another spin configuration X1 is produced from Y0 by assigning random spins to the connected components, i.e. by the random mapping of Corollary 6.3. This procedure is then iterated, producing a new edge configuration I11 and a new spin configuration X2, etc. By combining the two corollaries, we see that if X0 is chosen according to the target distribution, then the same holds for X1, and consequently for X2, X3 . . . . . In other words, the target distribution is stationary for the chain {Xk}~=0, and by the (easily verified) ergodicity of the chain we have a valid Markov chain Monte Carlo algorithm. Although it is not exact in the sense of the Propp-Wilson algorithm, it appears to converge much faster, thus in practice allowing simulation of systems that are orders of magnitude larger. Heuristically, the reason for this faster convergence is that large chunks of spins may flip simultaneously, allowing the chain to tunnel through any bottlenecks in the target distribution. However, rigorous upper and lower bounds on the time taken to come close to equilibrium are to a large extent lacking, although Li and Sokal (1989) have provided a lower bound demonstrating the phenomenon of "critical slowing down" as/~ approaches fie. The Swendsen-Wang algorithm has, since its introduction in 1987, become the standard approach to simulating Ising and Potts models. Interesting variants and modifications of this algorithm have been developed by Wolff (1989) and Machta et al. (1995); the last paper is an interesting attempt at combining the original approach of Swendsen and Wang with ideas from so-called invasion percolation (see Chayes et al., 1985) to get an algorithm specifically aimed at sampling from a Gibbs distribution at the critical inverse temperature fie, i.e. where the use of other algorithms have proved to be most difficult. Generalizations of the Swendsen-Wang algorithm for various models other than Ising and Potts models have also been obtained, see e.g. Campbell and Chayes (1998),
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H.-O. Georgii et aL
Chayes and Machta (1997, 1998), and H~iggstr6m et al. (1999).
6.7
Random-cluster representation of the Widom-Rowlinson model
The random-cluster model can be seen as a perturbation of Bernoulli bond percolation, where the probability measure is changed in favour of configurations with many (for q > 1) or few (for q < 1) connected components. A fairly natural question is what happens if we perturb Bernoulli site percolation in the same way. For lack of an established name, we call the resulting model the site-randomcluster model. Let G be a finite graph with vertex set/2 and edge set B. For a site configuration 7/6 {0, 1}s we write k(q) for the number of connected components in the subgraph of G obtained by deleting all vertices x with r/(x) - 0 and their incident edges. 6 f o r G with parameters Definition 6.21 The site-random-cluster measure grP,q
p 6 [0, 1] and q > 0 is the probability measure on {0, 1}z; which to each q {0, 1 } s assigns probability
where Zp,q a is a normalizing constant.
Analogously to the usual random-cluster model living on bonds, taking q = 1 gives the ordinary Bernoulli site percolation Op, while other choices of q lead to dependence between vertices. Taking q -- 2 is of particular interest because it leads to a representation of the Widom-Rowlinson model which is similar to (and slightly simpler than) the usual random-cluster representation of the Ising model. Let #G be the Gibbs measure for the Widom-Rowlinson model with activity ~. on G, i.e. # ~ is the probability measure on { - 1, 0, + 1}z; which to each ~ ~ { - 1, 0, + 1}z; assigns probability proportional to
H I{~(x)~(y)#-l} I-I ~l~(x)l. (xy)~13 xcs The following analogues of Corollaries 6.3 and 6.4 are trivial to check. Proposition 6.22 Let p -
x and suppose we pick a random spin configurai-4-2'
tion X ~ { - 1, O, + 1 }s as follows.
(1) Pick Y ~ {0, l} z; according
to ~rpG2.
(2) Set X ( x ) = Of o r each x ~ E such that Y ( x ) - O.
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Random geometry of equilibrium phases
71
(3) For each open cluster C of Y, flip a fair coin to decide whether to give spin + 1 or - 1 in X to all vertices of C. Then X is distributed according to the Widom-Rowlinson Gibbs measure # G. )~
Proposition 6.23 Let p = V4-~' and suppose we pick a random spin configuration Y E {0, l} s as follows. (1) Pick X ~ { - 1, 0, + 1}s according to #G.
(2) Set Y ( x ) -- I X ( x ) l f o r each x E •. Then Y is distributed according to the site-random-cluster m e a s u r e
~pG2.
We remark that for q 6 {3, 4 . . . . }, these results extend in the obvious way to a connection between ~pGq and the generalized Widom-Rowlinson model whith q types of particles rather than just 2 (and strict repulsion between all particles of different type). Many of the arguments applied to Ising and Potts models in Section 6.3 can now be applied to the Widom-Rowlinson model in a similar manner. To apply Theorem 4.8, we need to calculate the conditional probability in the site-randomcluster model that a given vertex is open given the status of all other vertices. For x ~ s and 77 ~ {0, 1}s we get pql-x(x,o)
gt 6 (x is open It/) P'q
-
pq
1-K(x,r/)
+ 1
_
p
(6.7.1)
where x ( x , ~) is the number of open clusters of 0 which intersect x's neighborhood {y 6 /2 9 y ~ x}. If the degree of the vertices in G is bounded by N, say, then 0 < x ( x , rl) < N for any x 6 s and 17 6 {0, 1}s For fixed q and any p* 6 (0, 1) , we can thus apply Theorem 4.8 to show that ~G p , q stochastically dominates ~p, for p sufficiently close to 1, and is dominated by ~pp, for p small enough. The arguments of Section 6.3 leading to a proof of Theorem 3.1, with the random-cluster model replaced by the site-random-cluster model, therefore go through to show Theorem 3.4. One thing that does not go through in this context, however, is the analogue of (6.3.2). The reason for this is that, in contrast to (6.2.1), the conditional probability in (6.7.1) fails to be increasing in ~, so that Theorem 4.8 is not applicable for comparison between site-random-cluster measures with different values of p. In fact, the analogue of (6.3.2) for site-random-cluster measures sometimes fails, and moreover the occurrence of phase transition for the Widom-Rowlinson model on certain graphs fails to be increasing in )~, as demonstrated by Brightwell et al. (1999).
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al.
Another consequence of the failure of the conditional probability in (6.7.1) to be increasing is that the FKG inequality (Theorem 4.11) cannot be applied to ~rpGq. AS a consequence, the proof of Theorem 6.10 cannot be adapted to the case of the multitype (q > 3) Widom-Rowlinson model. Such a WidomRowlinson analogue of Theorem 6.10 is known to be false, as shown by Runnels and Lebowitz (1974); see also Chayes et al. (1997) and Nielaba and Lebowitz (1997).
7
Uniqueness and exponential mixing from non-percolation
In Section 6 we saw examples where phase transition in one system was equivalent to the existence of infinite clusters in another, suitably defined, system. In this section we shall discuss various approaches where conclusions about the phase transition behavior can only be drawn from non-existence (and not from existence) of infinite clusters. On the other hand, these approaches typically apply to a much wider range of models. We address two problems: the uniqueness of the Gibbs measure, and the decay of correlations for a given Gibbs measure. The general theme of this section can be stated as: to what extent can a given spin be influenced by a configuration far away? If such an influence disappears in the limit of infinite distance, it follows (depending on the setting) that either there is no long-range influence of boundary conditions at all (implying uniqueness of the Gibbs measure), or that a specific low temperature phase exhibits some mixing properties. In both cases, the decreasing influence comes from the absence of infinite clusters of suitable type which could transport a dependence between spins. So, both uniqueness and mixing will appear here as a consequence of non-percolation. First we will address the problem of uniqueness. We will encounter conditions which not only imply the uniqueness of the Gibbs measure, but also lead us into a regime where "all good things" happen, i.e., where the unique Gibbs measure exhibits nice exponential mixing properties and the free energy depends analytically on all relevant parameters. (In general, the uniqueness of the Gibbs measure does not imply the absence of other critical phenomena, which might manifest themselves as singularities of the free energy or other thermodynamic quantities. For example, in Section 9 we will see that in the so-called Griffiths regime of a disordered system there is a unique Gibbs measure, but the free energy is not analytic.) The "nice regime" above is usually referred to as the high temperature, or weak coupling, low density, or also analytic regime, and is usually studied by high temperature cluster expansions. Dobrushin and Shlosman (1985b, !987) developed a beautiful and general theory describing a regime of "complete analyticity" by various equivalent properties. One of these ranks at the top of a hierarchy
1 Random geometry of equilibrium phases
73
of mixing properties. While complete analyticity makes precise what actually the "nice regime" is, and applies mainly to high temperatures or large external fields, it is not limited to this case (van Enter et al., 1997). The relationships between this and related notions and also with dynamical properties have been studied in many papers. Although some of these have an explicit geometric flavor, we do not discuss them here because of limitations of space. We rather refer to the sources (Dobrushin and Shlosman, 1985b, 1987; Stroock and Zegarlinski, 1992; Martinelli and Olivieri, 1994; Martinelli, 1999) and also to the references following condition (9.2.2). In Section 7.3 we shall discuss an application of the percolation method to the low temperature regime, and see how percolation estimates for the covariance between two distant observables, combined with contour estimates, give rise to exponential mixing properties.
7.1 Disagreementpaths Let (/2, ~) be an arbitrary locally finite graph, and suppose we are given a neighbor interaction U : S x S --+ R N {cx~}and a self-potential V : S ~ R. Consider the associated Gibbs distributions/z~,Ao introduced in (2.6.2). More generally, we could consider an arbitrary Markov specification (GA)Aes in the sense of Section 2.6. Such specifications appear, in particular, if we have an interaction of finite range R, say on Z d, and draw edges between all sites of distance at most R. However, for definiteness and simplicity we stick to the setting described by the Hamiltonian (2.5.1). We will often consider the inverse temperature ~ as fixed and then simply write/z~ instead of/z~,A-" If A is a singleton, we use the shorthand x for {x}. We look for a condition implying that there is only one Gibbs measure/z for the Hamiltonian (2.5.1), i.e., a unique probability measure on fla = S s satisfying # (. IX -= r/off A) = # ~
for/z-almost all r/ E fla.
Since this property needs only to be checked for singletons A = {x} (cf. Theorem 1.33 of Georgii (1988)), it is sufficient to look for conditions on the singlespin Gibbs distributions #x~ with x 6 s Intuitively, we want to express that /z~x(X(x) = a) depends only weakly on 77 (which can be expected to hold for small fl). This dependence can be measured by the maximal variation Px = max II~x~ r/,r/tEf2
-
~ x ~'
IIx,
(7.1.1)
where
IIvlIA =
sup Iv(A)l A~rA
(7.1.2)
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H.-O. Georgii e t al.
is the total variation norm on the sub-o--algebra 3cA of events which depend only on the spins in A. We write p as a shorthand for the family (Px)x~Z;. Given two configurations ~, ~' 6 f2, a path in s will be called a path of disagreement (for ~ and ~,) if ~(x) r ~'(x) for all its vertices x. For each finite region A C 12 and any two configurations r;, r / o n A c we will construct a coupling P of/z~x and # ~ describing the difference of these measures in terms of paths of disagreement running from the boundary 0A into the interior of A. Intuitively, these paths of disagreement indicate the range of influence of the r boundary conditions. We write {A < ~ 0A} for the event in S A • S A that there exists a path of disagreement from some point of a set A C A to some point of 0A. Although the coupling P to be constructed is not best suited for direct use, it has a useful special feature: its disagreement distribution is stochastically dominated by a Bernoulli measure. This will allow us to conclude that absence of percolation for the latter implies uniqueness of the Gibbs measure for the Hamiltonian (2.5.1). We write gro for the Bernoulli measure on {0, 1}s with 7zp(X(x) - 1) = Px for all x E s and 7zp,A for the analogous product measure on {0, 1}A. As in Section 4, we use the notation X (x) and X' (x) for the projections from S2 x f2 to S. The following theorem is due to van den van den Berg and Maes (1994). Theorem 7.1 For each finite A C 17, and each pair rl, rl' E ~2 there exists a
coupling P = PA,o,o' ~ lz~ and lz~ having the following properties: (i) For eachx c A , {X(x) C X t ( x ) } - - { x
# < ~ OA}P-a.s.
(ii) For the distribution "~ of (l{(x(x)#X'(x)})x~A under P, we have P 2 Z3p 1/fp, A 9
(iii) For each A C A,
I I ~ - ~A77! IIzx~ P ( A ~
0A) < ~ p ( A + - > - o q A ) .
(7.1.3)
Proof: We construct a coupling (X, X') of # ~ and # ~ by the following algorithm. In a preparatory step we introduce an arbitrary linear ordering on A , set A = A, and define X ( x ) = rl(x), X'(x) = r/(x) for x ~ A c. For fixing the main iteration step, suppose that (X, X') is already defined on the complement of a non-empty set A C A and is realized as a pair (~, ~,) off A, where (~, ~') = (r;, r/t) off A. Consider the Gibbs distributions # ~ and # ~ obtained by conditioning # ~ and # ~ on X = ~ resp. ~t off A. If ~ = ~, on OA then # ~ -
# ~ on ~A by the Markov property, so that we can take the
1
Random geometry of equilibrium phases
75
obvious optimal coupling for which X -= X' on A, and we are done. Otherwise we pick the smallest vertex x = x(~, ~') 6 A for which there exists some vertex y 6 A c with y ~ x and ~(y) # ~'(y). Consider the single vertex distributions /z~, x -- # ~ (X (x) -- .) and #~,x - # ~ (X (x) -- .) on S. Conditionally on (X, X') = (~, ~') off A, we then let (X (x), X' (x)) be distributed according to an optimal coupling (as in Definition 4.3) of #~,x and #~,x" The coupling (X, X') is then defined on the set x U A c, so that we can replace A by A \ x and repeat the preceding iteration step. It is clear that the algorithm above stops after finitely many iterations and gives us a coupling of #~x and #7(" Property (i) is evident from the construction, since disagreement at a vertex is only possible if a path of disagreement leads from this vertex to the boundary. For (ii), we note that the measures #~,x and #~,x are mixtures of the Gibbs distributions #x~ with suitable boundary conditions or, by the consistency of Gibbs distributions. Hence
By construction, this means that in each iteration of the main step we have
P ( X ( x ) r X ' ( x ) l ( X , x ' ) - (~, ~')off A) _< p~ for x -- x(~, ~'), so that (ii) follows by induction. Finally, (iii) follows directly from (i) and (ii) because for each A C A 7] ! II#~x- ~AI A ~ P(X(x) # X'(x)for some x c A)
by the coupling inequality (4.1.4). The proof is therefore complete. Although the algorithm in the proof above is quite explicit, it is not easy to deal with directly. In particular, it is not clear in which way the coupling depends on the chosen ordering, because the site x to be selected in each step depends on (~, ~') and is therefore random. Nevertheless, if the Gibbs distributions are monotone (in the sense of Definition 4.9), we get some extra properties. R e m a r k Suppose S is linearly ordered and the conditional distributions #~x are stochastically increasing in ~. Then, if 77 _ r/', the coupling P of Theorem 7.1 can be chosen in such a way that, in addition to properties (i) to (iii), X _~ X' P-a.s. and, for each x c A, , r (7.1.4) This is because in each step of the algorithm proving Theorem 7.1 we can achieve X (x) _< X' (x), and for the second inequality in (7.1.4) it is sufficient to note that
P(x
#-
.~ O A ) -
P(X(x) < X'(x)) _< Z acS\{m}
[P(X(x) <_a) - P(X'(x) <_a)],
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H.-O. Georgii e t al.
where m is the maximal element of S. For details see van den Berg and Maes (1994). In particular, for ISI = 2 we have equality in (7.1.4). Let us apply this remark to the ferromagnetic Ising model with external field h -- 0 and any inverse temperature fl, with boundary conditions r/ -- + 1 and r/ _--- - 1 outside of some finite region A 6 g. Then, by the spin flip symmetry and stochastic monotonicity, .7.A(X(x))
= .J.A(X(x)
= I) -- .7.A(X(x)
= I) = II"~.A -- "~.A II~
and therefore, by (7.1.4), #
# - ~ , A ( X ( x ) ) - P(x < > 0A). We emphasize that this relation is completely similar to what we obtained for the random-cluster representation, viz.
#~-,A(X(x)) -- q~I,2,A(X+->0A) for p = 1 - e-2r cf. (6.3.3). The coupling P, however, is less explicit, and the geometric event involves site percolation rather than bond percolation as for the random-cluster measure, but the exact correspondence between the magnetization for the spin system and the percolation probability of the geometric system is the same. Let us now turn to the main result of this subsection, the uniqueness theorem. Let # , / x ' be any two Gibbs measures for the Hamiltonian (2.5.1) at some inverse temperature ft. Inequality (7.1.3) then shows that
r/f Iltt- lz'llA _< sup II#,~- #A IIA _< ~pp(A<->OA) r/,r/ef2 whenever A C A 6 g. Letting A 1" L; we find
Ilu- ~'IIA ~ ~ p ( A o ~ ) which gives the following uniqueness result. Theorem 7.2 If ~p (:t an infinite open cluster ) = 0 then the set ~(fl H) of Gibbs measures for the Hamiltonian (2.5.1) at inverse temperature fl is a singleton. In particular, this holds if SUpx Px < Pc, the critical density for Bernoulli site percolation on (s ~). A weaker version of Theorem 7.2 was obtained first by van den Berg (1993) using a product coupling instead of Theorem 7.1; see Proposition 7.10 below and
1 Random geometry of equilibrium phases
77
also van den Berg and Steif (1994). In some cases, the simple product coupling nevertheless gives equivalent results; cf. the discussion in van den Berg and Maes (1994). For a large class of regular graphs such as Z a, the assumption of Theorem 7.2 not only implies the uniqueness of the Gibbs measure but even yields certain exponential mixing properties. This can be seen almost immediately by combining inequality (7.1.3) with Theorem 5.6 on the exponential tail of the distribution of the cluster diameter in sub-critical Bernoulli percolation. We will use similar arguments in Section 9.2 in the context of random interactions. Let us discuss now some special cases. Clearly, the conditions of Theorem 7.2 hold when 12 -- Z with the usual graph structure, since then Pc = 1. This gives uniqueness of the Gibbs measure for one-dimensional nearest-neighbor systems. Next we consider the case s - Z d, d > 2. Recall the bound (5.1.4) for the percolation threshold pc when d - 2, and the large-dimensions asymptotics of Pc in (5.1.5). E x a m p l e 7.3 The Isingferromagnet. Let fl > 0 be any inverse temperature and h an external field. Then, for any x, we obtain from (4.3.1) by a short computation + - # h , ~,x Ilx = [ t a n h ( f l ( h + 2 d ) ) - t a n h ( / ~ ( h - 2 d ) ) ] / 2 . Px = II#h,~,x
Hence, the Gibbs measure is unique when h - 0 and tanh(2dfl) < pc, or if Ih[ > 2d is so large that 2d < pc cosh2(fi(Ihl - 2d)), for example. E x a m p l e 7.4 The hard-core lattice gas. Setting fl = 1, we see that Px = )~/(1 + k) for any x, so that uniqueness of the Gibbs measure follows for k < p c / ( 1 - p c ) . (This can also be obtained by using the product coupling mentioned above, cf. van den Berg and Steif (1994).) E x a m p l e 7.5 The Widom-Rowlinson lattice gas. We take again fl = 1 and set ~+ = ~._ = )~. It turns out that the maximum in equation (7.1.1) is attained for the boundary conditions r / = 0 and r / e q u a l to + 1 and - 1 on (at least) two different neighbors of x, whence px = 2~/(1 + 2~) for any x. It follows that the Gibbs measure is unique when ~. < pc~(2(1 - Pc)). It is interesting to compare the uniqueness condition of Theorem 7.2 with the celebrated Dobrushin uniqueness condition, cf. Georgii (1988)and the original papers (Dobrushin, 1968b, 1972). This condition reads sup~ x
y
max
q ~ r f offy
II~x~ -~x~'lix < 1.
(7.1.5)
The constraint "r/ = r/' off y" means that the configurations r/, r/' differ only at the vertex y. For systems with hard-core exclusion or in certain antiferromagnetic
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models it often happens that, for every y ~ Ox, the maximum in (7.1.5) is actually the same as that in (7.1.1), see van den Berg and Maes (1994). Dobrushin's uniqueness condition then takes the form SUPx IOxlPx < 1. For L -- Z d and px = Po independently of x, this means that P0 < 1/(2d), while Theorem 7.2 only requires P0 < Pc, and it is known that Pc > 1 / ( 2 d - 1) f o r d > 1. However, if the constrained maximum in Dobrushin's condition is much smaller than the unconstrained maximum in (7.1.1), then Dobrushin's condition will be weaker than that of Theorem 7.2. For example, for the Ising ferromagnet on Z d with external field h - 0, Dobrushin's condition requires that 2d tanh/5 < 1 which, in view of (5.1.5), is less restrictive than the condition obtained in Example 7.3. Thus, roughly speaking, Theorem 7.2 works best for "constrained" systems with strong repulsive interactions and low-dimensional lattices (or graphs with small 10xls) for which reasonable lower bounds of the critical probability Pc are available. Examples are the hard-core lattice gas and the Widom-Rowlinson lattice gas on Z 2 considered above. There is also another reason why Theorem 7.2 is useful. Namely, its condition of non-percolation is a global condition: the absence of percolation does not depend on the value of px at any single site x. In particular, Px could be large or even be equal to 1 for all xs in an infinite subset (say, a periodic sublattice) of L; once the pxS are sufficiently small on the complementary set, there is still no infinite open cluster. This can be applied to non-translation invariant interactions where, in general, it is impossible to obtain uniform small bounds on the pxS (or on the strength of the interaction, as would be required by the Dobrushin condition or for some standard cluster-expansion argument). We will return to this point in Section 7.3.
7.2
Stochastic domination by random-cluster measures
Recently, Alexander and Chayes (1997) introduced a variant of the randomcluster technique that applies to a substantially greater class of systems than those considered in Section 6. This approach involves a so-called graphical representation of the original system. The graphical representation is stochastically dominated by a random-cluster model, and absence of infinite clusters in this random-cluster model implies uniqueness of the Gibbs measure for the original spin system. The price to pay for the greater generality is that the implication goes only one way: percolation in the random-cluster model does not, in general, imply non-uniqueness of Gibbs measures. We assume that the state space S is a finite group with unit element 1; the inverse element of a 6 S is denoted by a - l , so that a - l a - aa -1 - 1. For simplicity we assume that the underlying graph is/2 - Z d (although this will not really matter). We consider the Hamiltonian (2.5.1) for a pair potential U and with
1
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no self-energy, V = 0. By adding some constant to U (which does not change the relative Hamiltonian) we can arrange that U _< 0. The basic assumption is that U is left-invariant, so that U(a, b) - u ( a - l b ) (7.2.1) for all a, b E S and the even function u - U(1, .) < 0. Note that this setting includes the q-state Potts model for which S = Zq and u - -21{0}. For any finite A C Z d we consider the Gibbs distribution
,)
(xy)EBA
at inverse temperature fl with boundary condition r] E ~. Here we write ~A for the set of all bonds b E 13 with at least one endpoint in A. The graphical representation of #~,A will be based on bond configurations co E {0, 1 }t~A. Each such co will also be viewed as a subset of BA, and the bonds in co will be called open. The key idea of this representation is taken from the classical high temperature expansion. For fixed fl > 0 and any a e S we introduce the difference
Ra -- e -fu(a) - 1 >_ 0.
(7.2.2)
With this notation we can write
#f,A (or)
-
l{~=n offA} V I (1 + Ra(x)-l~r(y)) ZA(fl,/7) (xy)E/3h
Z I-I R(r(x)-'cr(Y)" l{a-O off A} ZA(fl, 77) o~e{o,1}~A(zy)e~o This shows that #r
is the first marginal distribution of a probability measure
Pfl, A on I2 • {0, 1 }t3A, namely
P;,A ((7, co) : l{a-n off A} V I R~(x) 1 ZA (fl, 17) (xy)~w - (~(Y)' cr E s
co E {0, 1 }t3A. The second marginal distribution of P;,A is equal to
•
(OJ) -- W~, A (~o)
ZA (f, n),
where W~, A(co) =
Z
I-I Rcr(x)-'cr(y)
o'--1/ off A (xy)Ew
(7.2.3)
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is the "graphical weight" of any co c {0, 1}I3A. The probability measure Y~O,A on {0, 1}I3A is called the graphical distribution or the grey measure (since it ignores the spins which are considered as colors). The graphical representation of #~,A thus obtained is analoguous to the random-cluster representation of the Potts model and can be summarized as follows. L e m m a 7.6 In the set-up described above, the Gibbs distribution lz3, o A can be derived from the graphical distribution Yfl,A by means of the conditional probabilities
P;,A ((9"[co) = W~,A(o9)-1
H
Rcr(x)-lcr(Y)"
b=<xy>~w That is,
fl,A oJ~{O,1}~A For the Potts interaction u = -21{0} with state space S -- Zq, the graphical representation above is easily seen to coincide with the random-cluster representation studied in Section 6. One important feature is that the graphical weights factorize into cluster terms. Indeed, each bond configuration co divides A into connected components called open clusters (which may possibly consist of isolated sites). The set of bonds belonging to an open cluster C is denoted by wc. Writing C(co) for the set of all open clusters we then obtain
W~,A (co) =
H W~,A (C, coc) CcC(co)
(7.2.4)
with
~'r;, A(C' coC) -
Z
H
cr~SC:a=-rl onCAOA (xy)~wc
ecr(x)-l~r(Y)"
(We make the usual convention that the empty product is equal to 1; hence W~,A (C, ~oc) - ISI if C is an isolated site.) Together with Lemma 7.6, equation (7.2.4) shows that the spins belonging to disjoint open clusters are conditionally independent. In particular, we can simulate the spin system by first drawing a bond configuration co with weights (7.2.3) and then obtain in each open cluster a spin configuration according to Pfl,A (or Io9). Suppose we knew that there is no percolation in the graphical representation, in the sense that max0 Y;,A (0 ++ 0A) ~ 0 as A 1' s The conditional independence of spins in different open clusters would then suggest that there is only one Gibbs measure for the spin system. Unfortunately, this is not known (though weaker statements have been established by Chayes and Machta (1997)). However, one can make a stochastic comparison of the graphical distributions
1 Random geometry of equilibrium phases
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with wired random-cluster distributions (Lemma 7.7 below), and the absence of percolation in the dominating random-cluster distribution will then guarantee that the original system has a unique Gibbs measure. This will be achieved in Theorem 7.8 allowing to bound the dependence on boundary conditions in terms of the connectivity probability in a random-cluster model. To this end we also need to consider Gibbs distributions #f,A
with free bound-
ary condition. These admit similar graphical representations Y~,A based on bond configurations inside A; that is, the bonds leading from A to A c are removed. In the following, the superscript f will refer to this case. The stochastic comparison with random-cluster distributions will be formulated using R * = m aa6S xRa,
~ _ IS--1[ E
Ra,
p
-- R* /(1 + R* ),
q =
R*/[~. (7.2.5)
aES Note that these quantities depend on/3 since the Ra in (7.2.2) do. In the case of the r-state Potts model when u -- -2I{0}, we have R* -- 1 - e -2/~ and q -- r; that is, in this case the parameters p and q are nothing but the standard parameters of the random-cluster representation. For p and q as above we consider now the wired (resp. free) random-cluster distribution r (resp. r in A as introduced in Section 6.2.
For any A ~ g, fl > 0 and p, q as above, Y;,A -<79 ~p,q,A1 YfA %79 ~O,q,A.
Lemma 7.7
and
Proof: We only prove the first statement since the second is similar and simpler. 1 According to Section 6.1, the weights of the random-cluster distribution ~p,q,A are proportional to ( k (P~
) 1~~ q
with Icol the number of open bonds and k (co, A) the number of open clusters meeting A (where all clusters touching 0 A are wired together into a single cluster). Up to a constant factor, the Radon-Nikodym density of Yfl,A relative to 4>~,q,A is thus given by
F(co) = W~,A (co)/(R*)l~
k(~
Since ~bl,q,A has positive correlations, the lemma will therefore be proved once
we have shown that F is a decreasing function of co. To this end we let co ___co' be such that col _ co t3 {b} for a bond b c BA \ co.
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We first consider the case when b -- (xy) is not connected to 0A and joins two open clusters Cx, Cy e C(co). For each open cluster C let W(C) = W~,6 (C, coc) be as in (7.2.4). Suppose we stipulate that the spin cr (z) at any site z E C is equal to some a 6 S. It is then easy to see that the remaining sum in the definition of W(C) does not depend on a and thus has the value W(C)/[S[. Prescribing the values of o" (x) and cr (y) in this way we thus find that ~ V ( C x (-J Cy (.) b) --
lfV(Cx)W(Cy)lS1-2
R~r(x)_lcr(y),
cr(x),cr (y) and therefore Wg, A (w') - / ~ W g , A (co). Since k(co', A) -- k(co, A) - 1 and
I~dl -
[co[ + 1, it follows that F (co') = F (co), proving the claim in the first case. If b links some cluster to the boundary which otherwise was separated from the boundary, then the argument above shows again that F (co~) = F (co). Next we consider the case when b = (xy) closes a loop in co but is still not connected to the boundary. Since clearly
Rcr(x)-lcr(y) < max Ra aES
-
-
R*,
we find that Wg, A (co') _< Wg, A (w)R*. On the other hand, in this case we have [w'[ = Icol + 1 and k(co', A) = k(co, A), so that F(co') < F(co). As we are considering the wired random-cluster measure, this argument remains valid if b joins two clusters already attached to the boundary. [] We are now in a position to state the main result of Alexander and Chayes (1997), an estimate on the dependence of Gibbs distributions on their boundary condition in terms of percolation in the wired random-cluster distribution. Recall the notation (7.1.2) for the total variation norm on the sub-~-algebra .T'A of events in some A. T h e o r e m 7.8 Consider the spin system with pair interaction (7.2.1) at some in-
verse temperature fl > O, and let p, q be given by (7.2.5). Then, for any A C A E C and any pair of boundary conditions rl, ~ E f2, /7
!
--
p,q,A ( A <--->0 A ) .
Proof: (This proof is different from the one that appeared in Alexander and Chayes (1997).) Let A be any event in .Yzx. From Lemma 7.6 we know that #~,A (A) -- Z (.o
Yfl,A (o9)Pfl, A (A[w).
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83
To control the 7-dependence of this probability we will proceed in analogy to the argument for the implication (ii) =r (iii) of Theorem 6.10. If o9 6 {A ~ 0A} then equation (7.2.4) shows that the conditional distribution Pfl,A (Alog) does not depend on 7. So we need to control the q-dependence of yC/,A(A /--> 0A). This, however, does not seem possible directly. So we will replace Y~,A by the 7independent ~blp,q,A by using a suitable coupling trick. By Lemma 7.7 and Strassen's theorem (Theorem 4.6) there exists a coupling (I?, I?') of ?'~,A and ~bl,q,A such that I? • I?' almost surely. If I?' 6 {A ~ 0 A }, there exists a largest (random) set F = F (I? ~) such that (a) A C F C A ,
and
(b) I?'(b) = 0 for all bonds connecting F with F c. For I?~ 6 {A +-> 0 A} we set F = 0. Conditional on F, Lemma 7.7 and Strassen's theorem provide us further with a coupling (I?v, I?{.) of Y f r and r176 such that ~l f'r • Yr'.
It is then easy to see that the pair of random variables (Y, Y~) defined
by (I?r,, I?I~)(b)
(Y, Y')(b) -
(f', f")(b)
if b is contained in F, otherwise
is still a coupling of ?'fl,A and 4~pl,q,A such that Y ~_ Y' almost surely. (Notice that also I? (b) = 0 for all bonds from F to Fc.) We denote the underlying probability measure by Qo. Now we can write #~,A(A)
-
Qo(P~,A(AIY))
1 0A). We The first term in the last sum is at most Q0(F - 0) - - ~p,q,A(t claim that the second term does not depend on 7. Indeed, if 1-' ~: 0 then, by only depends on the restriction YF of Y to the (7.2.4), p,7 /?,A (AIY) - Pfv(AIYr) ~_ set of bonds inside F. The second term can thus be written explicitly as
G#0
w in G
which is obviously independent of 7. The theorem now follows immediately.
[]
To apply the theorem we consider the limiting random-cluster measure 4)P,q 1 with arbitrary parameters p e]O, 1[ and q >_ 1 and wired boundary condition;
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recall from Section 6.2 that this limiting measure exists. By Corollary 6.7(d), it makes sense to define the percolation threshold Pc(q) -- inf{P " q~lp,q(O +-~ c~) > 0}.
We also consider the threshold Wc(q) for exponential decay of connectivities, which is defined as the supremum of all ps for which ~b;,q(O +-~ OA) < Ce -cd(O'OA) uniformly in A (or, at least, for A in a prescribed sequence increasing to/3) with suitable constants c > 0 and C < cx~. It is evident that We(q) < Pc(q); for large q it is known that we(q) = Pc(q) (van Enter et al., 1997). Theorem 7.8 then gives us the following conditions for high-temperature behavior; compare with Theorem 7.2. Corollary 7.9 Whenever ~ is so small that p < pc(q), there is a unique Gibbs measure f o r the Hamiltonian H with pair interaction (7.2.1). Furthermore, if in [act p < We(q) then the spin system is exponentially weak-mixing in the sense that there are positive constants C < cx~, c > 0 such that f o r all A C A E g and all boundary conditions q, rlt E f2 !
IIz~,A - Zz~,AIIA ~ ClOAle -cd(/x'Ac). In fact, Alexander and Chayes (1997) go a little further in their exploration of "nice" high temperature behavior, showing that for p < p c ( l ) the unique Gibbs measure satisfies the condition of "complete analyticity" (investigated by Dobrushin and Shlosman (1987), for example).
7.3
Exponential mixing at low temperatures
In the previous subsections we have seen how stochastic-geometric methods can be used to analyze the high temperature behavior of a spin system and, in particular, for establishing exponential decay of correlations. Here we want to demonstrate that similar percolation techniques can also be used in the low-temperature regime in the presence of phase transition. We will present a method to show that, for a given phase #, the covariance # ( f ; g) of any two local observables f and g decays exponentially fast with the distance between their dependence sets. (The problem of phase transition at low temperatures will be addressed in Section 8.) As a matter of fact, the problem of exponential decay of covariances (or truncated correlation functions) arises in many physical situations. Correlation functions are related to interesting response functions or to fluctuations of specific
1 Random geometry of equilibrium phases
85
order parameters. Exponential decay of covariances also provides estimates on higher order correlation functions, eventually providing infinite differentiability of the free energy with respect to an external field (von Dreifus et al., 1995). Motivated by these interests, a variety of techniques have been developed. The most familiar approach are cluster expansions which apply equally well to both the high-temperature (or low-density) regime and the low-temperature (or highdensity) regime. Although they often employ geometric concepts, it seems useful to combine them with ideas of percolation theory to make geometry more visible. An example of this is the method to be described below which is taken from a paper by Burton and Steif (1995), where it was used to show that certain Gibbs measures exhibit a powerful mixing property called "quite weak Bernoulli with exponential rate". We consider a spin system on an arbitrary graph (/2, ~) with Hamiltonian (2.5.1). As before, the essential feature of this Hamiltonian is that only adjacent spins interact, so that the Gibbs distributions #~,A in finite regions A have the Markov property. The inverse temperature/~ > 0 does not play any role for the moment, so we set it equal to 1 and drop it from our notation. Our starting point is the following estimate on the q-dependence in terms of disagreement paths for two independent copies of #~. This result is a weak version (and, in fact, a forerunner (van den Berg, 1993)) of Theorem 7.1. It is a pleasant surprise that although developed with high-temperature situations in mind, it also provides a useful alternative to some aspects of the standard lowtemperature expansions. P r o p o s i t i o n 7.10 For any A C A ~ 8 and q, ~ c f2,
,
,
r
For brevity let P - #~x x # ~ , and write X, X ~ for the two projections from f2 x f2 to f2. Then for any A E ~A we have
Proof:
Lt~A(A)- # ~ ( A ) -
P(X E A ) - P(X' E A).
We decompose the probabilities on the right-hand side into the two contribu# tions according to whether the event {A ~ ~ 0A} occurs or not. In the latter case, there exists a random set F C A containing A such that X -- X f on 01-'. (The union of all disagreement clusters in 3, meeting A is such a set.) Let F be the maximal random subset of A with this property. Then for each G the event {F = G} only depends on the configuration outside G, and X = X f on OG. The Markov property therefore implies that, conditionally on {F = G} and ! ! (X6 c, XGc), X6 and X 6 are independent and identically distributed, and this
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# r shows that P ( X c A, A 1t+ OA) -- P ( X ' c A, A y/+ OA). The proposition now follows immediately. [] What is gained with the disagreement estimate above? First, let us observe that this estimate provides bounds on covariances of local functions in terms of disagreement percolation. Corollary 7.11 Fix any A c g and rl ~ S2. Let f and g be any two local functions depending on the spins in two disjoint subsets A resp. A ~ of A. Then I # ~ ( f ; g)l < 3(f)a(g)#~a x # ~ ( A ", # , A' in A) where 6 ( f ) - max~ f (~) - min~ f (~) is the total oscillation of f .
Proof: By rescaling and addition of suitable constants we can assume that 0 < f < 1 and 0 < g _< 1. Proposition 7.10 then shows that
< ~(f)~(g)
f
#~(d~)
f
# ~ ( d ~ ' ) # ~ \ A, x #~\A,(A .-
'
> A in A)
,
because ~ - ~' - ~ on 0A. By carrying out the last integration we obtain the result. [] The bounds above leave us with the task of estimating the probability of disagreement paths in a duplicated system. In contrast to the situation in Section 7.1, we are looking now for estimates valid at low temperatures. If a cluster expansion works, there is no need to look any further. For instance, a low-temperature analysis and estimates of semi-invariants for the Ising model can be obtained using standard contour representations; see Dobrushin (1996). It needs to be emphasized, however, that the main step of cluster expansions consists in expanding the logarithm of the partition function. Only afterwards, by taking ratios of partition functions, does one obtain expressions for covariances and higher order correlation functions. Therefore, a point to appreciate is that the bound of Corollary 7.11 provides a direct geometric bound on covariances which avoids the machinery of cluster expansions and, in particular, the problems coming from taking logarithms. As a consequence, this estimate also applies to some cases where standard cluster expansions are doomed to fail. Alternative stochastic-geometric methods for estimating covariances in the absence of standard cluster algorithm arguments have been discussed in Bricmont and Kupiainen (1996, 1997).
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87
Let us illustrate this for the case of the Ising ferromagnet. This might not be the best example because other methods can also be applied to it. Nevertheless, it is useful to demonstrate the technique in this simple case. Afterwards we will discuss a case where cluster expansion techniques cannot be used equally easily. Consider the low-temperature plus phase #~- of the ferromagnetic Ising model on the square lattice 12 - Z 2 with zero magnetic field. By taking the infinite volume limit in Corollary 7.11 with r/=__ + 1 we obtain the estimate I # f ( f ; g)l _< 8 ( f ) 8 ( g ) # f x #~-(A ~
A')
(7.3.1)
for the covariance of any two local functions f, g with disjoint dependence sets A, A'. Next we observe that the event {A < r > A' } is clearly contained in the event that there exists a path from A to A f along which (X, X') ~ (+, +). The > A' }. Burton and Steif (1995) have shown latter event will be denoted by {A #<(+' +) how to estimate the probability of this event to occur in the duplicated plus phase. The result is the following.
Theorem 7.12 For the Ising ferromagnet on Z 2 at sufficiently large fi, there exist constants c > 0 and C < oc (depending on fi) such that for any two disjoint sets A, A' E g, < > A') < Cmin{]0iA], ]OiA'l}e -cd(A A'), #~- x #~-(A +(+'+)
where Oi A = O(A c) is the inner boundary of A. Combining this theorem with (7.3.1) we obtain an exponential bound for the covariance of any two local observables in the low-temperature Ising plus phase. While this result is well known, its proof below shows how one can proceed in more general cases.
Sketch proof of Theorem 7.12: It is sufficient to prove the statement with #~replaced by #~-,A, where A is a sufficiently large square box containing A, A'. For brevity, let PA -- #~,a + x #~-,a" Suppose that [0A'I -< IOAI , fix an arbitrary X E Oi At, and suppose that the event {x #<(+' +) > A} occurs. Let F0 -- F0(X, X') be the maximal connected set containing x on which (X, X') ~ (+, +). Also, let F be the union of F0 and all finite components of F0c. F is enclosing in the sense that both 1-' and 1-'c are connected. In fact, F can be identified with a contour, the broken line which separates F from its complement. Consider the set C of all enclosing sets containing x and contained in A. For any integer e > 2, let Ce be the set of all C 6 C such that IOiCI -- ~. Since the number of contours with length g surrounding a given site on the square lattice is bounded by g3 e and since each enclosing set with [OiC] - ~ uniquely defines a
88
H.-O. Georgii e t al.
contour with length between e and 4e, we have ICeI 5 3(4e + 1)/2(81) e growing exponentially with ~. Now we can write > A)-- PA(FAA #0) PA (x r <(+' +)
<
~
PA(F = C).
e>d(A,A') CECg
Furthermore, for C E Ce we have
PA(F = c)
_<
+
E ~fl,A (X _= - 1 on D, X -- + 1 o n 0 C ) DCOiC •
(X = - 1
on
OiC \ D, X =- +1
on OC). (7.3.2)
Now, a standard Peierls estimate (see Sinai (1982)) shows that #~-,A (X --= -- 1 on D, X = + 1 on OC) <_ C(fl)e -c(~)tDI
(7.3.3)
with constants c(fl), C(fl) independent of A satisfying c(fi) -+ cx) as/3 -+ e~. Substituting (7.3.3) into (7.3.2) we obtain the theorem by simple combinatorics and summations. [] We emphasize that the specific properties of the plus phase #~- are used only in the last step, the Peierls estimate (7.3.3). Before, we needed only the Markov property. Therefore it is useful to note that the Peierls estimate is not limited to the Ising model; it remains valid under the conditions of the standard Pirogov-Sinai theory (Sinai, 1982). In particular, it follows that the results of Burton and Steif (1995) on the ergodic properties of the Ising model carry over to more general Markovian models of Pirogov-Sinai type. Let us finally discuss a case in which a Peierls estimate of the form (7.3.3) is not available. Namely, we ask for covariance estimates of local functions, still for the ferromagnetic Ising model in a large square A, but now for some boundary condition ~ not identically equal to + 1. This question arises, for example, in the context of correlations atop of a disordered surface, or in the problem of establishing a Gibbsian description of non-Gibbsian measures. In fact, in Maes and Vande Velde (1997) the method to be described below is used to prove that the projection to a line of the low-temperature plus phase of the two-dimensional Ising model is weakly Gibbsian. To be specific, suppose that the boundary condition rl on OA is not identically plus but contains a large proportion of plus spins; we stipulate that rI = + 1 on three sides of A while on the remaining side only a large fraction of the spins is plus. In this case, (7.3.3) cannot be true because D can be small and close to the boundary of A. For example, if E = OD A 0A ~ 0 and ~7 happens to be minus on E then it is not very unlikely that all spins in D are minus, and (7.3.3) will not hold. However, one can take advantage of the fact that for small D its
1
Random geometry of equilibrium phases
89
complement in Oi C is large, and vice versa. In other words, to estimate the righthand side of (7.3.2) one should not apply (7.3.3) separately to each factor, but rather one can hope to estimate their product. To make these general remarks precise we consider the Ising model on the half-plane Z x Z+. For any n we consider the square An - - {(Xl, x2) E Z 2 : --n <_ xl < n, 0 < x2 < n} touching the boundary line Z x {0}. On this line we fix a configuration ~ E {+ 1, --1 }Z, thereby defining a boundary condition on one part of 0 A. On the remaining part of 0 A we impose a plus boundary condition. That is, we choose the boundary condition t/=_ +1 on OAn \ (Z x {0}) and 77 = on Z x {0}. We ask for the correlation of the spins at the sites x - (0, 1) and y - ( k , 1) w i t h 0 < Ikl < n . Theorem 7.13 In the situation just described, suppose ~ is such that m
E
-1
~ (j' O) > 8m/9
and
j--O
E
~ (j' O) > 8m/9
j =--m
for sufficiently large m, and let 71 be defined as above. Then there are constants c > 0 and C < r (not depending on n) such that [#~,A, (X(0, 1); X(k, 1))] _< Ce -clkl
whenever fi and [k] are sufficiently large and n > [kl. Sketch proof: We proceed as in the proof of Theorem 7.12. In dealing with the right-hand side of (7.3.2) we must take into account that possibly 0C n 0A ~: 0. We therefore replace OC by OC \ 0A in the product term and also estimate the probabilities of intersections by conditional probabilities, yielding the upper bound +'~ (X --=-1 on D)#~+'uc (X = - 1 o n OiC \ D) f~,8,c +,rl
for the summands on the right-hand side of (7.3.2). Here, #~,c stands for the Gibbs distribution in C with boundary condition equal to +1 on OC N A and equal to r/on OC n 0A. To derive the theorem we need to replace the Peierls estimate (7.3.3) by a similar bound on the last product. The exponential decay of correlations then again follows by simple combinatorics and summations. To make the influence of the boundary condition r/explicit we exploit a contour representation leading to the estimate +,rl(X ##,c
-~
-lonD)#~+'~(X=-lonOiC\D)
F , F t inside C F compatible with D F ! compatible with 0i C\D
y EF
y;EF ~
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H.-O. Georgii et al.
The right-hand side is defined as follows. For any configuration a E {+ 1, -1} a we draw horizontal resp. vertical lines of unit length between neighboring sites of opposite spins, as if the boundary spins were all plus. We then obtain a disjoint union of closed non-self-intersecting polygonal curves. Each of these curves is called a contour F, and a set F of contours arising in this way is called compatible. We thus have a one-to-one correspondence between spin configurations o- and compatible sets F of contours. If a -- - 1 on D, then each component of D is surrounded by some contour y (i.e., belongs to the interior Inty of F); the smallest contours surrounding the components of D are collected into a set F of contours. Each set F arising in this way is called compatible with D. The probability that a given set F of contours occurs is not larger than 1-I• w~(y), where
xEIntg
yEOA:y,~x
and [gl is the length of ?'; this can be seen by comparing the probability of a configuration containing 1-" with the probability of the configuration obtained by flipping the spins in U g c r Inty. These observations establish the inequality (7.3.4). Note that the weight w,7(y) of a contour y depends on the boundary configuration ~; this is because we have chosen to draw the contours for plus boundary conditions rather than 9. It follows that w,l(y) does not necessarily tend to zero when Igl grows to infinity; this is in contrast with the case ~7 = 4-1. The standard low-temperature expansion would therefore become much more complicated. However, if the density of plus spins in r/is sufficiently large, or if 0Intg AOA is rather small, the standard weight exp[-2fllg l] of the Ising contours will dominate, and the right-hand side of (7.3.4) can be estimated, as we will show now.
We unite the sets 1-', F' in (7.3.4) into a single set of contours [" = F U 1-". The contours in F can overlap, but a site of 0i C can only belong to the interior of at most two contours. On the other hand, every site of OiC is in the interior of at least one contour of 1-', and IOiC] > k, the distance of the two spins considered. These ingredients allow us to control the sum on the fight-hand side of (7.3.4). If k is so large that the density of plus spins in ~ between 0 and (k, 0) exceeds 8/9 then we find for any collection of contours [" = 1-" U 1-" as above Z
Z
~EI~ xEInt~
~ ( 1 - r/(y)) _< 5/9 Z I~[. yEOA:y"~x ~'EF
This yields
H wo(Y)H
yEF
y~EF ~
wo(y')<
H
VEF
exp[-8/9fllYl]
H
exp[-8/9fl]y']].
y~EF ~
1 Random geometry of equilibrium phases
91
At this point the standard arguments take over (with/~ replaced by 4/~/9), leading to an exponential estimate of (7.3.4). For example, one can conclude the proof along the lines of Lemma 2.5 of Burton and Steif (1995). []
8
Phase transition and percolation
Typically, two ends of the phase diagram are amenable to mathematical analysis. One is the high-temperature, or low-density, regime which was discussed in Section 7 and in which the system can be viewed as a small perturbation of an independent spin system. The other end is the low-temperature regime which we will now consider. At low temperatures, the energy dominates over the entropy which comes from the thermal fluctuations of the spins. One therefore expects that the spin configuration is typically similar to some frozen zero-temperature state, which is a configuration of minimal energy and thus called a ground state. The similarity of a low-temperature state with a ground state is conveniently described in geometric terms: one imagines that the spins which agree with the given ground state form an infinitely extended sea, whereas those spins which have chosen to deviate from the ground state are confined to interspersed finite islands. This is, of course, a picture of percolation theory: spins that agree with the ground state form a unique infinite cluster. We are thus led to the concept of agreement percolation, which will be discussed in the first part of this section. Agreement percolation is intimately related to the existence of a phase transition. If several distinct ground states exist, we may also hope to find, at low temperatures, several equilibrium phases which can be distinguished by agreement percolation with respect to the different ground states. One may ask further whether the geometric picture that applies to low temperatures remains valid throughout the whole non-uniqueness region. Physically, this is a matter of the stability of the ground states. Mathematically, it means looking for conditions under which distinct Gibbs measures allow distinct stochastic-geometric characterizations. We will approach this question from two different sides. In Sections 8.1 to 8.4 we investigate whether "phase transition implies percolation". We study a fixed equilibrium phase # in the non-uniqueness region which, by its very construction, can be viewed as a random perturbation of some ground state rl. We then will see that, in many cases, spins that agree with 7/ do percolate. After a general discussion of agreement percolation in Section 8.1, we investigate this concept in the subsequent subsections for some specific models including the Ising ferromagnet and the Potts model. In the case of the planar lattice Z 2 with its limited geometric possibilities we will also see that conversely, the absence of phase transition sometimes implies an absence of percolation, and that in the case of phase transition one has restrictions on the number of phases. (Methodologically,
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these results still run under the heading "phase transition implies percolation".) In Section 8.5 the converse will be treated more systematically and under a different aspect. We will show that at low temperatures one has percolation of bonds along which the interaction energy is minimal, and we will see that such a groundenergy bond percolation often implies a phase transition. Taken together, these results will show that in various models phase transition comes along with the existence of a ground-state sea with finite islands (deviation islands) on which the spins deviate from the ground state, and vice versa. A theory developing this picture in much more detail is that of Pirogov-Sinai (1976) which describes the low-temperature phase diagram in the presence of several stable ground states. One basic idea of this theory is to treat the finite deviation islands of a low-temperature system as the constituents of a low-density gas of hard-core particles. While the Pirogov-Sinai theory is intimately related to the subject of this section, it is much too involved to be developed here. There are, however, a number of expositions which may serve as general introductions and in which many additional references can be found. We mention only Sinai (1982); Zahradn~ (1984); Dobrushin and Shlosman (1985a); Slawny (1986); Z a h r a d n ~ (1987); Bricmont and Slawny (1989); van Enter et al. (1993); Koteck2~ (1994) and Zahradnik (1996). Here we will concentrate on more specific results which are partly beyond the Pirogov-Sinai theory, in that they are not limited to low temperatures but rather apply to the full non-uniqueness region, and comment occasionally on some relationships. In particular, the results of Section 8.5 are similar in spirit to this theory.
8.1 Agreement percolation from phase coexistence We consider again the general setting of Section 2. (12, ~ ) is an arbitrary locally finite graph, S is a finite set, and f2 - S s Suppose # is a random field and 0 E f2 a fixed configuration. We consider the event {x < ~ cx~} that x E s belongs to an infinite cluster of the random set R(0) = {y E s : X(y) = 0(Y)}, and we say that/z exhibits agreement percolation for O if # ( x < ~ cx~) > 0 for some x E 12. In short, we will then simply speak of 0-percolation. To visualize such an agreement, it may be convenient to think of a reduced description of # in terms of its image under the map s o 9f2 --+ {0, 1}s which describes local agreement and disagreement with 7, and is defined by (s~
1 0
if o'(x) - q(x), otherwise.
(8.1.1)
With this mapping, we can write {x < ~ cxz} - s o- 1{ x ~ } . We are interested here in the case when # is a Gibbs measure for the Hamiltonian (2.5.1), and 0 is an associated ground state. We say that a configuration
1 Random geometry of equilibrium phases
93
~ f2 is a ground state or, more explicitly, a ground state configuration for the relative Hamiltonian (2.5.2), if H(crlO) > 0 for any local modification (or "excitation") cr of r/. In other words, 77is a ground state if, for any region A ~ g, the configuration 7/minimizes the energy in A when rlac is fixed. One should note in this context that, in the low-temperature limit fi 1" oc, the finite-volume Gibbs distribution IX~,A from (2.6.2) tends to the equidistribution on the set of all configurations cr of minimal energy H(crlO). This suggests that, at least in some cases, the low-temperature phase diagram is only a slight deformation of the zero-temperature phase diagram describing the structure of ground states. This is precisely the subject of Pirogov-Sinai theory which provides sufficient conditions for this to hold, proposes a construction of low-temperature phases as perturbations of ground states, and also shows that the size distribution of the deviation islands has exponential decay. Suppose next that the Gibbs measure IX is related to the ground state ~ in some way. For example, IX might be obtained as the infinite volume limit of the finite volume Gibbs distributions IX~,A with boundary condition r/, possibly along some subsequence. (Under the conditions of the Pirogov-Sinai theory such a limit always exists.) In the case of a phase transition, when other phases than # exist and one is interested in characteristic properties of IX, one expects that the relationship between IX and ~ becomes manifest in a macroscopic pattern of the typical configurations, in that IX shows 0-percolation. In short, we ask for the validity of the hypothesis IG(C3H)I > 1, Ix is extremal in G ( f i H ) and related to a ground state ~ 6 S'2 > # ( x < > ec) > 0
Vx~/2.
(8.1.2)
In the specific cases considered below it will always be clear in what sense # and rl are related; typically, # will be a limiting Gibbs measure with boundary condition rl. We emphasize that (8.1.2) does not hold in general; a counter-example can be constructed by combining many independent copies of the Ising ferromagnet to a layered system, see the discussion after Proposition 8.3 below. Also, even when (8.1.2) holds, it does not necessarily imply that the phase IX is uniquely characterized by the property of 0-percolation. How can one establish (8.1.2)? In the context of the Ising model, Coniglio et al. (1976) and Russo (1979) developed a convenient citerion which is based on a multidimensional analog of the strong Markov property and thus can be used for general Markov random fields (Bricmont et al., 1987; Giacomin et al., 1995). One version is as follows. Theorem 8.1 Let (/2, ~) be a locally finite graph, IX a Markov field on f2 - S s and tl ~ f2 any configuration. Suppose there exist a constant c E R and a local function f 9s -+ R depending only on the configuration in a connected set A,
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H.-O. Georgii et aL
such that lz ( f ) > c but Ix(fiX
=~ ~ on OF) < c
(8.].3)
for all finite connected sets F ~ A and all ~ E f2 with so(~ ) =_ 0 on OF. Then /z(A <
77
~ o~) > 0, i.e., IZ exhibits agreement percolation for rl.
Proof: Suppose by contraposition that # ( A
<
r/
~ cx~) -- 0. For any e > 0
we can then choose some f i n i t e A D A such t h a t # ( A
<
/7
~ A
r
< e
.
For
r {A < ~ A c } , t h e r e e x i s t s a c o n n e c t e d s e t F such t h a t A C F C A and s0(~) ~ 0 on 0 F; we simply let F be the union of A and all 0-clusters meeting 0 A. As in the proof of Theorem 6.10, we let F (~) be the largest such set. For E {A < ~ A C } we put F(~) -- 0. Then, for each finite connected set F # 0, the event {~ 9F(~) - F} depends only on the configuration in A \ F, whence by the Markov property # ( f l F(.) = F) is an average of the conditional probabilities that appear in assumption (8.1.3). From this we obtain # ( f ) < c # ( F ( . ) :/= 0) + tz(lflI{r(.)=0}) < c + e Ilfll. Letting e -+ 0 we find # ( f )
< c, contradicting our assumption.
In most applications we will have a natural candidate for the function f . Whenever distinct phases do exist, they can be distinguished by some order parameter, viz. a local function f having different expectations for the two phases. If, in addition, some stochastic monotonicity is available then we can hope to establish (8.1.3). In fact, the percolation phenomena stated in Examples 5.115.15 can be deduced from a slight modification of Theorem 8.1. We will not go into the details of these examples which are treated in Bricmont et al. (1987), but rather apply Theorem 8.1 to our standard examples.
8.2
Plus-clusters for the Ising ferromagnet
The idea of agreement percolation was first developed in the context of the ferromagnetic Ising model (Coniglio et al., 1976; Russo, 1979). Let us apply Theorem 8.1 to this standard case. We only consider the case of no external field, i.e., we set h = 0, so the only parameter is the inverse temperature/3 > 0. We are interested in agreement percolation for the constant configurations 77 = + 1 resp. r / = - 1 , which are the only periodic ground states of the model. We write +
< ; resp. < ~- for the corresponding connectedness relation. Our first result shows that if there is a phase transition then there is plus-percolation for each
1 Random geometry of equilibrium phases
95
lo Gibbs measure except the minus-phase/x~, that is, assertion (8.1.2) holds for 77 --- + 1. This result (due to Russo (1979)) is valid for an arbitrary locally finite graph (/2, "~) with finite critical inverse temperature/3c.
Theorem 8.2 Let Iz be an arbitrary Gibbs measure f o r the ferromagnetic Ising +
model with parameters ~ > O, h - O . I f lz (= lz-~ then lz(x ~
~ c~) > Of o r all
+
x E s
In particular, ift~ > ~c then # ~ (x ~
~ c~) > O f or all x E s
Proof: By the sandwiching inequality (4.3.3) and Proposition 4.12, there exists a site x E /2 such that # ( X ( x ) - 1) > c - # - ~ ( X ( x ) - 1). On the other hand, the analogue of inequality (4.3.2) for the minus boundary condition shows that t z ( X ( x ) - 11X ~ --1 on o r ) - / Z ~ , r ( X ( x ) -- 1) <__c
for every finite 1-" ~ x. Theorem 8.1 thus gives the result for the x at hand. In view of the finite energy property of #, this extends easily to all other x E /2. [] Let us rephrase the last statement of Theorem 8.2 as follows: below the critical temperature the plus spins percolate in the plus phase and, by symmetry, the minus spins percolate in the minus phase #~. In the case of graphs with symmetry axes, this statement allows an interesting refinement. Proposition 8.3 Suppose (/2, ~ ) admits an involutive graph automorphism r which maps a subset 7-( C 12 onto its complement 7-{c, and that f o r x E 7-(, y E 7-[c either x 7c y or y = rx. For x E 7-[ let {x (+r+ (X) in 7-[} be the event that there exists an infinite path y in ~ starting from x such that all spins along both y and its reflection image r g are positive. I f lZ; 7L lZ-~ then l z f (x <+r+ CO in 7-[) > 0 for all x E 7-[.
One natural case to think of is when/2 -- Z a for d ___2, ~ a halfspace with boundary orthogonal to an axis, and r the associated reflection. The proposition then asserts that #~--almost surely there exists an infinite connected mirror-symmetric set of plus spins. Another interesting case is when s consists of two disjoint copies of a graph ~ which are not connected to each other by any bond. In this case, #~- - #~,~+ x #~,~,+ and the statement is that two independent realizations of #~,~+ exhibit simultaneous plus-percolation; in this case, the preceding proposition was observed by Giacomin et al. (1995). It is, however, not possible to take an arbitrarily large number k of independent realizations X1 . . . . . X~ of #~,~,+ at least when ~ has bounded degree N. For, if Pc is the Bernoulli site percolation threshold of 7-[ and k is so large that + / ( x (x) = 1)~ < pc sup u~,/x x ET-(
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H.-O. Georgii e t al.
then the set {x 6 7-g : X1 (x) . . . . . X k ( x ) = 1} does not percolate. This follows from a standard domination argument. Since the layered system consisting of k independent copies of the Ising model with I3 > tic certainly exhibits a phase transition, we see that hypothesis (8.1.2) does not hold in general.
Proof of Proposition 8.3: We identify each ~ ~ f2 with (~(x), ~(rX))x~7-t S 7-g, where S - {-1, 1 }2. The event under consideration then corresponds to r/percolation for the configuration 7/ 6 S 7-t with q(x) = (1, 1) for all x 6 7-/. Let f = X (x) + X (rx). Then #~- ( f ) -- 2/z~- (X (x)) > 0 by the r-invariance of/z~-. On the other hand, let F C 7-[ be a finite set containing x, and r" - F u r F. If (~, ~') ~ S ~ - f2 with s~(se, ~') _= 0 on O~F -- OF N 7-g, then se' -< - s e on O~F, and therefore (se, ~') -< (~, - s e) (as elements of f2) on 0[" = a ~ r u r a~r. We can thus write # ~ - ( f l ( X , X') = (~,~') on OF)
=
#~,p ~'~')(X(x))
+ #~,~ ~'~') ( X ( r x ) )
_< /z/~,(~-~)(X(x)) + #~,(~7~)(X(rx)) -- 0 by Lemma 4.13 and the symmetry under r and simultaneous spin flip. proposition thus follows from Theorem 8.1.
The []
In the remaining part of this subsection we consider the case of the square lattice s -- Z 2, for which we can obtain much stronger conclusions. The following result gives a complete characterization of the non-uniqueness regime of the parameter space in terms of percolation of plus spins in the Gibbs measure #~-. It is due to Coniglio et al. (1976); see also Higuchi (1985). Corollary 8.4 For the Ising ferromagnet on the square lattice Z 2 with no external field and inverse temperature fl, the lz~-probability of having an infinite pluscluster is 0 in the uniqueness regime fl < fie, and 1 in the non-uniqueness regime
Proof: The existence of an infinite cluster of plus spins is a tail event and thus, by the extremality of/z~-, has probability 0 or 1. The case/3 > tic is thus covered by Theorem 8.2. For/3
1 Random geometry of equilibrium phases
97
example, the Ising model on the cubic lattice Z 3. For/~ = 0 uniqueness certainly holds, and plus-percolation is equivalent to Bernoulli site percolation on Z 3 with parameter 1/2. But a result of Campanino and Russo (1985) states that pc(Z 3) < 1/2. The plus spins thus percolate at/~ = 0. In view of Proposition 4.16, this is still the case for sufficiently small/3, so that plus-percolation does occur in a non-trivial part of the uniqueness region. For the planar graph Z 2, however, Theorem 5.18 not only implies the equivalence of phase transition and percolation, but also gives some information on the number of phases in the non-uniqueness region. As a warm-up let us show that, for the Ising ferromagnet on Z 2 at inverse temperature/~ >/3c, there are no other translation and rotation invariant extremal Gibbs measures than #~- and #~. For, suppose another such phase/z existed. By Theorem 8.2 and the Burton-Keane uniqueness theorem 5.17, there exist unique infinite plus- and minus-clusters with /z-probability 1. As an extremal Gibbs measure,/z has positive correlations; recall the paragraph below Proposition 4.14. Proposition 5.19 thus shows that/z cannot exist. The statement just shown is a weak version of the following result which characterizes all translation invariant Gibbs measures. In fact, it is sufficient to assume periodicity, which means invariance under the translation subgroup (Ox)xcpZ 2 for some p > 1. Proposition 8.5 Any periodic Gibbs measure/Z for the Ising ferromagnet on s = Z 2 with no external field and inverse temperature 1~ > ~c is a mixture of the two phases/Z-~ and/z-~. Under the condition of translation invariance, this proposition was first derived for large/3 by Gallavotti and Miracle-Sole (1972), and later for all/3 > /~c by Messager and Miracle-Sole (1975) using some specific correlation inequalities; it follows also from the Onsager-formula for the free energy density and a result of Lebowitz (1972). We will give a geometric proof below. Remarkably enough, one can go one step further: each (not necessarily periodic) Gibbs measure for the Ising model on the square lattice is a mixture of the plus-phase and the minus-phase, and thus automatically automorphism invariant. This beautiful result was obtained independently by Aizenman (1980) and Higuchi (1981) based on the work of Russo (1979). A fairly short proof was found quite recently by Georgii and Higuchi (2000). For more general twodimensional systems the absence of non-translation-invariant Gibbs measures at sufficiently low temperatures was proved by Dobrushin and Shlosman (1985a). In three or more dimensions, however, non-translation-invariant phases of the Ising model do exist; this is a famous result by Dobrushin (1972), see also van Beijeren (1975) for a short proof.
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T h e o r e m 8.6 ( A i z e n m a n - H i g u c h i ) For the Ising ferromagnet on s = Z 2 with no external field and inverse temperature ~ > 1~c, Ix~ and IX~ are the only phases, and any other Gibbs measure is a mixture o f these two. To give an idea of some of the geometric arguments which lead to this theorem we will now provide a proof of Proposition 8.5. This argument is part of the recent proof of Theorem 8.6 by Georgii and Higuchi (2000). We need to consider infinite clusters in half-planes. Here, we say that a set C Z 2 is a half-plane if 7-[ is a translate of either the upper half-plane {x (Xl, x2) 6 Z 2 : x2 > 0} or its complement, the lower half-plane, or a translate of the fight- and left-half-planes which are similarly defined. The next lemma provides a first step in the proof of Proposition 8.5. L e m m a 8.7 Consider the Ising ferromagnet o n 2 2, and let D be the event that for at least one half-plane 7-[ in Z 2, both 7-( and 7-{c contain an infinite cluster of the same sign. Then IX(D) = 1 f o r all tx c G ( g H ) and g > t~c. Proof: Since each Gibbs measure is a mixture of extremal Gibbs measures, we only need to show that # ( D ) -- 1 for any extremal #. Suppose the contrary. Since D is tail measurable, it then follows that # ( D ) -- 0 for some extremal #. We will show that this is impossible. Step 1: Let 7-/be any half-plane, r the reflection of Z 2 which maps 7-/onto 7-/c, and r : cr -+ -or the spin flip on S2. We show that # = # o r o r . Since # ( D ) = 0, at least one of the half-planes 7-[ and ~ c contains no infinite minus-cluster, and this or the other half-plane contains no infinite plus-cluster. In view of the tail triviality of IX, we can assume that 7-r contains no infinite minus-cluster Ix-almost surely. Hence, for any given A E g and Ix-almost every ~ E f2, there exists an r-symmetric region F (~) ~ g such that F (~) D A and ~ -- 1 on 0 F (~) N 7-/. The last property implies that ~ _~ r o r (~) on 01-"(~), and using Lemma 4.13 and the flip-reflection symmetry of H we find that #~,r(~)---z)
" r~
tz~,r(~) = IX ,r(~) o r o r on f'A.
Assuming that F (~) is maximal in a large box A D A, we can apply the Markov property of # in the same way as in the proof of Theorem 8.1. This yields that IX ___D I x o r o r on .T'A for any A, and thus # _>_-z~# o r o r . (The preceding argument is a variant of an idea of Russo (1979).) Using the absence of infinite plus-clusters in ~ or 7-[c we find analogously that IX ___D IX o r o r. Hence IX = IX o r o r as claimed. Step 2: Here we use a variant of Zhang's argument which was explained in the proof of Theorem 5.18. To begin, we observe that the composition of two
1 Random geometry of equilibrium phases
99
reflections in parallel axes is a translation. Step 1 therefore implies t h a t / z is periodic with period 2. The flip-reflection symmetry of # implies further that /z is different from/z~- a n d / z ~ , so that (by Theorem 8.2) there exist both an infinite plus- and an infinite minus-cluster/x-almost surely. By the Burton-Keane uniqueness theorem 5.17, these infinite clusters are almost surely unique. We + now choose a square A = [ - n , n - 1] 2 n Z 2 SO large t h a t / z ( A .~ ~- cx~) > 1 10 -3. Since/x is extremal, # has positive correlations. By the argument + leading to (5.4.2) we thus obtain that #(O~A .~ ~- cx~) > 1 - 10 -3/4 for some k E {1 . . . . . 4}, where 0~ A is the intersection of OA with the kth quadrant (relative to the axes {x2 = - 1 / 2 } and {xl = - 1 / 2 } ) . For definiteness, we assume that k - 1. By the flip-reflection symmetry, it follows that the intersection 4{01A
-~
-
~" o c ,
02A
-~
+
~ oc,
03A
<
-
~" ~ ,
04A
<
~ oo}
has probability at least 1 - 4 910 -3/4 > 0, which is impossible because of the uniqueness of the infinite clusters. This contradiction concludes the proof of the lemma. [] P r o o f of Proposition 8.5" Let # be any Gibbs measure invariant under (Ox)xEpZ2 for some p > 1. Using the ergodic decomposition, we can assume that # is ergodic with respect to this group of translations. By Lemma 8.7, there exists a pair ( ~ , 7-(c) of half-planes such that, with positive probability, both ~ and ~ c contain infinite clusters of spins of the same constant sign. For definiteness, suppose ~ is the upper half-plane, and the sign is plus. In view of the finite energy property, it then follows that #(A0) > 0, where for k E Z A~ = {(k, 0) ~
+
~ ec both in 7-( and ~c}.
Let A be the event that A~ occurs for infinitely many k < 0 and infinitely many k > 0. The horizontal periodicity and Poincar6's recurrence theorem (or the ergodic theorem) then show that #(A0 \ A) = 0, and therefore # ( A ) > 0. Next, let B be the event that there exists an infinite minus-cluster. We claim that # ( A n B) -- 0. Indeed, suppose # ( A N B) > 0. Since A is tail measurable and horizontally periodic, we can use the finite energy property and horizontal periodicity of # as above to show that the event C - A n {(k, 0) .~ ~ oc for infinitely many k < 0 and infinitely many k > 0} has positive probability. But on C there exist infinitely many minus-clusters, which is impossible by the Burton-Keane theorem. To complete the proof, we note that # ( B ) < #(A c) < 1, and thus # ( B ) 0 by ergodicity. In view of Theorem 8.2, this means that # - #~-. In the case considered, the proposition is thus proved. The other cases are similar; in particular, in the case of negative sign we find that # = # ~ . []
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Constant-spin clusters in the Potts model
Consider the q-state Potts model on the lattice s -- Z a introduced in Section 3.3, q, d > 2, and recall the results of Section 6.3 on the phase transition in this model. The periodic ground states are the constant configurations/7i ~ i, 1 < i < i q. We write < ~ for the agreement connectivity relation relative to r/i, and we consider the limiting Gibbs measure #~,q i at inverse temperature/3 associated to r/i, which exists by Proposition 6.9. As a further illustration of assertion (8.1.2), exhibits/-percolation whenever there is a phase transition. we show that This is a Potts-counterpart of Theorem 8.2. For its proof, we use the randomcluster representation rather than Theorem 8.1 because for q > 2 there is no stochastic monotonicity available in the spin configuration.
/s
Theorem 8.8 For the Potts model on Z d at any inverse temperature fl with
i
1~(/3H)I > 1, we have [Z~,qi (X < ~ CXZ) > Of o r all x ~ Z d and i ~ {1 . . . . . q}.
Proof: By translation invariance we can choose x -- 0. In Theorem 6.10 we have seen thatq51p,q (0 ~ ec) = c > 0 f o r f l > tic , w h e r e p - 1 - e -2~ as usual. In view of (6.3.1), this means that qS~,q,a(0~ ~
A c) > c for all A ~
A1 0. But for the Edwards-Sokal coupling pip,q,A o f /Z~,q,A and dpp,q,
before Proposition 6.9) we have {0 ~ A c} C {0 <
i
(defined
~" A c} almost surely, so that
i i AC i i AC lZfl,q,A (0 < ~ ) > C. In particular, lZfl,q,A (0 < ~) > C whenever 0 9
A C A. Letting first A 1" Z d and then A 1" Z d we find that #~,q (0 < and the theorem follows.
i
~- ec) >_ c, []
Next we ask for a converse stating that "agreement percolation implies phase transition". As we already noticed in the case of the Ising model, this can be expected to hold only in the case of a planar lattice. But then a counterpart of Corollary 8.4 does indeed hold, as was shown by L. Chayes (1996). T h e o r e m 8 . 9 For the unique Gibbs m e a s u r e lzfl,q o f the q-state Potts model on the square lattice Z 2 at inverse temperature ~ < tic, we have /z/~,q (B an infinite/-cluster) - Of o r all i E {1 . . . . . q}. The strategy of proving this theorem is the same as that in the proof of Corollary 8.4. Suppose the/-spins percolate in #t~,q for some i. Then, by symmetry, this holds for all i, so that in particular the 1-spins and the other spins percolate. Hence, Theorem 5.18 leads to a contradiction, provided we can show that the set of ls has positive correlations. Theorem 8.9 thus follows from the following lemma.
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L e m m a 8.10 Consider the phase lZifl,q o f the q-state Potts model at any inverse temperature fl > O, and let the mapping si be defined by (8.1.1) with ~ = rli, i E { 1 . . . . . q }. Then the measure V~,q = lZ~,q o s~- 1 has positive correlations.
Sketch of proof: By symmetry, l)fl,q does not depend on i. For definiteness we set i - 1 in the following. Since the property of positive correlations is preserved under weak limits, it is sufficient to consider the finite volume Gibbs distribution #~,q,A and its image V~,q,A - #~,q,A o s~-1. By the FKG inequality, Theorem 4.11, it is further sufficient to show that V~,q,A is monotone. In terms of #~,q,A and the random field Y - Sl (X), this means that the conditional probability
qx(~) - #1/~,q,A (Y(x) - 1 I Y -- ~ off x) is increasing in ~ 6 {0, 1}zd for any x E A. Since the boundary condition is fixed to be equal to 1 off A, we can assume that ~ is equal to 1 off A, and it is sufficient to prove the inequality qx(~) < qx(~') for any two such ~, ~' that differ only at a single site y E A and are such that ~(y) = 0 and ~'(y) = 1. For such ~, ~', the inequality qx(~) < qx(~') simply means that Y(x) and Y ( y ) are positively correlated under the conditional distribution lZx,yl~ o f #~,q,A given that Y -off {x, y}. To show this we fix x, y, ~. For lZx,yl~, w e have X - 1 on the complement ofA = {x,y}t2{v 6 A\{x,y} : ~(v) = 0}. We thus consider the graph G with vertex set A and edge set/3(A) consisting of all edges of 13 with both endpoints lying in A. If we knew that Y ( x ) - Y ( y ) = 0, then lZx,yl~ would be the distribution of a (q - 1)-state Potts model on G with state space {2, 3 . . . . . q }. Now that we don't know Y(x) and Y(y), lZx,yl~ is still a modification of this (q - 1)-state Potts model, in which x and y are allowed to have the qth spin value 1. To describe this modification we suppose first that x and y are not adjacent. Let nx be the number of neighbors v of x with ~ (v) = 1, and define ny accordingly. The probability weight of #x,yl~ then contains the additional biasing factor
exp[2fl(nx I{x(x)=l} + nyI{x(y)=l})] which acts like an external field at x and y. For this modified Potts model, we can still define a modified random-cluster representation which gives any edge configuration ( E {0, 1}t3(A) a probability proportional to (q-
1 ) k ( ( ) ( q - 1 + e2~n~)k~(~)(q- 1 + e2t~nY)ky(()
- I p((e)(1 _ p)l-((e) eEB(A)
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Here p = 1 - e -2/~, k(~') is the number of connected components excluding singletons at x or y, and kx(~) and ky(~) are the indicator functions of having a singleton connected component at x resp. y. A spin configuration with distribution #x,yl~ is then obtained from the edge configuration by assigning spins at random uniformly from {2 . . . . . q} to connected components, except for a singleton at x, where the spin is taken from {1 . . . . . q } with probabilities proportional to (e 2flnx, 1, . . . , 1), and similarly for a singleton at y. Just as in Corollary 6.5, this representation gives the desired positive correlation of Y(x) and Y(y) under #x,yl~, provided we can show that kx and ky are positively correlated in the modified random-cluster model. Since these indicator variables are decreasing, it suffices to check that the modified random-cluster model has positive correlations, which follows from Theorem 4.11 by verifying that it is monotone; this, however, is similar to Lemma 6.6. The case when x and y are neighbors is handled analogously; in fact, the positive correlation can only become stronger when x and y have an edge in common. []
8.4 Further examples of agreement percolation Here we treat the Ising antiferromagnet, the hard-core lattice gas, and the W i d o m Rowlinson lattice model, and shortly mention the Ashkin-Teller model. The Ising antiferromagnet: Consider the setting of Section 3.2. We need to assume that the underlying lattice s is bipartite, and thus splits off into two p a r t s , l~,even and s If Ih[ is small enough, there exist two periodic ground 1 on states, 17even and 17odd ~ --17even, where ~]even ~ 1 o n l~,even and 17even ~ s (For/2 = Z d, this is the case when Ihl < 2d, and there are no other periodic ground states, see for example Dobrushin et al. (1985).) The phase transition in this model has been studied by Dobrushin (1968b) and Heilmann (1974). Because of the bipartite structure, we can flip all spins on a sublattice as in (3.2.1), which turns the model into an Ising ferromagnet in a staggered magnetic field of alternating sign o n s and s The latter model still satisfies the FKG inequality. As was pointed out in Section 3.2, for h - 0 there is a one-to-one correspondence between all Gibbs measures for the Ising ferromagnet and the Ising antiferromagnet. In particular, both models then have the same critical inverse temperature fie. For general h, we still have two limiting Gibbs 9~.... and #~odd and these measures have positive correlations relative measures tzt~ to the "staggered" ordering a _ a ' iff a(x)rleven(X) < a'(X)Oeven(X) for all x E s Relative to this ordering, an analogue of the sandwiching inequality (4.3.3) holds; for more details see Section 9 of Preston (1976). Here is a version of statement (8.1.2) for this model.
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Theorem 8.11 Consider the Ising antiferromagnet on a bipartite graph (/2, ~ ) in an external field h at any inverse temperature ~ > O. I f l G ( ~ H ) t > 1, we have JLrleven (X
oo) > 0
for all x ~ s
This follows from Theorem 8.1 in the same way as Theorem 8.2. For/2 = Z 2, the obvious counterparts of Corollary 8.4 and Proposition 8.5 are also valid since the proofs of these results carry over to the case of a staggered external field; see Georgii and Higuchi (2000) for an analogue of Theorem 8.6. The hard-core lattice gas: As we have seen in Section 3.4, this model has state space S = {0, 1} and corresponds to setting U(a, b) = ~ I { a = b = l } and V ( a ) = - a log)~ in (2.5.1), a, b ~ S; )~ > 0 is an activity parameter. For/2 -- Z d, the hard-core lattice gas can be seen as a gas of hard (i.e., non-overlapping) diamonds. In general, we still assume that/2 is bipartite. For )~ > 1, the hard-core model then has two periodic ground states of chessboard type, namely 17even which is equal to 1 o n ~ e v e n and 0 otherwise, and rlodd = 1-17eve n. As noticed in Section 4.4, the associated limiting Gibbs s t a t e s lzex yen and #~dd exist. So, following the program stated in (8.1.2), we may ask whether these Gibbs measures exhibit agreement percolation in the case of phase transition. The answer is again positive.
Theorem 8.12 For the hard-core model on a bipartite g r a p h / 2 we have f o r any "even ?]even> oo) > O f o r all x ~ /2, and activity )~ > O: I f IX x r lz~ dd then tx.even x (x < similarly with 'odd' in place o f 'even '.
This result is completely analogous to Theorem 8.11, and was conjectured by Hu and Mak (1989, 1990) from computer simulations. In these papers, the authors also discuss the case of hard-core particles on a triangular lattice, the hard hexagon model. While Theorem 8.12 does apply to the hard-triangle model on the hexagonal lattice (which is bipartite), the non-bipartite triangular lattice with nearest-neighbor bonds is excluded. The results of Hu and Mak (1989, 1990) suggest that Theorem 8.12 still holds for the triangular lattice. A geometric proof of this conjecture would be of particular interest. The hard-core model on the square lattice Z 2 admits an analogue to Corollary 8.4, in that non-uniqueness of the Gibbs measure is equivalent to rlevenpercolation for the Gibbs measure/x9 even x ; see Giacomin et al. (1995) or H~iggstr6m (1997a) for more details. A counterpart to Theorem 8.6 can be found in Georgii and Higuchi (2000). The W i d o m - R o w l i n s o n lattice model: Consider the set-up of Section 3.5, with equal activities )~+ = )~_ = ~. > 0 for the plus and minus particles. For ~. > 1 we have two distinct periodic ground states ~/+ ~ + 1 and r/_ ~ - 1 . From Sect/+ tion 4.4 we know that the associated limiting Gibbs measures/z + = limAl,s #Z,A
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and/z~- exist. Moreover, Theorem 4.17 asserts that a phase transition occurs for some activity ~. ifand only i f # + ( X ( x ) - 1) > # + ( X ( x ) = - 1 ) for somex 6 s Now, it turns out that in this model not only hypothesis (8.1.2) holds, but that the non-uniqueness of the Gibbs measure is equivalent to agreement percolation, not only for the square lattice but for any graph. This comes from the nature of the random-cluster representation of Section 6.7, which is related to the sites rather than the bonds of the lattice, and is a curious exception from the fact that, on the whole, the Widom-Rowlinson model is less amenable to sharp results than the Ising model. However, by the reasons discussed in Section 6.7, this result does not carry over to the multitype Widom-Rowlinson lattice model with q > 3 types of particles. Theorem 8.13 Consider the Widom-Rowlinson lattice model on an arbitrary graph (s "~) for any activity ~ > O. Then the following statements are equivalent. (i) The Gibbs measure for the parameter k is non-unique. /7+
(ii) /z + (x < ~ ~ ) > Ofor some, and thus all x ~ s Sketch of Proof: Consider #z,Ao+ for some finite A. In the same way as the random-cluster representation of Section 6.1 was modified in Section 6.2 to deal with boundary conditions, we can modify the site-random-cluster representation r/+ of Section 6.7 to obtain a coupling of/zz, A and a wired site-random cluster distribution ~pl,Z,A, so that analogues of Propositions 6.22 and 6.23 hold. As a counterpart to (6.3.3) and by the specific nature of the site-random-cluster representation, we then find that ~+ ( X ( x )
/~.,A
--
1)
o+ ( X ( x ) = - l ) - - ~ l
-- /~).,A
p,2,A
/)+
(x+-~0A)
0+ (x < ~ 0A)
- - JZ~.,A
for all x c s In the limit A 1' Z; we obtain by an analogue to (6.3.1)
~(X(x)
/7+
- 1 ) - ~*~ ( X ( x ) = - 1 ) = ~*~ (x ~
~ oo),
and the theorem follows immediately. To conclude this subsection, we note that hypothesis (8.1.2) also holds in other models. We mention here only the Ashkin-Teller model (Ashkin and Teller, 1943), a four-state model which interpolates in an interesting way between the four-state Potts and the so-called four-state clock model, which is also accessible to random-cluster methods; we refer to Salas and Sokal (1996); Chayes and Machta (1997); Pfister and Velenik (1997), and Chayes et al. (1998).
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105
Percolation of ground-energy bonds
So far in this section we considered a number of models which are known to show a phase transition, and asked whether this phase transition goes hand in hand with agreement percolation. These results run under the heading "phase transition implies percolation", even though for the square lattice we established results of converse type coming from the impossibility of simultaneous occupied and vacant percolation on Z 2. We now take the opposite point of view and ask whether "percolation implies phase transition". More precisely, we want to deduce the existence of a phase transition (at low temperatures or high densities) from a percolation result. In fact, such an idea is already implicit in Peierls' (1936) and Dobrushin's (1965) proof of phase transition in the Ising model, and is an integral part of the PirogovSinai theory. For models with neighbor interaction as in the Hamiltonian (2.5.1), the underlying principle can be sketched as follows. At low temperatures (or high densities), each pair of adjacent spins (or particles) tries to minimize its pair interaction energy. Note that this minimization involves the bonds rather than the sites of the lattice. So, one expects that bonds of minimal energy - the groundenergy bonds - prevail, forming regions separated by boundaries that consist of bonds of higher energy. Such boundaries, which are known as contours, cost an energy proportional to their size, and are therefore typically small when fl is large. This implies that the ground-energy bonds should percolate. Now, the point is that if the spins along a bond can choose between different states of minimal energy then this ambiguity can be transmitted to the macroscopic level by an infinite ground-energy cluster, and this gives rise to phase transition. In other words, the classical contour argument for the existence of phase transition can be summarized in the phrase: ground-energy bond percolation together with a (clear-cut) non-uniqueness of the local ground state implies non-uniqueness of Gibbs measures. We will now describe this picture in detail. We consider the cubic lattice/2 = Z d of dimension d > 2 with its usual graph structure. For definiteness we consider the Hamiltonian (2.5.1) for some pair potential U : S x S --+ R. We can and will assume that the self-potential V vanishes; this is because otherwise we can replace U by m
1 U ' ( a , b) -- U(a, b) + ~-~[V(a) + V(b)],
a, b ~ S,
(8.5.1)
which, together with the self-potential V t - 0, leads to the same Hamiltonian. Let m = min U (a, b) a,bES
be the minimal value of U.
(8.5.2)
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Given an arbitrary configuration o- 6 f2, we will say that an edge e = {x, y} 13 is a ground-energy bond for tr if U(cr(x), ~r(y)) = m. The subgraph of Z d consisting of all vertices of Z d and only the ground-energy bonds for cr splits then off into connected components which will be called ground-energy clusters for or. We are interested in the existence of infinite ground-energy clusters, and we also need to identify specific such clusters. Unfortunately, the Burton-Keane uniqueness theorem 5.17 does not apply here because, for any Gibbs measure, the distribution of the set of ground-energy bonds fails to have the finite-energy property. We therefore resort to considering ground-energy clusters in any fixed two-dimensional layer of Zd; the uniqueness of planar infinite clusters can be shown in our case. (An alternative argument avoiding the use of planar layers but requiring stronger conditions on the temperature has been suggested by Fukuyama (2000).) In fact, we have the following result. Theorem 8.14 Consider the Hamiltonian (2.5.1) on the lattice s = Z d, d > 2, with neighbor interaction U and no self-potential, and let 79 be any planar layer in s (So 79 = s for d = 2.) If ~ is large enough, there exists a Gibbs measure lz ~ G(~H) which is invariant under all 79-preserving automorphisms of s and all symmetries of U such that #(3 a unique infinite ground-energy cluster in 79) - 1. In the above, a symmetry of U is a transformation r of S such that U(ra, rb) = U (a, b ) f o r all a, b E S; such a r acts coordinatewise on configurations.
Theorem 8.14 is a particular case of a result first derived by Georgii (1981 a) and presented in detail in Chapter 18 of Georgii (1988). We will sketch its proof below. The remarkable fact is that this type of percolation often implies that # has a non-trivial extremal decomposition, so that there must be a phase transition. This happens whenever the set Gu = {(a, b) 6 S x S : U(a, b) --m}
(8.5.3)
of bond ground states splits into sufficiently disjoint parts. To explain the underlying mechanism (which may be viewed as the core of the classical Peierls argument, and a rudimentary version of Pirogov-Sinai theory) we consider first the standard Ising model. Example 8.15 The Ising ferromagnet at zero external field. In this model, we have as usual S = {-1, 1}, U ( a , b ) = - a b for a , b c S, m = - 1 , and Gu = { ( - 1, - 1), (1, 1) }. Hence, either all spins of a ground-energy cluster are negative, or else all these spins are positive. In other words, each ground-energy cluster is either a minus-cluster or a plus-cluster. This implies that {3 a unique infinite ground-energy cluster in 79} C A_ U A+,
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107
where A_ and A+ are the events that there exists an infinite cluster of negative, resp. positive, spins in 79. For the Gibbs m e a s u r e / z of Theorem 8.14 we thus have/Z (A_ U A+) = 1 and, by the spin-flip symmetry of U and thus/Z,/Z (A_) = /Z(A+). H e n c e / z ( A _ ) > 0 and/Z(A+) > 0, so that the measures/Z- = / z ( . I A - ) and/Z+ -- /Z(.IA+) are well defined. Since A_, A+ are tail events, it follows immediately that/Z-,/Z+ are Gibbs measures for/~H. Also, A _ N A + is contained in the event that there are two distinct ground-energy clusters in 79 , and therefore has/z-measure 0. H e n c e / Z - and/Z+ are mutually singular, whence IG(/~H)I > 1. The same argument as in the preceding example yields the following theorem on phase transition by symmetry breaking. A detailed proof (in a slightly different setting) can be found in Georgii (1988, Section 18.2). T h e o r e m 8.16 Under the conditions of Theorem 8.14, suppose that the set Gu defined by (8.5.3) admits a decomposition G u - G1 U ... U GN such that (1) the sets Gn, 1 < n < N, have disjoint projections, i.e., if ( a , b ) 6 Gn, (a', b') E Gn', and n ~= n', then a 7/= a', b ~= b', and (2) for any two distinct indices n , n f ~ {1 . . . . . N} we have r ( G n ) - Gn' for some transformation f of S x S which is either the reflection, or the coordinatewise application of some symmetry of U, or a composition of both.
Then, if 13 is sufficiently large, there exist N mutually singular Gibbs measures /ZN E G(fl H), invariant under all even automorphisms of 7-9 and such that /z l . . . .
/z n (3 an infinite n-cluster in 79) -- 1
[or all 1 < n < N. In particular, there exist N distinct phases for ~ H. In the above statement, an infinite n-cluster for a configuration o- is an infinite cluster of the subgraph of 79 obtained by keeping only those edges e 6 /3 with (~r(x), ~r(y)) c Gn, where x is the endpoint of e in the even sublattice s and Y 6 s is the other endpoint of e. Also, an even automorphism of 79 is an automorphism of Z d leaving s n "]') invariant. We illustrate this theorem by applying it to our other standard examples. E x a m p l e 8.17 The Ising antiferromagnet in an external field. We have again S - { - 1 , 1}, but the interaction is now U(a, b) - ab - ~ ( a + b) for some constant h 6 R. (Here we applied the transformation (8.5.1).) If Ihl < 2d then m = - 1 and G u - {( - 1, 1), (1, - 1) }. G u splits up into the singletons
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G 1 = {(1, - 1) } and G2 = {( - 1, 1) }. This decomposition meets the conditions of the theorem; in particular, G1 and G2 are related to each other by the reflection of S • S. Consequently, there exist two mutually singular Gibbs measures # 1 and lZ2 which are invariant under even automorphisms and have an infinite cluster of chessboard type, either with plus spins on the even cluster sites and minus spins at the odd cluster sites, or vice versa. Example 8.18 The Potts m o d e l . In this case, S = {1 . . . . . q} for some integer q > 2 and U ( a , b) -- 1 - 2l{a=b}. Again m = - 1 , and Gu = {(n, n) 9 ! < n < q }. Theorem 8.16 is obviously applicable, and we recover the result that for sufficiently large fl there exist q mutually singular, automorphism invariant Gibbs measures, the nth of which has an infinite cluster of spins with value n. Example 8.19 The h a r d - c o r e lattice gas. This model has state space S -- {0, l} and neighbor interaction U of the form U ( a , b ) = ~ i f a = b = 1, and U(a, b ) = log~.Zd(a + b) for all other (a, b) 6 S 2. Here ~ > 0 is an activity parameter, and we have again used the transformation (8.5.!). For ~ > 1 we have Gu = {(0, 1), (1, 0)}, so that Theorem 8.16 applies. Since multiplying U with a factor/3 amounts to changing ~, we obtain that for sufficiently large ~ there exist two distinct Gibbs measures with infinite clusters of chessboard type, just as for the Ising antiferromagnet at low temperatures. Example 8.20 The W i d o m - R o w l i n s o n lattice model. Here we have S = {-1 ' 0, 1} and U ( a , b ) -- cx~ i f a b -- - 1 ' U ( a , b ) = - log~ 2d ( [ a [ - ~ - [ b f ) o t h e r w i s e , a , b 6 S. I f ~ > 1 t h e n G v -- { ( - 1 , - 1 ) , ( 1 , 1 ) } . Theorem 8.16 thus shows that for sufficiently large ~. there exist two translation invariant Gibbs measures having infinite clusters of plus- resp. minus-particles. Although the results in the above examples are weaker than those obtained by the random-cluster methods of Section 6 (when these apply), the ideas presented here have the advantage of providing a general picture of the geometric mechanisms that imply a phase transition, and Theorem 8.16 can quite easily be applied. Moreover, the ideas can be extended immediately to systems with arbitrary state space and suitable interactions. In this way one obtains phase transitions in anisotropic plane rotor models, classical Heisenberg ferromagnets or antiferromagnets, and related N-vector models; see Georgii (1988, Chap. 18). One can also consider next-nearest neighbor interactions, and thus obtain various other interesting examples; for this one has to consider percolation of ground-energy plaquettes rather than ground-energy bonds, which is the set-up used by Georgii (1988). Last but not least, the symmetry assumption of Theorem 8.16 can often be replaced by either some direct argument, or a Peierls condition in the spirit of the Pirogov-Sinai theory (see Georgii (1988, Chap. 19)). One such extension will be used in our next example.
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Example 8.21 First-order phase transition in the Potts model. Consider again the Potts model of Example 8.18, and suppose for simplicity that d = 2. Any translate in Z 2 of the quadratic cell {0, 1}2 is called a plaquette. For a given configuration cr 6 f2, a plaquette P is called ordered if all spins in P agree, disordered if no two adjacent spins in P agree, and pure if one of these two cases occurs. If q (the number of distinct spin values) is large enough then, for arbitrary fi, there exists an automorphism invariant Gibbs measure # supported on configurations with a unique infinite cluster of pure plaquettes. This variant of Theorem 8.14 is due to Koteck3~ and Shlosman (1982), see also Georgii (1988, Section 19.3.2). Clearly, each cluster of pure plaquettes only contains plaquettes of the same type, either ordered or disordered. For some specific critical value fie(q) both possibilities must occur with positive probability; this follows from thermodynamic considerations, namely by convexity of the free energy as a function of/~ (Koteck2~ and Shlosman, 1982; Georgii, 1988). Conditioning on each of these two cases yields two mutually singular Gibbs measures with an infinite cluster of ordered resp. disordered plaquettes. Furthermore, all spins of a cluster of ordered plaquettes must have the same value, so that by symmetry the "ordered" Gibbs measure can be decomposed further into q Gibbs measures with infinite clusters of constant spin value. As a result, for large q and fi = fie(q) there exist q + 1 mutually singular Gibbs measures which behave qualitatively similar to the disordered phase for fi < fie(q) resp. the q ordered phases for fi > ~c(q). This is the first-order phase transition in the Potts model for large q. For further discussions we refer to (Koteck3~ and Shlosman, 1982; Wu, 1982; Lannait et al., 1986, 1991) and the references therein. We now give an outline of the proof of Theorem 8.14. Sketch proof of Theorem 8.14: For simplicity we stick to the case d -- 2. For any inverse temperature/~ > 0 and any square box An -~ [ - n , n - 112 N Z 2 we 9per write ~/~,n for the Gibbs distribution relative to fi H in the box An with periodic boundary condition. The latter means that An is viewed as a toms, so that (i, n 1) ~ ( i , - n ) and ( n - 1, i) ~ ( - n , i ) f o r / c [ - n , n - 1] N Z; the Hamiltonian Hnper in An with periodic boundary condition is then defined in the natural way. per Let I~ p e r be an arbitrary limit point of the sequence ~#~,n)n>-l" Evidently, lZpe r has the symmetry properties required of tz in Theorem 8.14, and ll~per E ~ ( ~ H ) . To establish percolation of ground-energy bonds we fix some ot < 1 and consider the wedge W = {x = (Xl,X2) E /~ : Xl ~ 0, Ix21 < O~Xl}. Let A W be the event that there exists an infinite path of ground-energy bonds in W starting from the origin. We want to show that ll~per (Aw) > 3/4 when/~ is large enough. Suppose ~ r Then there exists a contour crossing W, i.e., a path V in the dual lattice s - Z 2 + (1, 89 which crosses no ground-energy bond for and connects the two half-lines bordering W. For each such path y we will
110
H.-O. Georgii e t al.
establish the contour estimate
#per (y is a contour) __<(ISle-fiB) I•
(8.5.4)
where [gl is the length (the number of vertices) of y, and 6 > 0 is such that m + 26 is the second lowest value of U. Assuming (8.5.4) we obtain the theorem as follows. The number of paths of length k crossing ~V is at most ck 3~ for some c < c~ depending on or. Hence 1 k (3tSle-~) k < k>l 4
#per ( A ~ ) < c Z
for sufficiently large /~. By the rotation invariance of ~per, it follows that lZper(AO) > 0, where A0 is the intersection of AW with its three counterparts obtained by lattice rotations. Roughly speaking, A0 is the event that the origin belongs to two doubly infinite ground-energy paths, one being quasi-horizontal and the other quasi-vertical. Since #per is invariant under horizontal and vertical translations, the Poincar6 recurrence theorem (or the ergodic theorem) implies that the event A ~ = {~ E f2 9Ox~ ~ Ao for infinitely many x in each of the four half-axes}
has also positive #per-probability. Each configuration in A ~ has infinitely many quasi-horizontal and quasi-vertical ground-energy paths in each of the four directions of the compass, and by planarity all paths of different orientation must intersect. Therefore all these paths belong to the same infinite ground-energy cluster which has only finite holes, and is therefore unique. Hence A ~ is contained in the event B that there exists a ground-energy cluster surrounding each finite set of /2, and #per ( B ) > 0. As B is a tail-event and invariant under all automorphisms of s and all symmetries of U, the theorem follows by setting # - - lzper( 9I B ) . It remains to establish the contour estimate (8.5.4). For this it is sufficient to show that per #/~,n (V is a contour) _< (IS e-/~) I• (8.5.5) when n is so large that 9/is contained in An. This bound is based on reflection positivity and the chessboard estimate, which are treated at length in Georgii (1988, Chap. 17). Here we give only the principal ideas. The basic obserper. for any vation is the following consequence of the toroidal symmetry of #/~,n i 6 {0. . . . . n - 1}, the configurations on the two parts An,+ i -
{x ~ A n "
x~ >_ i or xl <_ i - n} and An, / -- {x E An " i n < Xl _< i} o f An are conditionally independent and, up to reflection, identically distributed given the spin values on the two separating lines {xl = i} and {xl = i - n}. Hence, if f, g are real functions on S A" depending only on the configuration in A /7,1' +
1 Random geometry of equilibrium phases
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and g(i) is the function obtained from g by reflection in these two separating lines (and thus depending on the configuration in An, i), then the bilinear form 9p e r
(f, g) ~ i~, n (fg(i)) is positive definite and thus satisfies the Cauchy-Schwarz inequality. Similar Cauchy-Schwarz inequalities hold for vertical reflections. The chessboard inequality is obtained by suitable combinations of all these, as we will illustrate next. Let us mark the plaquettes around the vertices of ~, with a o; this gives [y[ marked plaquettes. Marking a plaquette indicates that at least one of its bonds has non-minimal energy; leaving it unmarked does not say that it consists of groundenergy bonds, but that we don't need any information on its spins. In the case n = 2, this might lead to the picture
I I
To estimate its probability we repeatedly use the Cauchy-Schwarz inequality relative to suitable pairs of reflection lines. Indicating each time only the pair of lines used next, and omitting the event that no plaquette is marked (which has probability 1), we obtain 1/4 9
~fl,n
per --l~fl,n<
t
9
<
per
9
9
~
~n
9
9
9
1/8 9
9
9
9
.o . 9. . 9 9
9
9t
9
per
9
~ fl,n
9
9
t
9
<
per
9
~
~H
9)
9
1/8
9
<
9
~
per ~
9
9
9
9
9
9
( ....
9
9
9
)
3/16
9
9
9
9
9
9
9
9
9 9 9 9
In general, we obtain in this way
per
per
#~,n (Y is a contour) _< #~,n (Cn
)[y[/[An , I
where C n is the event that all plaquettes in An contain at least one bond of nonminimal energy. But if Cn occurs then at least BAn1/2 of the 2lAne edges in An are no ground-energy bonds. The Hamiltonian H per with periodic boundary condition is therefore at least (2m + 6)JAn1 on Cn. Since there is at least one cr c S A" with H p e r (o-) -- 2]Anlm, it follows that
per
l~fl,n(Cn)
<
~
e -~(2m+~)[An[
/
e -~2mlAnl
<
([S[e-~6)
IAn[
I'his gives estimate (8.5.5) and completes the proof of Theorem 8.14.
[]
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Random interactions
So far in this review, the spin systems considered had an interaction which was invariant under all automorphisms of the underlying graph s Here we will assume for convenience that s is the cubic lattice (Z a, B), d > 2, but the interaction between adjacent spins will no longer be translation invariant. That is, instead of the Hamiltonian (2.5.1) we now consider a modified Hamiltonian of the form
H(a)
Z b=(xy)c13
JaU(cr(x),or(y)) + Z
hxV(a(x))
(9.1)
x~Z d
where the Jb and the hx may vary from bond to bond, resp. from site to site. We are interested in the case where these coupling coefficients show no regular structure, and thus we assume that they are random. Such systems of spins interacting differently depending on their position and in a way governed by chance are known as disordered systems. We will not elaborate on the physical origins of such random interactions. We merely mention that they can be related to the presence of impurities or defects in an originally homogeneous system, and are used to model quenched alloys of magnetic and non-magnetic materials like FeAu. For details we refer to Fr6hlich (1984); Binder and Young (1988) and Fisher and Hertz ( 1991). We assume that the family J - (Jb)bc13 of coupling coefficients and the external fields h -- (hx)xeZd are independent, and each collection constitutes a family of mutually independent and identically distributed real random variables. Hence, while no realization of the coupling coefficients is translation invariant, we still have translation invariance in a statistical sense. We will not specify the underlying probability space, except that the letter P will be used to denote the probability measure and the associated expectation. The random families J and h are often referred to as the disorder. The disorder is called bounded if P(]Jb[ > c) -- 0 for some finite c. Physically, this is the most relevant case. (In some physical applications it is natural to assume that the Jb are not independent but rather have some finite-range dependence structure, but we will not include this case here. We also assumed for simplicity that the disorder is real valued, although some of the following also applies to the case when Jb or hx are allowed to take the value + c o with positive probability.) In Section 9.1 we will discuss dilutedferromagnets. A bond-diluted Ising or Potts ferromagnet on Z d can be viewed as an Ising or Potts model on the open clusters for Bernoulli bond percolation on Z d. As observed by Aizenman et al. (1987b), these models can quite easily be understood using their random-cluster representation. They form just about the only class of disordered systems where the phase transition can be investigated in such detail. Then, in Section 9.2, we study the so-called Griffiths regime which is the only non-trivial regime for disordered systems where by now quite general results are
1 Random geometry of equilibrium phases
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available. It occurs at intermediate temperatures if the disorder is bounded, or at arbitrary high temperatures if the disorder is unbounded, and is characterized by the fact that the Gibbs measure is still unique but fails to have a nice high temperature behavior uniformly in the disorder. The study of random Gibbs measures in the Griffiths regime started in the early 1980s and has only recently reached a satisfactory stage. The simplest, and also most powerful, methods use stochastic-geometric representations and will be presented here. As representative for the large literature on the subject we refer to Griffiths and Lebowitz (1972); Fr6hlich (1984); Fr6hlich and Imbrie (1984); Berretti (1985); Bassalygo and Dobrushin (1986); Fr6hlich and Zegarlinski (1987); Campanino and Klein (1991); Aizenman et al. (1992); Perez (1993); Gielis and Maes (1995); von Dreifus et al. (1995); Petritis (1996). Dynamical problems (which are not touched upon here) are treated e.g. in Cesi et al. (1997a, b, c); Guionnet and Zegarlinski (1997); Alexander et al. (1998); Gielis (1998) and Gielis et al. (1999).
9.1
Diluted and random Ising and Potts ferromagnets
The random Potts model is defined as follows. Spins take values in the state space S -- {1. . . . . q }, and the interaction is given by the Hamiltonian (9.1) with U(cr(x), or(y)) = 2 I{cr(x)V:~r(y)}
and V - 0. Note that this choice of U and V coincides with that used in Section 6.1 for the standard Potts model and differs from that in Section 3.3 only by constants which cancel in the definition of Gibbs distributions. The Ising model corresponds to the choice q -- 2. As for the disorder, we make the essential assumption that the random coupling coefficients Jb are non-negative, so that the interaction is still ferromagnetic. Of course, we also make the general assumption of this section that the Jb are independent with the same distribution, say re. A particular case of special interest is that of dilution, in which the Jb take the values 1 and 0 with probabilities p and 1 - p, respectively, which means that Jr p31 + (1 - P)30. For p = 1 we then recover the homogeneous Potts model of Section 3.3. In the following, the distribution Jr of the Jo will enter only through the quantities /5(/3, J r ) - - P ( 1 - e - 2 ~ S b ) ,
p(/3, :rr ) -
P
1 - e -2~Jb ) 1 + ( q - 1)e -2/~Jb
for/3 > 0. Note that they do not depend on b 6 /3, and that 0 < p(fl, Jr) < /5(/3, Jr) < p with p - P (Jb > 0). I
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For a given realization J = (Jb)b613 of the disorder and inverse temperature r , we can introduce the Gibbs measure/z~j,q i obtained from the Gibbs distributions with constant boundary condition i ~ {1. . . . . q} in the infinite volume limit. This limit exists by the arguments of Proposition 6.9, since these use only the stochastic monotonicity coming from Corollary 6.7(a) as well as the randomcluster representation, which both remain valid in the non-homogeneous case. The key quantity for phase transition, the order parameter, is the "quenched magnetization" m(fl, y r ) =
q-
q
1
p(
i
/zflj,q
(X(0)=i)
1) q .
Indeed, an inhomogeneous version of equation (6.3.3) shows that m(fl, Jr) P(Oq(flJ)), where 0q(flJ) -- qSl,q(0 ++ cx~) is the percolation probability for the wired infinite-volume random-cluster measure with bond probabilities Pb -1 -- e -2~Jb. Hence, we have m(fl, 7r) > 0 if and only if 0q (flJ) > 0 with positive P-probability. But whether or not Oq (r J) > 0 does not depend on the value of Jb for a single bond b. So, Kolmogorov's zero-one law implies that m (fl, 7r) > 0 if and only if Oq (r J) > 0 P-almost surely, and by an inhomogeneous version of Theorem 6.10 the latter means that multiple Gibbs measures for flJ exist with P-probability 1. Moreover, an inhomogeneous version of relation (6.3.2) shows that Oq (flJ) is an increasing function of flJ. It follows that m (/3, Jr) is increasing in/3 and also in Jr (relative to %z~). In particular, for each Jr there exists a critical inverse temperature flc(rr), possibly = +c~, such that m (/3, re) > 0 for/3 > tic (Jr) and m (/3, Jr) = 0 for fi < tic (re). It remains to investigate the quenched magnetization m (/3, 7r). The following lemma shows how m (/3, 7r) can be estimated in terms of Bernoulli bond percolation (recall the end of Section 5.1). L e m m a 9.1 Let O(p) = d?p(O +-~ c~) be the Bernoulli bond percolation probability on Z d with parameter p. Then
O(p(fi, yr)) < m(fl, ~r) < O(~(fl, yr)). Proof: We will use an inhomogeneous limiting version of the domination bounds (b) and (c) of Corollary 6.7. Although they were stated only in the case of a homogeneous interaction, they do extend to the inhomogeneous case. Define two families p - (Pb)bel3 and p' -- (P'b)b~13 in terms of a realization J by Pb --
1--e--2flJb' P~b -- (1--e--2flJb )/(l +(q -1)e-2~Jb ) = pb/(pb+q(1--pb)). Let~ppl,q be the associated wired random-cluster measure, and cpo, cpo, the corresponding product measures on {0, 1}/3. An inhomogeneous version of Corollary 6.7 then shows that
q~p'(0 +-~ OO) < ~)p,q(O ~ (X)) 5 ~pp(0 +-~ ~ ) .
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We now take the expectation with respect to P. In view of the preceding remarks, the middle term has P-expectation m(/~, rr), while the P-integration of the Bernoulli measures q~p, and 4~p again leads to Bernoulli measures, namely the homogeneous Bernoulli measures 4)p(~,~) and q~P(t~,~)- The lemma follows immediately. [] Combining the lemma with the discussion before we arrive at the following result on phase transition in the random Potts model. T h e o r e m 9.2 Consider the random Potts model on Z a at inverse temperature fl > 0 with coupling distribution Jr. Set p(rc) - zr(]0, oo[) = P(Jb > 0), and let Pc be the Bernoulli bond percolation threshold of Z a" so Pc - 1/2 when d = 2. (i) If/3(/5, rr) < pc then with P-probability 1 there exists only one Gibbs measure with interaction flJ. In particular, this holds when p(rc) < pc or 15 is small enough. (ii) If p(fl, Jr) > Pc then m(fl, Jr) > O, and with P-probability 1 there exist q distinct phases for the interaction flJ. In particular, this holds when p(rr) > Pc and t5 is large enough. Another way of stating this result is the following. Suppose zr = (1 - P)30 + prr+ with Jr+ = zr(-I]0, ~ [ ) , and let L+(fl) - f o e-ftrr+ (dt) be the Laplace transform ofzr+. (Note that then/5(/5, rr) = p ( 1 - L + ( 2 / 3 ) ) and p(fl, re) > p ( 1 q L+(2/3)).) Then there is no phase transition for p < Pc, whereas for p > Pc the critical inverse temperature tic(P, rr+) - / 3 c ( r r ) is finite (and decreasing in p) and satisfies the bounds p - Pc
< L+(2 tic(P, Jr+)) <_
P-
Pq
Pc P
If O(pc) = 0 (which is known to hold for d = 2, and is expected to hold for all dimensions) then uniqueness holds when p = Pc or/3 = ~c(P, rr+). In physical terminology, the preceding bounds on fie(P, re+) imply that the so-called crossover exponent is 1. Example 9.3 The case of dilution. If Jb is 1 or 0 with probability p resp. 1 - p then L+(fl) -- e - f . Hence, for p > Pc the critical inverse temperature satisfies the logarithmic bounds P-
pc
- In ~
P
_< 2 / 3 c ( p , 3 1 ) ~ - - In
p-
pc Pq
For q - 2, the diluted Ising model, assertion (ii) of Theorem 9.2 gives the slightly sharper upper bound ~c(p, ,~l) < tanh -1 (Pc~p).
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Example 9.4 The case o f p o w e r law singularities. Suppose 7r+(dt) = F(a)-lta-le-tdt is the Gamma distribution with parameter a > 0. Then L+ (/3) -- (1 +/3) -a, so that for p > Pc the critical inverse temperature satisfies a power law with exponent - 1 / a "
( Pc ) 1/aP-
Pc
1 < 2/~c(p, zr+) < -
( Pcq ) 1/a
-
P-
Pc
Examples with other kinds of singularities can easily be produced (Georgii, 1984). Theorem 9.2 is due to Aizenman et al. (1987b). Earlier, a generalized Peierls argument was used by Georgii (198 l b) to show that for the diluted Ising model (q -- 2) in d = 2 dimensions a phase transition occurs almost surely when p > pc - 1/2 and fl is large enough. In fact, this paper dealt mainly with the case of site dilution, in which sites rather than bonds are randomly removed from the lattice, and which in the present framework can be described by setting J(xy) - ~ ( x ) ~ ( y ) for a family (~(X))x~Zd of Bernoulli variables; the Jb are thus 1-dependent. This was continued in Georgii (1984, 1985) for a class of random interaction models including the random-bond Ising model as considered here, obtaining improved bounds on ~c(p, Jr+) for d = 2 as p $ 1/2. Extensions, in particular to d ___ 3, were obtained by Chayes, Chayes and Fr6hlich (1985). The diluted Ising model with a non-random external field h g= 0 does not exhibit a phase transition; this was shown by Georgii (1981b) for s = Z d and recently extended to quite general graphs by H~iggstr6m et al. (1999). For the diluted Ising model there is also a dynamical phase transition at the point p -- Pc. For p > Pc and/~c(1, 61) < /~ < ~c(P, 61) the relaxation to equilibrium is no longer exponentially fast (Alexander et al., 1998). This illustrates that uniqueness of the Gibbs measure does not in itself imply the absence of a critical phenomenon. Beside such dynamical phenomena, there are also some static effects of the disorder in the uniqueness regime, albeit these are perhaps less remarkable. These are the subject of the next subsection.
9.2 Mixing properties in the Griffiths regime As we have seen above, the diluted Ising ferromagnet shows spontaneous magnetization when p > Pc and fl > tic(P) - tic(P, 61), and multiple Gibbs measures for flJ exist almost surely. In the uniqueness region when still p > Pc but fl < tic(P) we need to distinguish between two different regimes. At high temperatures when actually fl < tic - tic(l), the critical inverse temperature of the undiluted system, we are in the so-called paramagnetic case. This is comparable to the usual uniqueness regime for translation invariant Ising models. At intermediate temperatures, namely when/~c < fl < tic(p), we encounter different behavior
1 Random geometry of equilibrium phases
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arising from the fact that the system starts to feel the disorder. This regime is called the Griffiths regime, since it was he (Griffiths, 1969) who discovered in this parameter region the phenomenon now called Griffiths' singularities. He studied site-diluted ferromagnets, but the arguments remain valid also in the bond-diluted case. The basic fact is the following: adding a complex magnetic field h to the Hamiltonian of the diluted Ising model we find that the partition function in a box with plus boundary conditions, as a function of h, can take values arbitrarily close to zero. The reason is that typically a large part of the box is filled by a huge cluster of interacting bonds, giving a contribution corresponding to an Ising partition function in the phase transition region. The radius of analyticity of the free energy around h = 0 is thus zero. In other words, the magnetization m(fl, p, h) cannot be continued analytically from h > 0 to h < 0 through h = 0 when p > pc and/3 > tic. So, the presence of macrosopic clusters of strongly interacting spins (on which the spins show the low-temperature behavior of the corresponding translation invariant system) gives rise to singular behavior. Related phenomena show up in a large variety of other random models, though not necessarily in the form of non-analyticity in the uniqueness regime; in general it may be difficult to pinpoint their precise nature. Nevertheless, we will speak of the Griffiths phase or the Griffiths regime whenever such singularities are expected to occur, even when a proof is still lacking. These terms then simply indicate that the usual high-temperature techniques cannot be applied as such. As another illustration we consider a random Ising model with unbounded, say Gaussian coupling variables Jb. Then fl Jb is also unbounded, even for arbitrarily small/3, and with high probability a large box contains a positive fraction of strongly interacting spins. In particular, there is no paramagnetic regime, and the whole uniqueness region belongs to the Griffiths phase. For this reason, it is a nontrivial problem to show the uniqueness of the Gibbs measure. For example, the standard Dobrushin uniqueness condition encountered in (7.1.5) (cf. Dobrushin (1968b) and Dobrushin and Shlosman (1985b)) is useless in this case; similarly, a naive use of standard cluster expansion techniques fails. These methods are bound to fail since they also imply analyticity which is probably too much to hope for (even though we cannot disprove it). In the following we will not deal with the singular behavior in the Griffiths phase. Instead, we address the problem of showing nice behavior, which we specify here as good mixing properties of the system. We shall present two techniques: the use of random-cluster representations, and the use of disagreement percolation.
Application of random-cluster representations: Consider a random Ising model. Spins take values ~(x) = 4-1, and the formal Hamiltonian is H ( o ) -- -
Jbcr (x)cr (y).
b=(xy)El3
(9.2.1)
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We setl3 boundary estimates measures
= 1. Letlzj, A be the associated Gibbs distribution in A ~ s with condition 7/ ~ ~2. For many applications it is important to have good on the variational distance ]]-[]/x on A C A (see (7.1.2)) of these with different boundary conditions.
Definition 9.5 The random spin system above is called exponentially weakmixing with rate m > 0 if f o r some C < oo and all A E s and A C A P
(
)
m a x []P'J,A -- P],A ][A rl,r/~f2
< C[AI e-md(A'Ac)
(9.2.2)
where d(A, A c) is the Euclidean distance o f A and A c.
Various other mixing conditions can also be considered. A stronger condition requires that the variational distance in (9.2.2) is exponentially small in the distance between the set A and the region where the boundary conditions 17 and q' really differ. One could also restrict A and/or A to regular boxes. See Dobrushin and Shlosman (1985b, 1987); Stroock and Zegarlinski (1992); Martinelli and Olivieri (1994); Cesi et al. (1997a, b); van den Berg (1997) and Alexander et al. (1998). Let us comment on the significance of the exponential weak-mixing condition above. Remarks: (1) Suppose condition (9.2.2) holds. A straightforward application of the Borel-Cantelli lemma then shows that for any m ~ < m m a x [[#J,A -- # ,A [[A _< Cj]A[ e-m d(A'AC)
with some realization-dependent Cj < cxz P-almost surely. Integrating over ~' ' for for any Gibbs measure # j we find in particular that # j -- lima l.zd #J,a all ~, implying that # j is the only Gibbs measure (and depends measurably on 0 J). Moreover, noting that #j(A]B) -- f #j,A(A)#j(dJTIB) for A 6 .T/x and B 6 fAC, we see that this realization-dependent Gibbs measure # j satisfies the exponential weak-mixing condition sup
] # j ( A I B ) - #j(A)I < CjIAIe -m'd(A'A~).
(9.2.3)
A~.T'A,BEf'AC,Uj(B)>O (2) Condition (9.2.3) above also implies an exponential decay of covariances. Let f be any local observable with dependence set A ~ [ and g be any bounded observable depending only on the spins off A, where A C A. Also, let 6 ( f ) = maxc~,~, I f ( o ) - f(cr~)l be the total oscillation of f and 6(g) that of g. The covariance # j ( f ; g) of f and g then satisfies P-almost surely the inequality ]#J(f; g)l < C j [ A l 6 ( f ) 6 ( g ) e - m ' d ( A ' a c ) / 2 .
1 Random geometry of equilibrium phases
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Indeed, a short computation shows that [ttJ,h(f; g)l is not larger than the lefthand side of inequality (9.2.3) times (~(f)3(g)/2, cf. inequality (8.33) in Georgii (1988). If g is local, a similar inequality holds for covariances relative to finite volume Gibbs distributions in sufficiently large regions with arbitrary boundary conditions. We will now investigate the conditions under which the random Ising system with Hamiltonian (9.2.1) is exponentially weak-mixing. We start from the estimate ! ,7 -- /'L~,A[IA <-- ~bp,2,A(A 1 I[~j,A ~ 0A) obtained in Theorem 7.8. As before, this bound is also valid in the inhomogeneous case considered here, and ~b~,2,A stands for the wired random-cluster distribution in A with bond-probabilities p - P(IJ]) - (Pb)bel3 given by Pb -1 - exp[-2lJbl]. Next we can use a recent concavity result of Alexander et al. (1998). L e m m a 9.6 Let K - (Kb)bc13 be a collection of positive real numbers and p p(K) denote the collection of densities Pb -- 1 - e -2Kb. For any increasing function f , the expectation @p,Z,A( f ) is then a concave function of each Kb. 1 Proof: For brevity we set ~bp,2, A -- qb. For any fixed bond b we consider the
functions F (Pb) = ~b( f ) and G (Kb) = F (1 - e -2Kb). Using equation (6.2.1) of Lemma 6.6 we then find that F' (Pb) -- C])(f gpb) -- (/)(f)qb(gpb)
and F" (pb) = - 2 F ' (pa)dp(gpb),
where for r/b 6 {0, 1} gpb (rlb) --
2Oh- 1 p~)b ( 1 - Pb)l-"b
"
This implies G" (Kb) -- - 4 e -2Kb F' (pb) [ 1 4- 2e--2Kbqb(gpb)].
Now, F' is non-negative because f and g pb are increasing and r has positive correlations by Corollary 6.7(a). Another explicit computation shows that 1 4- 2e--ZKb~(gpb ) --2~b(~/b- 1 ) / p b -
1 > 2/(2-
Pa)-
1 > 0,
where the first inequality uses the fact that the random-cluster distribution for q ! 2 dominates an independent percolation model with densities Pb - pb/(2 - pb), see Corollary 6.7(c). It follows that G" < 0, proving the claimed concavity. []
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The preceding lemma implies that
x60A
where p = 1 - e x p [ - 2 P (IJbl]. The weak mixing property (9.2.2) thus follows provided we can show an exponential decay of connectivity in the wired random1 cluster distribution 4)1,2,^. The latter certainly holds when p < Pc because 4~p,2,h is dominated (on A) by the Bernoulli measure ~bp (cf. Corollary 6.7(b)), and the connectivity of a subcritical Bernoulli model decays exponentially fast (recall Theorem 5.6). So we arrive in particular at the following result. Theorem 9.7 Consider a random Ising model on Z d with Hamiltonian (9.2.1). If 2 P(IJbl) < - ln(1 - Pc) for the Bernoulli bond percolation threshold Pc o f Z d then the system is exponentially weak-mixing.
As the proof above shows, the factor IAI on the right-hand side of (9.2.2) can be replaced by l0 A I. In the diluted Ising model at inverse temperature fl (see Example 9.3), the condition of the preceding theorem reads 2tip < - In(1 - Pc). Therefore, if d is so large that 2tic pc < - In(1 - pc) then the theorem covers part of the Griffiths regime. This fact is evident in the case of an unbounded, say Gaussian, disorder because then (as explained above) there is no paramagnetic phase. Exponential weak-mixing for random Ising models can also be shown by other applications of random-cluster domination. Let us sketch such an alternative route. Using the random-cluster representation, Newman (1994) showed that (pointwise in the disorder J)
, ~ + max II#j,n - #~I,A IIz~ _< 2 /zlji, h (X (x)), r/,rf xEA
(9.2.4)
+ A is the Gibbs distribution in A with plus boundary conditions for the where/zlji, Hamiltonian (9.2.1) with Jb replaced by IJbl. On the other hand, Higuchi (1993a) obtained the estimate
f
PlJI,A
/ZlJI,A
(X(x)X(z))
y~Az6A:z"y
(the superscript ' f ' referring to the free boundary condition), while Olivieri et al. (1983) proved that (9.2.5)
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121
with J = P(IJbl). We are thus back to the standard Ising Gibbs distribution in A with zero external field, free boundary condition and coupling constant J. Now we can take advantage of the second Griffiths inequality (which we have not discussed so far in this text) stating that correlation functions such as on the right-hand side above are monotone in the coupling coefficients, see Ellis (1985); Liggett (1985). This implies that the right-hand side of (9.2.5) is an increasing function of A and thus bounded above by its infinite volume limit. But for J less than Jc, the critical coupling, the Gibbs measure is unique and has an exponential decay of correlations, see Aizenman et al. (1987). We thus find that for J < Jc the right-hand side of (9.2.5) has an exponential upper bound C e x p [ - m l x - zl] for suitable constants C < c~ and m > 0. Together with the previous estimates, we conclude that under the condition P (I J b I) < Jc, the random lsing model on Z a with Hamiltonian (9.2.1) is exponentially weak-mixing. Again, this condition covers part of the Griffiths regime for the diluted Ising ferromagnet. As an alternative to the use of the Griffiths inequalities above we can also apply Theorem 6.2 and Corollary 6.7(b), giving
where 4~p is the bond Bernoulli measure with density p = P (1 - exp[--2lJbl]). As mentioned above, its connectivity function decays exponentially fast when P < Pc. We therefore conclude that the random Ising model on Z d is also exponentially weak-mixing when P (1 - exp[-2i Jb I]) < Pc. The above estimate (9.2.4) does not hold when we add a random magnetic field to the Hamiltonian (9.2.1), :-
2 _ , hx~(X),
_L b = ( x y ) El3
(9.2.6)
x EZ d
with i.i.d, real random variables hx independent from the Jb. However, if Jb, hx >_ 0 then we can take advantage of Higuchi (1993a, Section 2) to replace (9.2.4) with ,
max II/~J,h,A - LL~,h,A IIzx <__ 2 r/' r/f
+
x~A
/~J,0,A (X (x)),
and we can continue as above.
Application of disagreement percolation: As we have indicated at the end of Section 7.1, the idea of disagreement percolation can be applied to study the Griffiths regime for rather general random-interaction systems. To be specific we consider Ising spins with the Hamiltonian (9.2.6). We consider the finite volume ,7 in a box A with boundary condition 77; as before, the Gibbs distribution #J,h,h subscripts J, h describe the random interaction. We are going to apply Theorems
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7.1 and 7.2 pointwise in the disorder. As in (7.1.1) we thus have to consider the variational single-spin oscillations pJ,h
x
-
r/
'
max ]l#j,h,x -#~,h,xllx
r/, r/tel2
(9.2.7)
J,h which depend on the disorder J, h. Under the present assumptions, (Px)x~Zd is a 1-dependent random field; this is the only property of the disorder needed below. Now, from Theorems 7.1 and 7.2 we can conclude that if with P-probability 1 there is no Bernoulli site-percolation with densities (9.2.7) then, again with Pprobability 1, there is a unique Gibbs measure for J, h. (Similarly to the results of Section 7.2 the random system can be dominated by a random percolation system which more or less coincides with a stochastic-geometric representation of the diluted ferromagnet (Gielis, 1998).) The following theorem is an immediate consequence of the results of Section 7.1.
Theorem 9.8 An Ising system with random Hamiltonian (9.2.6) satisfying p (pJ,h) <
1
(2d - 1)2
is exponentially weak-mixing. Of course, we now have to add in (9.2.2) the subscript h referring to the random external field. Proof: Consider Theorem 7.1. Evidently, the right-hand side of (7.1.3) can only increase if the density for an open site is set to be 1 on the odd sublattice s of Z a. Due to the 1-dependence of the random field p = (pJ,h)xeZd noticed
above, the remaining pxJ'h with x ~ l~even are mutually independent. Taking the P-expectation in (7.1.3) we thus obtain on the right-hand side the Bernoulli percolation probability ![rp,even (A <----~0 A), where l[rp,even is the Bernoulli measure with density p - P (pxJ'h) on the even sublattice s and density 1 on s The exponential weak-mixing property now follows by an argument similar to that following (5.1.3); note that, relative to ~p,even, a path of length k is open D with probability at most plk/2J. Remarks: (1) Suppose we add a uniform magnetic field h to the random Hamiltonian (9.2.6). Under the conditions of Theorem 9.8 it is then not too difficult to show that the disorder-averaged expectation P(lzJ, h+h(f)) of any local function f is an infinitely differentiable function of h, see von Dreifus et al. (1995). (2) Following Theorem 7.2, Theorem 9.8 requires that the single-point densities (9.2.7) are globally small enough to prevent percolation in the associated
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Bernoulli model. This condition can be extended to so-called constructive conditions involving finite boxes rather than single sites (van den Berg, 1997). (3) Another very powerful and transparent treatment of the Griffiths regime (based on very similar percolation ideas) has been developed by von Dreifus et al. (1995). This paper proposes a technique similar to that of Fr6hlich and Imbrie (1984) and Bassalygo and Dobrushin (1986) of a resummation in the high-temperature cluster expansion. The bounds then allow a probabilistic interpretation of the expansion linking it with a bond percolation process.
10
Continuum models
All models considered so far lived on a lattice. Physical systems like real gases, however, are more realistically modeled by particles living in continuous space. This section is an outline of how some of the stochastic-geometric ideas developed in previous sections can be applied to a continuum setting. First, in Section 10.1, we consider the natural continuum analogues, based on Poisson processes, of the Bernoulli percolation models introduced in Section 5. Then, in Section 10.2, we consider a continuum variant of the Widom-Rowlinson model introduced in Section 3.5 and discuss its phase transition behavior. (As mentioned in Section 10.2, this continuum variant is the one originally considered by Widom and Rowlinson (1970), so it predates the lattice model.)
10.1 Continuum percolation Here we consider the basic models of continuum percolation. For a thorough treatment of the mathematical theory of continuum percolation we refer to Meester and Roy (1996). We first need to introduce the Poisson process on R a and its subsets. Heuristically, a Poisson process with intensity ~. > 0 on R d is a random set X of points in R d with the properties that (i) for any bounded Borel set A of R d with volume ]A l, the number of points of X in A is Poisson distributed with mean ~.IA l, i.e., for n -- 0, 1, 2 . . . . the probability of seeing exactly n points in A equals exp(-)~lAl)O~lA])n/n!; (ii) for any two disjoint such subsets A1 and A2, the numbers of points observed in A1 and in A2 are independent. For a construction of such a process, we first consider a bounded Borel set A of R d. Let f2A be the set of all finite subsets of A. A Poisson process on A with
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intensity ~. > 0 is then given by a random element of f2t, having distribution fez,A, where 7rz,A(F) = e -ztAt nZ= 0 ~ . I
"'"
I{{xl
.....
Xn}EF} dxl . . . dxn
for all F 6 f'A, the smallest a-field which allows us to count the number of points in each Borel subset of A. Next, let f2 be the set of all locally finite point configurations on Rd; locally finite means that any bounded set contains only finitely many points. The Poisson process on R d with intensity ~. is a random point configuration X distributed according to the unique probability measure rex on f2 which, when projected on any bounded Borel set A C R d, yields zrz,A. Properties (i) and (ii) above are easily verified, and make Poisson processes the natural analogues of Bernoulli measures for lattice models. To study percolation properties of the Poisson process X, we need to introduce some notion of connectivity. A natural way is to imagine a closed Euclidean ball B(x, R) of fixed radius R around each point x of the Poisson process, which leads us to considering the random subset .,Y = Ux~x B(x, R) of R d. Such random subsets are widely known as Boolean models. (More generally, B(x, R) could be replaced by a closed compact random set centered at x.) This particular Boolean model is often referred to as the Poisson blob model or lily pond model. Two points x, y E X are then considered as connected to each other if they are connected in )~, meaning that there exists a continuous path from x to y in X. By scaling, there is no loss of generality in setting R = 1/2, so that two balls centered at x and y intersect if and only if I x - Yl < 1, where I"I denotes Euclidean distance. The basic result on Boolean continuum percolation, analogous to Theorem 5.3 for ordinary site percolation, is the following. Theorem 10.1 Pick a Poisson process X on R a, d > 2, with intensity ~.. Let f( = UxEx B(x, 1/2) be the associated Boolean model, and let 0(~,) denote the probability that the origin belongs to an unbounded connected component of f(. Then there exists a critical value )~c -- )~c(d) E (0, ~ ) such that 0()0 -- 0 if < ~c and 0(~.) > 0 if~ > )~c. The standard proof of this result is based on a partitioning of R d into small cubes, reducing the problem to its lattice analogue, Theorem 5.3; see Meester and Roy (1996) or Grimmett (1999). Another continuum percolation model is the so-called random connection model, or Poisson random edge model, which was introduced by Penrose (1991). Let g : [0, cx~) --+ [0, 1] be a decreasing function with bounded support (that is g(x) = 0 when x exceeds some R < cx~). The random connection model with
1 Random geometry of equilibrium phases
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intensity X and connectivity function g arises by taking a Poisson process X in R d with intensity X and independently drawing an edge between each pair of points x and y of X with probability g(Ix - Yl). This setting includes the Boolean model, which corresponds to the choice g = l[0,1]. Theorem 10.1 extends to this model: For g as above with f o g ( x ) d x > 0 and dimensions d > 2, there is a critical value kc - Xc(d, g) such that infinite connected components almost surely occur (resp. do not occur) whenever )~ > ~.c (resp. ~. < )~c) in the random connection model with intensity ~. and connectivity function g. Much of the theory of standard (lattice) percolation has analogues for these continuum models. An example is the uniqueness of the infinite cluster (Theorem 5.17), which goes through for the Boolean and random connection models. See Meester and Roy (1996) for this and much more.
10.2 The continuum Widom-Rowlinson model The continuum Widom-Rowlinson model is a marked point process where the points are of two types" we call them plus-points and minus-points. (From now on and throughout this section, we drop the term "continuum" when refering to this model, and instead add the word "lattice" when talking about the model of Section 3.5.) For the model defined on a region A c R d realizations take values in S2A x f2A; the first coordinate describes the locations of the plus-points, and the second coordinate the minus-points. There is a hard sphere interaction preventing two points from coming within Euclidean distance R from each other; again, we set R -- 1 without loss of generality. This interaction corresponds to the Hamiltonian H(x, y) --
~
cx~ l{ix_yl<_l},
x~x,y~y x , y E f]A.
When A is bounded, the Widom-Rowlinson model on A with intensity ~ is obtained by conditioning the Poisson product measure ~X,A x JrX,A on the event that there is no pair of points of opposite type within unit distance from each other. The extension to R a is done in the usual DLR fashion: a probability measure # on ~2 x f2 is a Gibbs measure for the Widom-Rowlinson model at intensity ~. if, for any bounded Borel set A, the conditional distribution of the point configuration on A given the point configuration on R d \ A is that of two independent Poisson processes conditioned on the event that no point in A is placed within unit distance from a point of the opposite type, either inside or outside A. The resemblence with the lattice Widom-Rowlinson model of Section 3.5 is evident. We have the following analogue of Theorem 3.4.
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Theorem 10.2 For the Widom-Rowlinson model on R a, d > 2, with activity X, there exist constants 0 < X~c < X~ < cx) (depending on d) such that f o r X < Uc the model has a unique Gibbs measure, while f o r X > X~ there are multiple Gibbs measures. The proof of this result splits naturally into two parts: first, we need to demonstrate uniqueness of Gibbs measures for X sufficiently small, and secondly we need to show non-uniqueness for ~. sufficiently large. The first half can be done by a variety of techniques. For instance, one can partition R d into cubes of unit sidelength and apply disagreement percolation (Theorem 7.1). Two observations are crucial in order to make this work: that the conditional distribution of the configuration in such a cube given everything else only depends on the configurations in its neighboring cubes, and that the conditional probability of seeing no point at all in a cube tends to 1 as X --+ 0, uniformly in the neighbors' configurations. The more difficult part, the non-uniqueness for large ~., was first obtained by Ruelle (1971) using a Peierls-type argument. Here we shall sketch a modem stochastic-geometric approach using a random-cluster representation. This approach is due mainly to Chayes et al. (1995) (but see also Giacomin et al. (1995)), and works in showing both parts of Theorem 3.4. The so-called continuum random-cluster model is defined as follows. Definition 10.3 The continuum random-cluster distribution qbx,A with intensity X for the compact region A C R a is the probability measure on ~2A with density 1 2k(x) , f ( x ) -- ZX,A
x 6 f2A
(10.2.1)
with respect to the Poisson process rcx,A o f intensity X" here Zx,A is a normalizing constant and k(x) is the number o f connected components o f the set = Ux~xB ( x , 1/2).
In analogy to the correspondence between the lattice Widom-Rowlinson model and its random-cluster representation in Propositions 6.22 and 6.23, we obtain the continuum random-cluster model by simply disregarding the types of the points in the Widom-Rowlinson model, with the same choice of the parameter X. Conversely, the Widom-Rowlinson model is obtained when the connected components in the continuum random-cluster model are assigned independent types, plus or minus with probability 1/2 each. To see why this is true, note that for any x E f2A there are exactly 2 k(x) elements of f2A x ~A which do not contradict the hard sphere condition of the Widom-Rowlinson model and which map into x when we disregard the types of the points. Besides the random-cluster representation, we can also take advantage of stochastic monotonicity properties. Let us define a partial order _~ on f2 • f2
1 Random geometry of equilibrium phases
127
by setting (x, y) _ (x', yt)
if
x c x' and y _ y',
(10.2.2)
so that, a configuration increases with respect to % if plus-points are added and minus-points are deleted. A straightforward extension of Theorem 10.4 below then implies that the Gibbs distributions for the Widom-Rowlinson model have positive correlations relative to this order. The methods of Sections 4.1 and 4.3 can therefore be adapted to show that the Widom-Rowlinson model on R d at intensity )~ admits two particular phases #+ and #~-, where #+ is obtained as a weak limit of the Gibbs measures on compact sets (tending to R d) with the boundary condition consisting of a dense crowd of plus-points, and #~- is obtained similarly. We also have the sandwiching relation #x- ~_9 # •
/z+
(10.2.3)
for any Gibbs measure # for the intensity )~ Widom-Rowlinson model on R d, so that the uniqueness of the Gibbs measures is equivalent to having #~- - / z +. The Gibbs measure for the Widom-Rowlinson model on a box A with "plus" (or "minus") boundary condition corresponds to the wired continuum randomcluster model 4~,A on A where all connected components within distance 1/2 from the boundary count as a single component. Arguing as in Sections 6.3 and 6.7 we find that uniqueness of the Gibbs measures for the Widom-Rowlinson model is equivalent to not having any infinite connected components in the continuum random-cluster model. Let ~.c be as in Theorem 10.1. Theorem 10.2 follows if we can show that the continuum random-cluster model 4~),A with sufficiently large intensity ~ stochastically dominates 7rX1,A for some ~.1 > ~.c, whereas 4~1,A __59 7r)v2,A for s o m e ~.2 < ~,c when ~. is sufficiently small To this end we need a point process analogue of Theorem 4.8, which is based on the concept of Papangelou (conditional) intensities for point processes. Suppose bt is a probability measure on f2A which is absolutely continuous with density f (x) relative to the unit intensity Poisson process 7rl,A. For x ~ A and a point configuration x ~ f2a not containing x, the Papangelou intensity of # at x given x is, if it exists, f ( x U {x})
~.(xlx) = ~
/(x)
.
(10.2.4)
Heuristically, ~.(xlx)dx can be interpreted as the probability of finding a point inside an infinitesimal region dx around x, given that the point configuration outside this region is x. Alternatively, )~(. I') can be characterized as the RadonNikodym density of the measure f #(dx)Y]x~x 8(x,x\{x}) on A x f2A, the socalled reduced Campbell measure of #, relative to the Lebesgue measure times # (Georgii and Ktineth, 1997). It is easily checked that the Poisson process rrX,A has Papangelou intensity )~(xlx) = )~.
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The following point process analogue of Theorem 4.8 was proved by Preston (1977) under an additional technical assumption, using a coupling of so-called spatial birth-and-death processes similar to the coupling used in the proof of Theorem 4.8. Later, the full result was proved by Georgii and Ktineth (1997) by a discretization argument. Theorem 10.4 Suppose lZ and ft are probability measures on f2A with Papangelou intensities )~(. I') and ~.(. I') satisfying ~(xlx) _< ~(xl~)
whenever x ~ A and x, ~ ~ f2A are such that x c ~. Then # "
(10.2.5)
where to(x, x) is the number of connected components of [.-Jyex B(y, 1/2) intersecting B(x, 1/2). It is a simple geometric fact that there exists a constant Ir x - - K m a x (d) < oo such that x (x, x) < K m a x for all x and x; for d = 2 we may take Ir x - - 5 . It follows that
~. 21-'cmax < ~.(xlx) < 2,k
(10.2.6)
for all x and x. Hence, applying Theorems 10.4 and 10.1 we find that 4~l,A __Z~ rrZZ,A, so that for ~. < ~.c/2 we obtain the absence of unbounded connected components of Uxex B(x, 1/2) in the limit A t Rd of continuum random-cluster measures. On the other hand, taking ,k > ~.c 2Xmax-1 yields the presence of unbounded connected components in the same limit. Theorem 10.2 follows immediately. It is important to note that this approach does not allow us to show that the non-uniqueness of the Gibbs measures depends monotonically on s The reason is similar to that for the lattice Widom-Rowlinson model in Section 6.7" the righthand side of (10.2.5) fails to be increasing in x. It thus remains an open problem whether or not one can actually take Uc = s in Theorem 10.2. There are several interesting generalizations of the Widom-Rowlinson model. Let us mention one of them, in which neighboring pairs of particles of the opposite type are not forbidden, but merely discouraged. Let h 9[0, c~) -+ [0, c~] be an "interspecies repulsion function" which is decreasing and has bounded support. For A C R d compact and )~ > 0, the associated Gibbs distribution #h,~,A on f2A x f2A is given by its density f(x, y)
1 -
Zh,~.,A
exp(-
~ xex,y~y
h(lx-y,)).
1 Randomgeometry of equilibrium phases
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relative to zrz,A x rrz,A. Infinite volume Gibbs measures on ~ x f2 are then defined in the usual way. Lebowitz and Lieb (1972) proved non-uniqueness of Gibbs measures for large ~ when h (x) is large enough in a neighborhood of the origin. Georgii and H~iggstr6m (1996) later established the same behavior without this condition, and for a larger class of systems, using the random-cluster approach. This involves a generalization of the continuum random-cluster model, which arises by taking the random connection model of Section 10.1 with connectivity function g(x) = 1 - e - h ( x ) and biasing it with a factor 2 ~(z), where k(z) is the number of connected components of a configuration z of points and edges. To establish the phase transition behavior of this "soft-core Widom-Rowlinson model" (i.e., uniqueness of Gibbs measures for small ,k and non-uniqueness for large X) one can basically use the same arguments as the ones sketched above for the standard Widom-Rowlinson model. However, due to the extra randomness of the edges some parts of the argument become more involved. In particular, there is no longer a deterministic bound (corresponding to K m a x ) o n how much the number of connected components can decrease when a point is added to the random-cluster configuration; thus more work is needed to obtain an analogue of the first inequality in (10.2.6). To conclude, we note that the Widom-Rowlinson model on R d has an obvious multitype analogue with q > 3 different types of particles. This multitype model still admits a random-cluster representation from which the existence of a phase transition can be derived (Georgii and H~iggstr6m, 1996). There is, however, no partial ordering like (10.2.2) giving rise to stochastic monotonicity or an analogue of (10.2.3).
Acknowledgments
It is a pleasure to thank J. L. Lebowitz for suggesting (and insisting) that we should write this review. We are also grateful to L. Chayes, A. C. D. van Enter and J. L6rinczi who looked at parts of the manuscript and made numerous suggestions, and to Y. Higuchi for discussions on Proposition 8.5.
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Exact Combinatorial Algorithms: Ground States of Disordered
Systems
M. J. Alava Laboratory of Physics, Helsinki University of Technology, PO Box 1100, HUT 02015, Finland
P. M. Duxbury Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, MI 48824, USA
C. F. Moukarzel Instituto de Ffsica, Universidade Federal Fluminense, 24210-340 Niteroi, RJ, Brazil
H. Rieger Institut fiJr Theoretische Physik, Universit#t des Saarlandes, 66041 SaarbrScken, Germany
PHASE TRANSITIONS
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Connectivity percolation
6.3
Minimal path
6.4
G r e e d y algorithms and extremal processes . . . . . . . . . . . . . . . . .
7 R a n d o m Ising magnets
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195 196
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Interfaces in r a n d o m - b o n d magnets . . . . . . . . . . . . . . . . . . . . .
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Random-field magnets and D A F F
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Ising spin glasses and Euclidean matching . . . . . . . . . . . . . . . . .
.....................
8 Line, vortex and elastic glasses . . . . . . . . . . . . . . . . . . . . . . . . . .
221 234
8.1
Introduction and overview
. . . . . . . . . . . . . . . . . . . . . . . . .
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Disordered flux arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Arrays of directed polymers
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G a u g e (vortex) glass
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Disordered elastic media
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241
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9 Rigidity theory and applications
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9.2
Rigidity theory
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9.3
Rigidity percolation on triangular lattices
9.4
Connectivity percolation
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Applications to soft materials . . . . . . . . . . . . . . . . . . . . . . . .
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10 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Exact combinatorial algorithms
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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297 297
Overview
This review provides an introduction to combinatorial optimization algorithms and reviews their applications to ground-state problems in disordered systems. We focus on problems which are solvable in polynomial time, and for which exact large-scale (e.g. a million sites) solutions are possible. This excludes some old favorites such as the traveling-salesman problem for which no exact polynomial algorithm is known. Nevertheless, we demonstrate that many difficult ground-state problems, such as: the random-field Ising model and diluted antiferromagnets in a field; spin glasses in two dimensions; domain walls in randombond magnets; arrays of directed polymers; periodic elastic media; and rigidity percolation, are exactly solvable in polynomial time. In addition to their intrinsic interest, these problems provide a set of systems on which heuristic algorithms should be checked for accuracy and convergence. The intent of this review is to collect together and introduce the relevant combinatorial algorithms (Sections 2-5) and to illustrate and review their applications to physics problems (Sections 6-9). Our introduction and survey of the relevant optimization algorithms begins in Section 2, which introduces the basic graph terminology and some of the mathematical notation we use throughout the review. We define what is meant by a polynomial algorithm and emphasize the importance of such algorithms. Section 2 also introduces the basic search algorithms (breadthfirst search and depth-first search) and methods for finding minimal paths (e.g. Dijkstra's method) and minimal spanning trees (e.g. Prim's algorithm). Section 3 covers flow algorithms (augmenting path and preflow methods), starting with the maximum-flow problem and continuing to the minimumcost-flow problem. Section 4 introduces matching algorithms, with particular emphasis on bipartite matching. Section 5 completes the algorithm survey by outlining the connection between the integer optimization algorithms described in Sections 2-4 and the problem of mathematical (e.g. linear)
programming. We describe in detail some of the key algorithms, which we write in "pseudocode" which is typical of the computer science literature. These algorithms, which had their origins concurrently with the development of "computer science" as an independent discipline, are being continuously refined. Nevertheless the basic ideas about search, path, tree, flow and matching methods were developed in the 1950s. However, important conceptual developments have occurred recently such as preflow methods for flow problems and interior-point
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methods for linear programming. There are excellent accessible textbooks on algorithms (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) as well as more specialized books and reviews covering flow algorithms (Ford and Fulkerson, 1962; Ahuja et al., 1993a, b), matching (Lov~sz and Plummer, 1986) and mathematical programming (Papadimitriou and Steiglitz, 1982; van Leeuwen, 1990; Rockafellar, 1984; Wright, 1997). Efficient free software is available on the internet (see note after the References). Ones that we have downloaded and used include: the very broad LEDA library; the efficient flow algorithms by Goldberg; and the interior-point convex-cost programming codes by Wright. In some cases, e.g. rigidity percolation, the algorithmic theorems were available (Hendrickson, 1992) but the first implementations were carried out by physicists (Jacobs and Thorpe, 1995; Moukarzel and Duxbury, 1995). Sections 6-9 cover the applications of combinatorial-optimization methods to a wide variety of problems in disordered systems. Since these sections cover many different problems, it is not possible, in one review, to do justice to all contributions to each of them. However, we do strive to include all of the papers which use exact combinatorial algorithms, which are in the polynomial class, to study these problems. In a few cases we also allude to some important advances in non-polynomial, but exact methods, for example in the spin glass problem. There are an enormous number of useful heuristic (approximate) methods which we do not exhaustively cover, though we do refer to them when relevant. The applications discussed in Sections 6-9 rely upon mappings between optimization problems, discussed in Sections 2-4, and physics problems. Some of these mappings are quite easy to see, but others are quite sophisticated and require considerable formal development (e.g. rigidity theory). In many cases, we devote separate subsections to the formal development and to the description of the key mappings. In Section 6, we discuss the connectivity-percolation problem (Stauffer and Aharony, 1994) and the minimal-path problem (Halpin-Healy and Zhang, 1995; L~issig, 1998). Precise algorithms have traditionally played a key role in the analysis of these two problems. We clarify the relationship between these traditional algorithms and optimization methods. In particular: Prim's algorithm for minimum spanning tree is essentially the same as the invasion algorithm for percolation; while Dijkstra's algorithm finds minimal paths, as does the transfermatrix method, however, Dijkstra's method is more general as it allows overhangs. In addition, both the Prim and Dijkstra algorithms work by generating a growth (Barabasi and Stanley, 1995) or invasion front which sweeps through the lattice. The invasion rules are in both cases extremal or greedy, and they generate rough growth fronts without tuning any variables. That is, they display "self-organized" critical behavior (Bak et al., 1987).
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In Section 7, we discuss random Ising magnets (Young, 1998). In particular: interfaces in random-bond magnets; random-field ferromagnets (RFIM) and diluted antiferromagnets in a field (DAFF) and; spin glasses and Euclidean matching. In each of these problems, there is a different mapping to an optimization problem. The optimization methods are restricted to ground states or excitations generated in some way from the ground states. Nevertheless groundstate properties capture much of the most interesting and difficult aspects of disordered systems. For example, metastability, lack of self-averaging, chaos and other phenomena which are difficult to analyze precisely are most evident at zero temperature. In the jargon of the renormalization group, these problems are frequently governed by a zero-temperature fixed point. As in the directedpolymer problem, the functional renormalization group has been quite successful in predicting the exponents for random-manifold and random-surface problems. The first precise confirmations of these results in (2 + 1) dimensions have resulted from the application of optimization methods. However, the continuum theory of the RFIM and DAFF phase transitions in three dimensions remain ambiguous, despite some high precision data generated using optimization methods (Section 7.3.3). Section 8 covers models for disordered vortex matter (Blatter et al., 1994). Several limiting cases of this very general problem can be solved exactly using optimization methods. These limiting cases are derived in Section 8.2. All of the tools developed in Sections 2-4 find application in various limiting cases of disordered vortex matter. One particularly surprising result is that the minimum-cost-flow method finds the exact ground-state configuration of a set of N directed polymers with g-function interaction (Section 8.3). Another solvable limit is the problem of a periodic elastic medium. Until recently there was a controversy about whether the ground state of periodic elastic media was in the same class as conventional rough surfaces in (2 + 1) dimensions, or whether there is a super-rough phase in the ground state. Optimization methods have convincingly confirmed the renormalization-group prediction of a superrough phase (Section 8.5.5). However, further extensions of the optimization methods have allowed the inclusion of dislocations in the periodic elastic medium. This destroys the super-rough Bragg glass state at low temperatures (Section 8.5.6). Rigidity theory and its applications (Thorpe and Duxbury, 1999) to rigidity percolation, and soft condensed matter is the subject of Section 9. The basic model here is simple to define. Consider a lattice whose vertices are connected together with stiff Hooke springs. Actually they could be connected together by any central-force springs. The key point is that the only restoring force is axial. Two points connected together by a spring are free to move as long as they maintain their axial separation. Now we want to know how many such springs are needed to make a set of points rigid with respect to
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each other. Of course we can solve for the forces using Newton's laws but that is tedious. Instead there are combinatorial methods which just count degrees of freedom and compare that freedom with the number of constraints. Rigidity theory (Section 9.2) makes this counting precise and turns it into a powerful tool which identifies flexible, rigid and overconstrained regions of soft materials. This theory also enables a mapping of rigidity theory to an optimization problem (bipartite matching), which in turn has led to the precise analysis of rigidity percolation on triangular lattices. There are many analogies with connectivity percolation, which can be seen as one limit of the general rigidity problem (Moukarzel and Duxbury, 1999). Further extensions to soft condensed matter, such as proteins, are also starting to develop (see Section 9.5). We make a few closing remarks in Section 10.
2
2.1
Basics of graphs and algorithms
Graph notation and overview of network problems
A graph ~(V, A)(Harary, 1969; West, 1996; van Lint and Wilson, 1992)consists of a vertex set V and an arc set A (= edge set E). Each member of the edge set (i, j ) 6 A connects two of the vertices i 6 V and j 6 V in the vertex set. We use A(i) to denote the set of arcs connected to node i. We use bond, arc and edge interchangeably. Similarly with node, site and vertex. Bipartite graphs are a very important subclass of graphs, and frequently are easier from a computational standpoint. A bipartite graph/3(X, Y, A) is a graph in which the node set V = X U Y can be partitioned into two parts X and Y, such that all of the arcs in the graph connect nodes in X with nodes in Y. Nearest-neighbor hypercubic lattices are bipartite, while triangular and face-centered-cubic lattices are not. Most physics applications involve sparse graphs which have a small number of arcs incident on each node. Many of the upper bounds on algorithmic performance occur on dense or complete graphs where a very large number of arcs are incident on each node. Complete graphs correspond to infinite-range models where every node is connected to every other node. We emphasize the algorithmic methods and bounds for sparse graphs. We assign a cost, r and/or a capacity, Uij , t o each edge in the graph. The capacities, Uij, a r e restricted to be non-negative integers, while the costs may be arbitrary integers. In many of the applications discussed in this review U ij and r a r e integers which are randomly drawn from uniform, Gaussian or bimodal distributions. Once the graph nodes or edges have been assigned flow capacities or costs, they are frequently called networks, and they are studied using network-
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optimization methods (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Cormen et al., 1990; van Leeuwen, 1990). We consider directed arcs, in which an arc (i, j) has cost Cij and capacity Uij for a path which goes from node i to node j. An arc in the opposite direction (j, i) may have a different cost Cji and a different capacity Uji. Undirected arcs are the special, symmetric case, in which Cij -- Cji and Uij -- Uji. In the physics applications, it is usual to consider either the undirected case, or the special directed case where either U ij -- 0 o r u ji "- O. Note that, even in these special cases, the algorithmic procedures for flow problems (Ford and Fulkerson, 1962; Ahuja et al., 1993a, b) frequently generate a residual graph which has forward and reverse arcs with different costs or capacities. Thus, in the algorithms, we must keep track of a graph in which Cij ~ Cji and Uij ~ Uji. When we refer to a graph G(V, A), we mean a graph with both forward and reverse arcs which have, in general, different values for their costs, capacities and flows. In minimum-path and minimum-spanning-tree problems each bond is assigned a cost, Cij, and the problem is to optimize the appropriate cost function. In the maximum-flow problem, each bond is assigned a capacity Uij and the problem is to find the maximum flow which the network can sustain, given those capacities. In minimum-cost flow each bond is assigned a cost and a capacity. The problem is to find the lowest cost flow, given the capacity constraints and a set of sources and sinks. The sources and sinks are introduced at the vertices of the graph. Several other vertex and edge variables will be defined and used in the more detailed sections describing the algorithmic solutions to these network-optimization problems. In graph algorithms we frequently use the concept of a path, e.g. Pst, between two specified nodes s and t. Any such path consists of a connected sequence of edges. In many problems we are interested in optimal or lowest-cost paths. In flow (Ahuja et al., 1993b) and matching problems (Lov~isz and Plummer, 1986), we often iteratively find the optimal solution by selecting paths and augmenting the solution using them. We restrict all of the variables and parameters in the problems to the integers. This restriction maintains the property that an exact solution is possible in polynomial time, but also enables us to model problems like directed polymer arrays using integer flow lines. Before proceeding to the algorithms, we give a brief overview of the ideas and notation of computational complexity, in particular we give a more precise definition of what we mean by a polynomial algorithm.
2.2 Algorithmic complexity Given a graph, ~(V, A), a problem is polynomial (P) if its solution can be found in a time bounded above by a polynomial in the number of arcs and/or number
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of vertices in the graph (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990). The number of nodes in the graph is denoted by IVI, and the number of arcs is IAI. If an algorithm is in P, then it is O(IAI x IVIY), where x and y are finite. The big 0 denotes the scaling behavior of the upper bound on the computational time. The lower bound, which we shall not use, is denoted by big f2. The upper bound is not typical of most physics applications, and we use ~ IAlxlvI y to denote the scaling behavior of a particular application (this is the average computation time). The discovery of an exact polynomial algorithm for a hard combinatorial problem is usually of great theoretical and practical (in terms of computational efficiency) significance. This review is devoted to P problems and their applications. The notation for problems which do not have polynomial solutions is less intuitive. NP (non-deterministic polynomial) refers to problems for which a solution can be verified in polynomial time. Clearly the class NP contains the class P. NP-complete refers to NP problems which: (i) can be verified in polynomial time; (ii) for which there is no known exact polynomial algorithm (on "classical" computers) and; (iii) which are related to each other by polynomial reductions. This means that if a polynomial algorithm is found for one NPcomplete problem, polynomial algorithms exist for all the others. Three examples of NP-complete problems are: the traveling-salesman problem; satisfiability (whether a set of logical clauses contradicts itself); Hamiltonian circuits and; 3-matching (a generalization of bipartite matching). The NP-complete class is now rather large, so it appears very unlikely that the NP-complete problem class will eventually be proven to have P solutions. Nevertheless efficient algorithms can be found for some "typical" cases of NP-complete problems. On the other hand, certain realizations of NP-complete problems are very difficult to solve, and there is speculation that the difficult regimes are critical in the statisticalmechanics sense. (Proximity to a phase transition does frequently, but not always, increase the computational time in the ground-state problems discussed in Sections 6-9). Finally, it should be noted that in practice an algorithm which is not polynomial can still be competitive with P algorithms. The classic example is the simplex method for linear programming, which is not polynomial. Simplex methods remain competitive (in practice) with interior-point methods (which are in P) even for large sparse problems.
2.3
Basic Algorithms
We discuss three basic classes of graph algorithm: search (Section 2.3.1); minimal path (Section 2.3.2) and; minimal spanning tree (Section 2.3.3) (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990). These problems are solved exactly by greedy algorithms (Papadimitriou and Steiglitz, 1982; Lov~isz and Plummer,
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1986; Cormen et al., 1990) which work by choosing moves which are locally optimal. This locality condition makes their implementation very efficient. We show that for sparse graphs the computational time of efficient algorithms for these problems are O(IAI). In contrast, generic implementations are O(IVI2).
2.3.1
Search algorithms
Consider a connected graph {7(V, A) containing a vertex set V and arc set A. A connected graph has sufficient arcs such that a connected path exists between any two nodes i and j. We input the graph connectivity, i.e. a list of arcs which are present (the parameters Uij and Cij do not play any role in the calculation). There are two basic search strategies on {7(V, A)" breadth-first search and depthfirst search. These strategies are closely related, though depth-first search is more complicated. Breadth-first search (Cormen et al., 1990; West, 1996) labels each vertex with the minimum number o f bonds l(i) which must be crossed in order to go from a starting site s to site i. This is found by starting at s, which is assigned label l(s) - O, and by growing or burning outward from that site. In the algorithm below we define S to be the set of labeled sites, and S to be the set of unlabeled sites. Clearly S U S = V. We also use a vector called p r e d ( i ) , which identifies the site from which the breadth-first search arrived at the current site. This is important in reconstructing breadth-first paths and is used in the flow algorithms described in Section 3.
algorithm begin
Breadth-first
search
S-{s}, s=V\{s} l(s) -- O,
pred(s) = 0
while ISI < IVI do begin choose ( i , j ) 9 l(i) : mink{l(k)lk e S}, j e S, ( i , j ) e A then l ( j ) = l(i) + l , S - S \ { j } , S : S U { j } , pred(j) - i end m
end (Here we use a U b to denote the union of two sets a and b and a \ b to denote the set remaining after the elements of set b are removed from the set a.) The above algorithm is very inefficient if the choose operation above requires a search over all i 6 S at each step. However, it is easy to keep an ordered list of sites, those at the growth front, from which the next growth site is selected. An efficient algorithm generates and adds to the list all of the next candidates for burning each time a local advance of the growth front occurs. The sites which are added to
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this list have a monotonically increasing label (1(i)). Thus sites are added to the top of the list (which has highest label) and growth is to the site at the bottom of the list (which has the lowest label). This list is often called a "first-in-firstout" or FIFO list (Cormen et al., 1990). The algorithm ends when the FIFO list is empty. Using this procedure ensures that breadth-first search is O(IA[). In the percolation literature the label l(i) has been called the chemical distance (Stauffer and Aharony, 1994). This and other applications of breadth-first search will be discussed in Section 6.2. Depth-first search (Cormen et al., 1990; West, 1996) labels in the same way as breadth-first search, except it labels only one of the next level unlabeled sites at each iteration, and growth is always from the highest available label. Clearly a strategy for choosing which of the next level sites to label is needed. This choice depends on the application. A common and useful choice, for a planar graph, is to "stay as far to the right as possible" or "as far to the left as possible". These procedures identify the perimeter of a graph. Depth-first-search paths may lead to a "dead end", in which case it is necessary to "backtrack" to the highest available label which is not at a dead end. Depth-first search is the basis of algorithms for the hull in percolation. Finding the backbone in connectivity percolation is possible once a graph has been separated into its biconnected parts. An O(IAI) algorithm to find the biconnected parts (and hence the backbone) has been available since the early 1970s (Tarjan, 1972), but it was only noticed recently by the physics community (Grassberger, 1992a, 1999). There are several methods available for finding the backbone. The standard method in the physics community is the burning algorithm which does a forward and backward breadth-first search to iteratively remove dangling ends (Herrmann et al., 1984). However, a forward and reverse depth-first search of the whole graph finds the hull of the backbone in two sweeps of the graph, and is more efficient. Nevertheless, high precision results have been found using efficient implementations of burning (Rintoul and Nakanishi, 1992, 1994), at the percolation threshold. More recently matching (Moukarzel, 1998a) methods have also been used to find the backbone and have also lead to high-precision results. Nevertheless, all of the available backbone algorithms remain storage limited, as some information at each node of the graph remains necessary. For this reason the very large calculations carried out for the infinite cluster (see Rapaport (1992) and Lorenz and Ziff (1998)) remain out of reach for the backbone.
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2.3.2
Exact combinatorial algorithms
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Shortest path
Given a set of costs Cij o n each arc of a graph, we calculate the distance label d(i), which is the cost of a minimum-cost path (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) from a starting node s to the node i. Dijkstra's algorithm, which works for non-negative costs, is a label-setting algorithm to solve the shortest-path problem. It is a label-setting algorithm because it finds the exact distance label correctly at the first attempt. In contrast, label-correcting algorithms iteratively approach the exact distance label, and work even if some of the costs are negative, provided there are no negative-cost cycles in the graph. In order to reconstruct the set of bonds which make up the minimal-cost path from site s to site i, the algorithms also store a predecessor label, pred(i), which stores the label of the previous site from which the minimal path reached i. The set of minimal paths from a starting site s to all of the other sites in the graph forms a spanning tree, Tp. An example is presented in Fig. 2.1. As will be shown in Section 6, the minimal-path problem is closely related to a polymer in a random medium and hence to growth problems. Both label-setting and label-correcting algorithms use the key properties which shortest paths obey: (i) For each arc belonging to the shortest path tree from node s:
d(j) -- d(i) +
(i, j)inTp.
Cij
(ii) For each arc which does not belong to a shortest path:
d(j) <_d(i) +
(i, j)not inTe.
Cij
These properties are often discussed in terms of reduced costs, defined as, C d - - Cij + d ( i )
-
d(j).
(2.3.1)
Properties (i) and (ii) above are then, cd - O ,
if
( i , j ) E Tp,
(2.3.2)
and c/~ > 0,
otherwise.
(2.3.3)
The proof of these properties relies on the spanning-tree structure of the set of minimal paths, namely that each site of the tree has only one predecessor. Thus if there is a bond with c/~ < 0 which is not on the minimal-path tree, adding that bond to the tree and removing the current predecessor bond for site j (which from condition (2.3.1) has zero reduced cost), leads to a reduction in the cost of
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1271~ 7 [21[ 3 117t 2 ~19t ] ~12ol 5
13~
6
113t 2 ~lst 1 ~I~6~5 ~12~1 l'q
, F1
I]!to 1skis3 ~Izll~9 161191k~ Iz~9141 2
3
151
Io
k s'lsl Fig. 2.1 The tree of minimal paths from the source node (shaded) to all other nodes in a directed square lattice (The arcs of the graph only allow paths which are in the positive {01 } and positive {10} directions). All bonds between nearest neighbors are labeled with their costs, but only the tree of minimal paths is shown. Each node is labeled with the cost of a minimal path from the source to that node. (Generated using the demonstration programs from the LEDA library (see note at end of References).)
the minimal-path tree. Thus any tree for which there exists a bond with c/~ < 0, is not a minimal-path tree. For later reference we also note that a directed cycle, W, has the property,
Z cd-- Z Cij, (i,j)eW (i,j)~W
(2.3.4)
which follows from (2.3.1)-(2.3.3). We now discuss Dijkstra's method for the minimum path, which works by growing outward from the starting node s in a manner very similar to the breadthfirst search. At each step Dijkstra's algorithm chooses to advance its growth front to the next unlabeled site which has the smallest distance from the starting node.
algorithm Dij kstra begin S={s}, s : v\{s}
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d(s) = O, pred(s) = 0 while [SI < IVI do begin choose (i, j ) 9 d ( j ) -- mink,m{d(k) + Ckmlk E S, m E S, (k, m) E A} then S - - S \ { j } , S-SU{j}, pred(j)=i end end
The algorithm maintains the minimal distance growth front by adding the node j E S with minimal distance label d (j). The proof that d (j) generated in this way is actually a minimal-cost label proceeds as follows: (i) Assume that we have a growth front consisting of sites which are labeled with their minimal path lengths to the source s. (ii) The next candidate for growth is chosen to be a site which is not already labeled, and which is connected to the growth front by an arc (i, j) E A. (iii) We choose the site j for which d ( j ) is minimal d ( j ) = min~,m {d (k) + Clcm,k E S , m E S, (k,m) EA}. m
(iv) Because of (iii) and because all of the costs are non-negative there can be no path from the current growth front to site j which has smaller distance than d ( j ) . This is because any such path must originate at the current growth front and hence must use a non-optimal path to generate any alternative path to j (negative costs can compensate for locally non-optimal paths from the growth front and hence Dijkstra's method is restricted to non-negative costs.) Minimal path with positive costs is an example of a global-optimization problem which is exactly solved by local growth (greedy) dynamics. The name greedy was invented for the problem of a maximal weight forest (MWF), which is related to minimal spanning tree by a simple transformation. MWF works by choosing the highest-weight edges at each step, hence the term greedy. The term extremal has been used in the physics community to refer to algorithms that choose either largest-cost or smallest-cost possibilities at each step. We discuss this in a more general context in Section 6.4. The "generic" Dijkstra's algorithm scales as o(Ig12), if the choose statement in the above algorithm requires a search over all the sites in the lattice. It is easy to do much better than this by maintaining a list of active sites at the growth front (as in breadth-first search). However, now we must choose the lowestcost site from among this list. Thus the potential growth sites must be ordered according to their distance label. This ordering must be reshuffled every time a
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new growth site (with a new distance label) is added to the list. In the computer science community this is typically done with heaps which consist of a treelike data structure. Heap reshuffling is O(lnlAI) which reduces the algorithmic bound to O(IAIlnlAI). However, if the bond costs are integers(as we consider here), the site distance labels themselves can be used as pointers. Thus we can set up a queue with the distance label as pointers and the site labels with that distance in the queue(or, in computer science terminology, we use buckets). The number of buckets that is required, nb, is n b > 2C where C = max(i,j){Cij } is the maximum cost. For example if the costs are chosen from the set 1, 2 . . . . . 10, then C -- 10. As long as C is finite, and the graph is sparse, buckets are very efficient both in speed and storage (Dobrin and Duxbury, 1999; Duxbury and Dobrin, 1999). Using buckets, the next growth site is immediately given as one of the sites with smallest distance pointer. In addition, since each potential growth site has a distance label, we also know where to add any new growth sites in the queue. The queue constructed in this way contains up to z (z is the co-ordination number of a site) copies of each growth site. Only one of these copies is used in constructing Te. The redundant copies are deleted once a site has been added to Te. Using distance pointers, we have implemented a Dijkstra's algorithm for integer costs which scales as O(IAI). Note that breadth-first search is the special case of identical costs (i.e. Cij : constant) in Dijkstra's method. As we said, Dijkstra's algorithm is a label-setting algorithm. We now describe a label-correcting method, which solves the minimal-path problem when there are negative costs provided there are no negative cycles (i.e. closed directed paths W with E(i,j)~w Cij < 0). Equations (2.3.1) and (2.3.2) form a basis for the following label-correcting algorithm. The idea is to start with a non-optimal spanning tree and to iteratively remove bonds with negative reduced costs from it. This iteration proceeds until no negative reduced-cost bonds remain. algorithm L a b e l - c o r r e c t i n g begin d(s) : 0 a n d pred(s) : 0
d ( j ) : o o for each node j E V\{s} while some arc (i,j) s a t i s f i e s d(j)>
begin
d(i)-}-cij do
d ( j ) : d ( i ) -}- cij
pred(j) -- i end end Initially the distance labels at each site are set to a very large number (except the reference site s which has distance label d(s) = 0). This method requires that
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the starting distance labels d ( j ) > exact values and the choice of d ( j ) = cx~ ensures that. In practice it is efficient to grow outward from the starting site s. The algorithm may sweep the lattice many times until the correct distance labels are identified. The worst case bound on running time O(min{lVl2lAlC, IAl21Vl}) with C = max{Icijl}, which is pseudo-polynomial. An alternative procedure is to sweep the lattice once to establish approximate distance labels and then to iterate locally until local convergence is found. This FIFO implementation has complexity O(I VI IAI). Note that if there are negative cycles, the instruction d ( j ) = d(i) + Cij would decrease some distance labels ad (negative) infinitum. However, if there are negative cycles, one can detect them with an appropriate modification of the label-correcting code: One can terminate if d(k) < - n C for some node i (again C = max [cijl) and obtain these negative cycles by tracing them through the predecessor indices starting at node i. This will be useful in the negative-cyclecancelling method for minimum-cost flow (Section 3.2.2).
2.3.3
Minimal spanning tree
The minimal spanning tree (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) of a connected graph with arc costs Cij is a tree which: (i) visits each node of the graph and; (ii) for which Y~tree cij is a minimum. Prim's algorithm and Kruskal's algorithm are two methods for finding the minimal spanning tree (Cormen et al., 1990; West, 1996). Prim's algorithm is very similar in structure to Dijkstra's algorithm, although the physics is very different. In fact (see Section 6.2), as has been noted (Barabasi, 1996), Prim's algorithm is essentially equivalent to the invasion algorithm for percolation. In Prim's algorithm we start by choosing the lowest cost bond in the graph. The algorithm then uses the two sites at the ends of this minimal cost bond as the starting sites for growth. Growth is to the lowest cost bond which is adjacent to the growth front. The algorithm terminates when every site has been visited. The cost of the minimal spanning tree is stored in Cr, and the bonds making up the minimal spanning tree are stored in T.
algorithm Prim begin choose (s, r) 9 Csr then S - {s, r} , S while ISI < IVl do begin choose (i, j ) 9
- mink,m{Ckml(k, m ) e A} V\{s, r},
T -- (s, r ) ,
C r = Csr
ci] - mink,m{Ckmlk e S, m e S, (k, m ) e A}
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then S -- S \ { j } ,
S = S U {j},
CT -- CT + Cij,
T -- T U (i, j )
end end It is evident that Prim's algorithm is almost identical to Dijkstra's algorithm. The only difference is in the choose instruction, which gives the cost criterion for the extremal move at the growth front. In the minimal-path problem, one chooses the site at the growth front which has the minimum-cost path to the source, while in the minimal-spanning-tree problem one simply chooses the minimum-cost bond. Both of these problems lead to spanning trees, but Prim's has lower total cost (it is less constrained). In Dijkstra's algorithm one is checking the cost of the path from the tested site all the way back to the "source" or "root" of the tree. It is thus more non-local. As we discuss in Section 6.3, Dijkstra's algorithm leads to self-affine growth fronts, while Prim's method leads to self-similar growth fronts. Using the same distance pointer strategy as described above for Dijkstra's method, Prim's algorithm is O(IAI). Heap implementations of Prim's method are O(IAllnlAI) as for Dijkstra's method. An alternative method for finding the minimal spanning tree is Kruskal's algorithm, which nucleates many trees and then allows trees to merge successively into one tree by the end of the procedure. This is achieved by allowing growth sites to nucleate off the growth front. That is, growth occurs at the lowest cost unused bond which does not lead to a cycle, regardless of whether it is on or off the growth front. Efficient Kruskal's algorithms have similar efficiency to heap implementations of Dijkstra's method (i.e. O(IAllnlAI)).
3
3.1
Flow algorithms
Introduction
In this chapter we introduce flow problems and describe the algorithms which solve them (Ford and Fulkerson, 1962; Goldberg and Tarjan, 1988; Goldberg et al., 1990; Goldberg, 1992; Ahuja et al., 1993a, b). As their name suggests the characteristic feature of flow algorithms is that a quantity of flow is injected at one or more vertices of a graph(these are the sources) and the same quantity of flow is removed at another set of vertices(the targets or sinks). We define Xij >~ 0 to be the flow passing from node i to node j, and at each node in the graph flow is conserved. The vector x = {xij} describes the flow in all of the edges in the graph. Note that the requirement that Xij >_ 0 does not exclude flow from node j to node i. Any such flow is represented by Xji >__O. In general the flow algorithms generate asymmetric networks where the costs Cij ~ Cji. Although the flows are positive, the costs may be negative, provided there are no negative-cost cycles.
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Flow problems are familiar in disordered systems, in the form of resistor networks, flow in porous media etc. The flow algorithms described here solve efficiently problems which have non-linear (but convex) characteristics, and which can also have a threshold for perfect plastic response. A third important feature of combinatorial flow algorithms is the discreteness imposed by requiring integer flows. In particular, a flow of one unit along a path can be used to model a nonintersecting polymer. More surprisingly, arrays of polymers with delta-function interactions can be modeled using integer flows (see Section 8.3). In Section 3.2 we introduce the maximum-flow problem in which each edge in a graph has a maximum flow capacity (blij). In this problem we will find the maximum flow between two vertices s (the source) and t (the target) given that each bond in the network can transport a maximum flow of blij. Note that in the maximum-flow problem there is no optimization in the sense of a cost function. We are only interested in the maximum flow. The problem is defined by a set of constraints (0 < xij ~ uij), rather than by a cost. Associated with the maximum flow is a minimum cut (see Section 3.2.1) and frequently the minimum cut is relevant to the physics applications. In Section 3.3 we introduce the minimum-cost-flow problem. In this problem there is a cost function C(x) - Y~(i,j)hij(xij) associated with the flow, xij in each bond and a capacity constraint, blij , o n each bond. Minimum-cost flow thus includes maximum flow as the special case in which all of the bond costs are constant (independent of the flow). However, maximum flow is a sufficiently important special case that it is usually treated separately. More particularly it may be solved more efficiently than the general minimum-cost problem. The minimum-cost-flow problem has polynomial algorithms if the cost function h(x) is convex. One convex-cost function is very familiar namely P = y~ x 2 Rij, which is the power in a resistor network. Minimum-cost-flow algorithms thus find the current flow in resistor networks. This is not very exciting. However, two important generalizations which are also solved by minimum-cost algorithms are: flow in resistor networks in which each bond has a fixed maximum current Uij , and; non-linear resistors, which correspond to a non-linear cost function, h(x) ~ x ~, with c~ > 1 to preserve convexity, and ot integer to preserve the integer character of the flow. The constraint 0 < xij ~ blij can be interpreted as a perfectly-plastic response (i.e. a threshold after which the flow in a bond is constant) and is a hard problem if attempted by conventional means. Several other, not so obvious, physics applications are described in Sections 7 and 8. Flow problems have a wide variety of applications including: traffic, phone communications and water distribution networks. There are many excellent books and review articles devoted to this class of problem.. The classic work by Ford and Fulkerson (1962), who invented the augmenting path method, has been continuously refined over the years. The preflow methods introduced by Tarjan and Goldberg (Goldberg and Tarjan, 1988; Ahuja et al., 1993b) have provided new insights
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and have led to significantly improved algorithms. Recent stochastic algorithms which find the minimum cut by iterative contractions also look promising, but are beyond the scope of this review. Integer minimum-cost flow is related to integer programming and in engineering and operations-research textbooks is frequently discussed in that context (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984). However, the dedicated flow algorithms are significantly faster than convex-cost programming codes (see Section 5), so for large-scale analysis (as is typical of physics problems) it is important to use algorithms like those discussed below.
3.2 3.2.1
Maximum flow/minimum cut Basic ideas and definitions
Let us define a capacitated network as a graph ~(V, A), each of the arcs (i, j) 6 A has a capacity, lgij > O, which we take to be non-negative. A flow in the network ~(V, A) is a set of non-negative integers, Xij, subject to the capacity constraint, Igij , for each bond 0 < xij < uij u j) ~ A (3.2.1) and to a mass-balance constraint (i.e. flow conservation) for each node. In many applications (e.g. the physics applications in Sections 7.2 and 7.3) the special case in which there is one source, s, and one sink, t, arises. This is sometimes called the min(s-t)-cut/max-flow problem. In this case, the mass-balance constraints are,
Z Xij -- ~ Xji = {jl(i,j)Ea} {jl(j,i)Ea}
v --V 0
fori = s for i -- t else
(3.2.2)
where v is the value of the flow from the source s to the target (or sink) t. The maximum-flow problem for the capacitated network G is to find the flow x that has the maximum value v under the constraints (3.2.1) and (3.2.2). When the maximum flow is being transported from s to t, all of the flow capacity between s and t is being used. However, not all of the arcs carry flow equal to their capacity. In fact there exists a bottleneck or minimum cut on which the capacities Uij a r e saturated. In the physics applications, the relevant flow geometry is quite often like that shown in Fig. 3.1. To discuss the relation between maximum flow and minimal cut, we introduce some notation: A cut is a partition of the node set V into two subsets S and S - V\S denoted by [S, S]; We refer to a cut as an s-t-cut if s E S and t 6 S; The forward arcs of the cut [S, S] are those arcs (i, j ) 6 A with i E S and j 6 S, the backward arcs those with j E S and i E S; The set of all forward arcs of [S, S] is denoted (S, S); The capacity of an s-t-cut is defined to be u[S, S] --
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Fig. 3.1 A flow network. Each arrow has a capacity uij, and a flow 0 < Xij < uij. The "source" node is the leftmost node and the "target" node is the rightmost node. The saturated bonds (i.e. xij = uij ) on the "cut" have been removed. Actually only the "forward arcs" on the cut are saturated. (This figure was generated using the graphing routines provided with the LEDA package (see note at end of References).)
Y~(i,j)~(s,-s) Uij. Note that the sum is only over forward arcs of the cut. The minimum cut is an s-t-cut whose capacity u is minimal among all s-t-cuts. Now we state the max-flow/min-cut theorem of Ford and Fulkerson (Ford and Fulkerson, 1962; van Lint and Wilson, 1996), which is a key result in network flows. T h e o r e m (min-cut/max-flow) In a transportation network G(V,A) the maximum value of x over all flows {Xij} is equal to the minimum value u[S, S] over all cuts [S, S].
Proof: Let x be a flow, v its value and [S, S] an s-t-cut. balances for all nodes in S we have
Then, by adding the mass
l) ~ i~ES{ ~ Xij-Z Xji} --- Z Xij-- Z Xji" 9 {jl(i,j)6A(i)} {jl(j,i)6A(i)} (i,j)E(S,-S) (i,j)E(-S,S) (3.2.3) Since
xij ~ uij
and
Xji >__0
the following inequality holds
13 <
Z (i,j)E(S,S)
Uij = u[S, S]
(3.2.4)
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In these expressions A (i) is the set of arcs incident to or emanating from node i. From (3.2.4), the value of any flow v is less than or equal to the capacity of any cut in the network. Thus, if we discover a flow x whose value v is equal to the capacity of some cut [S, S], then x is a maximum flow and the cut is a minimum cut. [] m
At a superficial level, finding the minimum cut looks combinatorially hard as it seems to require searching over all possible cuts, which is worse than exponential in the number of nodes in the graph. However simple polynomial algorithms exist as shown in Sections 3.2.2 and 3.2.3. They rely on finding the maximal flow and then finding the minimum cut from the flow configuration. Most maximum flow algorithms rely on the concept of a residual network G(x) = ~(x, V, A), which is a graph that is constructed from the original network and the current flow vector x. The starting graph ~(V, A) = ~(0). In the case of maximum flow, the residual graph (G(x)) has arcs with residual capacity rij given by, rij "-" u i j -- x i j + x j i
(3.2.5)
for (i, j ) ~ A.
The residual capacity is the maximum flow that can be sent from node i to node j in residual graph G(x). Note that both the forward and reverse flows appear in this expression, and that the residual capacities, r i j , are always positive. We also have the conservation condition, rij -~- r j i : u i j -Jr- u j i . The residual graph is important in designing algorithms for maximal flow due to the following corollary to the minimum-cut/maximum-flow theorem.
Corollary (to minimum-cut/maximum-flow theorem) If x is a maximum flow with its associated minimum cut (S, S), then rij {(i, j) E A, with i E S, j E S}.
3.2.2
--
0 for all arcs
Augmenting-path method
Starting with zero flow in the network, the augmenting-path method finds paths from s to t and then puts the maximum flow possible onto these paths (Ford and Fulkerson, 1962; Cormen et al., 1990; Ahuja et al., 1993a, b). An augmenting path is thus a directed path from the node s to the node t in the residual network. The capacity of an augmenting path is the minimum residual capacity of any arc in this path. The key observation is that: if there is no augmenting path in the residual network G(x), then the flow x is maximal. Stated formally, we have the following theorem.
Theorem (augmenting path) A flow x* is a maximum flow if and only if the residual network G(x*) contains no augmenting path.
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At each iteration the augmenting path algorithm either performs an augmentation or terminates because no directed path exists from s to t in ~ ( x ) . In the latter case the flow is a maximum flow.
algorithm begin
Augmenting
path
x : 0 while G(x) c o n t a i n s do
a directed
path
f r o m n o d e s to t
begin
i d e n t i f y an a u g m e n t i n g node t
path
P
from n o d e
s to
= min{rij [(i, j) c P} augment end end
6 units
of f l o w a l o n g
P
and update
~(x)
A key part of the implementation of this algorithm is the identification of directed paths from s to t in the residual network. A breadth-first search (see Section 2.3.1) starting at s is an efficient way to do this. Since breadth-first search is O(IAI), and we augment flow by at least one unit per breadth-first search, the augmenting-path algorithm is bounded by O(IAIv). In addition, the maximal flow max(i,j)6a{ttij} is the maximum capacity of any obeys, v <_ UlVI, where U arc in the graph. Thus the upper bound on the complexity of the augmenting-path algorithm is O(IAIIVIU). This is called a pseudo-polynomial bound due to the appearance of U in the expression. Strategies for improving augmenting paths include ways to select the best possible paths for augmentation at each step (Bertsekas, 1995). The augmentingpath method (in combination with Dijkstra's algorithm) forms the basis for the successive-shortest-path algorithm for minimum-cost flow discussed in Section 3.3.3. Now we discuss the push-relabel algorithms which are significantly more efficient than augmenting-path methods (Goldberg and Tarjan, 1988). =
3.2.3
Push-relabel algorithms
The inherent drawback of the augmenting-path algorithms is the computationally expensive operation of repeatedly finding augmenting paths. In order to avoid this, a new strategy, invented in the 1970s by Karzanov, is to violate the massbalance constraints at intermediate stages of a flow algorithm, and to restore them at the end. This is a very general concept in combinatorial optimization: algorithms either operate within the space of admissible solutions and seek optimality
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during iteration, or they maintain optimality and restore feasibility in the end. The preflow-type algorithms (Goldberg and Tarjan, 1988; Ahuja et al., 1993b) begin at the source by "flooding" the network. This is implemented by "pushing" the maximum possible flow along the arcs adjacent to the source. This is a local operation which depends only on the arc capacities. The flow which has reached the sites adjacent the source is then "pushed" to their neighbors in a similar way. This procedure continues through the network, with the maximum possible flow being pushed at each step until the maximum flow is pushed to the target. In carrying out this procedure, some excess flow is left behind at some nodes. This excess is pushed back to the source until the mass-balance constraints are fulfilled. Now we describe this more formally (Goldberg and Tarjan, 1988; Ahuja et al., 1993b). A preflow is a function x that satisfies the flow constraint 0 < xij <_ uij and the following relation for the excess e(i) flow at each node i"
e(i) =
Z XjiE Xij > 0 {jI(j,i)EA} {jI(i,j)EA}
u ~ V\{s, t}.
(3.2.6)
Clearly e(i), which is non-negative, is the degree to which flow conservation is violated at each site. A node i is active if its excess is positive e(i) > 0, and the algorithm terminates when the excesses all reach zero. We again concentrate on the special case of one source s and one target t. The source, s, is assumed to be a perfect flow reservoir and the target, t, is a perfect sink. The algorithm works by pushing flow from the source toward the target. The basic procedure is to select an active node and push flow to its neighbors. To do this in a systematic and convergent manner, one defines a distance function d(i) from which one identifies admissible arcs on which it is possible to push flow. The distance function used here is not equivalent to shortest-path distances except when a global update is carried out. A distance function is valid with respect to a flow x, if it has the following properties, (i) d(t) = 0 and (ii) d(i) < d ( j ) + 1 for every arc (i, j) in the residual network G(x). An arc (i, j ) is admissible if d(i) = d ( j ) + 1. The idea of the algorithm is to either push, or to relabel. This means that if an active node under consideration has no admissible arcs, its distance label is increased by one until at least one arc becomes admissible. This procedure is necessary to get rid of "puddles" of excess flow that get trapped in the network. This relabel procedure preserves properties (i) and (ii). If d(i) (i ~ s, t) is valid then it also has the following properties. (i) d(i) <_ length of the shortest directed path from node i to t in G(x)
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(ii) d(i) >__IVl ~ ~(x) contains no directed path from s to t. We say that d (i) is exact if in condition (i), the equality holds. The push-relabel algorithm starts with an exact set of distance labels and then performs push and relabel operations until no more flow can be pushed from s to t, and until e(i) --- 0, except for s and t.
algorithm begin
Preflow-push/relabel
preprocess while e ( i ) > O
for
any i 6V\{s,t}
do
begin choose i e V \ { s , t } , push/relabel(i)
with
e(i) > 0
end end procedure preprocess begin x:0 F i n d e x a c t d(i) Xsj - - tlsj f o r (s, j) e A , d ( s ) - IVl
e(j) : ttsj.
end procedure push/relabel( i) begin if t h e n e t w o r k c o n t a i n s then p u s h to
j
else d(i)
end
an a d m i s s i b l e a r c (i, j) ~ : min{e(i),rij} u n i t s of f l o w f r o m n o d e
i
= minjev { d ( j ) + 11(i, j ) e A and rij > O}
The residual capacity is r i j = u i j - - X i j + Xji, as for the augmenting-path method. In the preprocess step, it is necessary to find the exact distance labels. This is efficiently performed by doing a breadth-first search with the target t as the starting site. The algorithm terminates when the excess e(i) = 0 for all sites i e V\{s, t}. A sum over the flow in the bonds leaving the source (or entering the sink) then gives the maximal flow. To find the minimum cut requires comparing the flow in
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each bond with its capacity. All bonds with X i j - - U i j are saturated. A breadthfirst search on the residual graph, from either the source or the sink identifies a minimum cut, as in the augmenting-path method. This procedure again shows how finding the maximum flow leads to the minimum cut. The complexity of the generic preflow algorithm is O(IVI2IA]), as the maximum number of relabel steps at each site is IVI, and the maximum number of push steps(between each relabel) at each site is ]A]. An important decision in implementing the push-relabel method is the order in which sites are chosen for the push-relabel operations. A queue of active sites is established. Sites are chosen from this queue and as new sites with a positive excess are born they are added to the list. There are several ways to order the active site list: highest label; maximum excess; last-in first-out; first-in first-out. The highest-label option produces the least non-saturating pushes. In general, one is interested in minimizing both the number of pushes and relabels during a run, which makes the choice of the best stack problem-dependent (Cherkassky and Goldberg, 1997). For problems related to Ising models, for which the capacities of the edges are of the same order of magnitude, the FIFO stack results in almost linear scaling (o(Igl 1"2) (Sepp~il~i, 1997), even though the theoretical upper bound is O(IVI3). Finally, a scaling (Ahuja et al., 1993b) algorithm which selects nodes with sufficiently large excess and chooses from among them the one with smallest distance label yields the bound O(JVIIAI + IVIZLog(U)), which is still much slower than the practical performance. Another practical trick which is known (in some cases) to increase the efficiency of the preflow algorithm is to perform global updates occasionally. A global update involves recalculating the exact distance labels (e.g. using breadthfirst search starting at the target). Although this is quite expensive, it increases the efficiency of the push operations. The optimal frequency of global updates depends on the problem.
3.2.4
Further uses of minimum cuts
In many of the physics applications, the minimum cut is of most physical interest. It corresponds to the interface structure in interface problems, and the domain structure of random magnets. The residual graph contains the minimum-cut structure and so contains the information about complex ground-state morphologies. The residual graph and its minimum-cut structure are used in two additional ways in Section 7 namely: testing the sensitivity of cuts to small perturbations and; finding all of the cuts in degenerate systems. Sensitivity analysis (Ahuja et al., 1993b) is a useful tool in operations research as well as in physics. The general idea is to study the variation of the maximum flow and the minimum cut as a function of perturbations to the local capacities.
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Often the capacities close to the source are the focus of interest. In the context of random magnets, variation of bulk or surface capacities are possible and are similar to studies of chaos (Bray and Moore, 1987) in the ground state of random magnets (see Section 7.2.4). However, the most direct application has been to the analysis of the sensitivity of ground states to variations in an external field (Angl6s d'Auriac and Sourlas, 1997). Random Ising magnets with bond dilution (Bastea, 1998), diluted antiferromagnets (Morgenstern et al., 1981; Hartmann and Usadel, 1995; Bastea, 1998; Hartmann, 1998c) in a field and the 1/0-problem of transportation networks are example of graphs with degenerate cuts (see Sections 7.2.4 and 7.3.5). Counting all of the possible degenerate cuts in a graph can itself be reduced to a much smaller graph which contains a directed arc for each segment of the degenerate cut structure. To enumerate all cuts (Bastea, 1998), it is necessary to count all directed partitions of the reduced graph into two parts, one of which contains the source s, and the other which contains the target t. Although this graph partitioning problem is computationally hard, the reduced graph is quite small and hence the algorithm is quite efficient in practice.
3.3 Minimum-cost-flow problems 3.3.1
Definition of the minimum-cost problem
Let G(V, A) be a network with a cost r and a capacity Uij associated with every arc (i, j) E A. Moreover we associate with each node i E V a number b(i) which indicates its supply or demand depending on whether b(i) > 0(a source) or b(i) < 0 (a target). The minimum-cost-flow problem is (Ahuja et al., 1993b): minimize
z(x)--
~
hij(xij),
(3.3.1)
(i,j)EA subject to the mass-balance constraints,
Z
xij --
{j[(i,j)EA}
Z Xji = b(i) {jI(j,i)EA}
u E V,
(3.3.2)
and the capacity constraints
0 ~ Xij <_ Uij
g(i, j) E A.
(3.3.3)
The parameters b(i), Cij are integers, while Xij, Uij are non-negative integers. Note that we have allowed for a general set of sources and sinks b(i), though we shall restrict attention to one source, s, and one sink, t, in the applications. In fact, it is easy to convert a problem with a general set b(i) to one with just one source s and one target t in the following way:
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(i) Connect all of the nodes with b(i) > 0 to s by arcs with capacity Usi = b(i) (ii) Connect all of the nodes with b(i) < 0 to t by arcs with capacity Uit -- b ( i ) . Flow conservation requires ~-~-i b(i) = 0 so that the flow into the network is equal to the flow out of the network. The quantity h ij (xij) is the cost function and may be a different function on each bond. The cost functions hij (xij) can be any convex function, that is, Yx, y, and 0 ~ [0, 1]
hij(Ox -Jr-(1 - O)y) < Ohij(x) 4- (1 - O)hij(Y)
(3.3.4)
Linear cost is the special c a s e , h ( x i j ) --- r There are faster algorithms for linear cost than for convex cost, but for the applications considered here, the difference is not large. The general convex-cost case is as simple to discuss as the linear-cost case, so we discuss the general algorithm. In minimal-cost algorithms, the residual network G(x), corresponding to a flow, x, again plays an essential role. The residual capacities are defined in the same way as in the maximum-flow problem. However, the residual cost is not so simple. We need to know the cost of augmenting the flow on arc (i, j ) , when there is already a flow Xij in that arc. In the general convex-cost problem, we always augment the flow by one flow unit. The costs incurred in adding or subtracting one unit of flow between nodes i and j are represented by residual costs. The key subtlety is that if a forward arc has positive cost Cij, the residual cost on the r may be negative. This arises because if a flow Xij > 0 exists reverse a r c , cji, on a positive-cost arc (i, j), injecting a compensating flow from node j to node i r in the cancels the flow Xij which requires that there be a negative residual cost, Cji' reverse arc. The residual costs defined below reflect this possibility. Because we have defined Xij >__0 and Xji >~ 0, we must treat three cases. Although this seems cumbersome now, in the later analysis it significantly simplifies the discussion. The residual costs for the general convex-cost problem are as follows. (i) If Xij >~ 1, =~ Xji = 0 -
hij(xij),
1) -
hij(xij).
crj(xij) "~ h i j ( x i j Jr- 1) Cji(Xi j) -- h i j ( x i j (ii)
If
xji >_
l , ==>
xij - - 0 c j i ( x j i ) = h j i ( x j i + 1) Cij(Xji) --- h j i ( x j i - l )
(iii)
If
xij -- O,
and
(3.3.5)
-
hji(xji), hji(xji).
(3.3.6)
xji -" 0 Ci~ (0) = hij (1) - hij (0), c~.i(O) -- h j i ( 1 ) - hji(O).
(3.3.7)
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As seen in the second of (3.3.5) and (3.3.6), negative residual costs may occur when reducing the flow in an arc. The residual network G(x) is then a graph with residual capacities rij = u i j -- x i j + x j i , and residual costs found from (3.3.5)(3.3.7). An intuitively appealing way of thinking about the convex-cost problem is to replicate each arc, (i, j) many times, with each replicated arc having capacity one. The kth replicated arc has cost h i j ( k ) - h i j ( k - 1). As flow is pushed along the arc (i, j), the first unit of flow goes into the first replicated arc, the second unit of flow in the second replicate etc. When the flow is reversed, the flow is cancelled first in the highest replicated arc p r o v i d e d the cost f u n c t i o n is convex. That is, we need h i j ( k ) - h i j ( k - 1) > h i j ( k - 1) - h i j ( k - 2) so that this replication procedure makes sense. Unfortunately no analogous procedure is possible when the convexity condition is violated. In the case of linear costs there is no need to replicate the arcs as the cost for incrementing the flow does not depend on the existing flow. As discussed in Section 8 many of the physics problems map to linear or convex minimum-cost flow with non-negative costs. A simple example of using minimum-cost flow to model two directed polymers in a random pinning potential is presented in Fig. 3.2, where each arc has capacity one. This ensures that only one unit of flow can pass through any arc and models a contact repulsion between polymers. Note that this contact repulsion is on the arcs, so that more that one polymer can still be incident on a node. Node disjoint paths are more difficult to model. Nevertheless, arc disjoint paths provide a strong contact repulsion and should capture the essence of non-intersecting polymers. We now discuss two methods for solving minimum-cost-flow problems, namely the negative-cycle-cancelling method (Section 3.3.2) and the successiveshortest-path method (Section 3.3.3), both of which rely on residual-graph ideas. The latter is more efficient, but is restricted to non-negative costs as it uses Dijkstra's method to find lowest-cost paths.
3.3.2
Negative-cycle-canceling
algorithm
The idea of this algorithm is to find a f e a s i b l e flow, that is, one which satisfies the mass-conservation rules, and then to improve its cost by cancelling negative-cost cycles. A negative-cost cycle in the original network is also a negative-cost cycle in the residual graph, so we can work with the residual graph. Moreover, flow cycles do not change the total flow into or out of the network, and they do not alter the mass-balance conditions at each node. Thus, augmenting the flow on a negative-cost cycle maintains feasibility and reduces the cost, which forms the basis of the negative-cycle-canceling algorithm. This is formalized as follows. Theorem (negative cycle) A feasible solution x* is an optimal solution of the
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Fig. 3.2 Finding the exact ground state of two non-intersecting directed polymers in a random pinning potential. The energy (cost) of each arc is indicated. The arcs attached to the source (top node) and target (bottom node) have zero cost. a) The flow computed using the minimum-cost-flow method. Note that in the central horizontal bond, flow cancels, b) I'he ground-state configuration of two directed polymers.
minimum-cost-flow problem, if and only if the residual network ~(x*) contains no negative-cost cycle.
Proof: Suppose the flow x is feasible and ~(x) contains a negative cycle. Then a flow augmentation along this cycle improves the function value z(x), thus x is not optimal. Now suppose that x* is feasible and G(x*) contains no negative cycles and let x ~ ~= x* be an optimal solution. Now decompose x ~ - x* into augmenting cycles, the sum of the costs along these cycles is c - x ~ - c . x*. Since G(x*) contains no negative cycles c . x ~ - c . x* _ O, and therefore c . x ~ = c . x* because optimality of x* implies c . x ~ _< c . x*. Thus x ~ is also optimal. [] A minimum-cost algorithm based on the negative-cost-canceling theorem, valid for graphs with convex costs and no negative-cost cycles in ~7(0), is given below.
2
Exact combinatorial algorithms
algorithm C y c l e c a n c e l i n g
begin
establish calculate
171
(convex costs)
a f e a s i b l e flow x the r e s i d u a l costs
cri as in eqs.
while ~(x) c o n t a i n s a n e g a t i v e cost cycle do
(3.3.5-3.3.7)
begin
end
end
use some a l g o r i t h m to i d e n t i f y a n e g a t i v e a u g m e n t one unit of flow in the cycle u p d a t e cri(xij) and r/j.
cycle
W
To begin the algorithm, it is necessary to find a feasible flow, which is a flow which satisfies the injected flow at each of the sources, the extracted flow at each of the sinks, and which satisfies the mass-balance constraints at each node. A robust procedure to find a feasible flow is to find a flow which satisfies the capacity constraints using the maximum-flow algorithm (e.g. using the preflow algorithm). To detect negative cycles in the residual network, (7(x), one can use the labelcorrecting algorithm for the shortest-path problem presented in Section 2.3.2. In the linear cost case, the maximum possible improvement of the cost function is O(IAICU), where C = m a x Icij] and U = max uij. Since each augmenting cycle contains at least one arc and at least one unit of flow, the upper bound on the number of augmenting cycle iterations for convergence is also O(IAICU). Negative-cycle detection is O(I vzl) generically, but for sparse graphs with integer costs it is O(IA)[). Thus for sparse graphs with integer costs, negative cycle canceling is O([AI2CU).
3.3.3
Successive-shortest-path algorithm
The successive-shortest-path algorithm iteratively sends flow along minimal-cost paths from source nodes to sink nodes to finally fulfill the mass-balance constraints. A pseudo-flow satisfies the capacity and non-negativity constraints, but not necessarily the mass-balance constraints. Such flows are called infeasible as they do not satisfy all the constraints. The successive-shortest-path algorithm is an infeasible method which maintains an optimal-cost solution at each step. In contrast, the negative-cycle-canceling algorithm always satisfies the constraints (so it is a feasible method) and it iteratively produces a more optimal solution. The imbalance of node i is defined as
e(i) - b(i) +
~ Xji ~ Xij. {jI(ji)EA} {jl(ij)Ea}
(3.3.8)
If e(i) > 0 then we call e(i) the excess of node i, If e(i) < 0 then we call it the deficit. The successive-shortest-path algorithm sends flow along minimal-cost
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paths from excess sites to deficit sites until no excess or deficit nodes remain in the graph. Dijkstra's algorithm (Section 2.3.2) is efficient in finding minimum-cost paths, but it only works for positive costs. The successive-shortest-path algorithm uses Dijkstra's method to find augmenting paths, but to make this work, we have to develop a different sort of residual network with positive reduced costs (remember that the residual costs can be negative, see (3.3.5)-(3.3.7)). Surprisingly, this is possible. To construct positive reduced costs from which the optimal flow can be calculated, we must first introduce the concept of a node potential 7r(i) and some beautiful results related to it. The reduced costs used in the successive-shortest-path problem are inspired by the reduced costs, c/~, introduced in the shortest-path problem (2.3.1)-(2.3.3). c/~ has the attractive feature that, with respect to the optimal distances, every arc has a non-negative cost. To generalize the definition (2.3.1) so that it can be used in the minimum-cost-flow problem, one defines the reduced cost of arc (i, j) in terms of a set of node potentials zr (i), C~j -- Cij -- 7r(i) + 7r(j).
(3.3.9)
We impose the condition that a potential is only valid if Cij > 0 as for reduced costs in the minimal path problem. Note that the residual costs defined in (3.3.5)(3.3.7) appear here. All of the quantities in (3.3.9) depend on the flow X i j , though we don't explicitly state this dependence. From the definition (3.3.9), it is evident that the reduced costs, c~j, have the following properties. (i) For any directed path P from k to l:
(i,j)EP
(i,j)EP
(ii) For any directed cycle W:
Z (i,j)EW
Z cirj" (i,j)EW
In particular, property (ii) means that negative cycles with respect to crj are also negative cycles with respect to c~j. We define the residual network GJr (x) to be the residual graph with residual capacities defined as before, but with reduced costs as given by (3.3.9). The next step is to find a way to construct the potentials zr(i). This is carried out recursively, starting with zr(i) - 0 when there is no flow in the network. The procedure for generating potentials iteratively relies on the potential lemma given below.
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Lemma (potential) (i) Given: a valid node potential zr(i)" a set of reduced costs, c~j; and a set of distance labels, d ( i ) (found using cTj > O) then the potential Jrt(i) -~t
rr(i) - d ( i ) also has positive reduced costs, cij > O. ~t
(ii) cij - 0 for all arcs (i, j) on shortest paths.
Proof: Properties (i) and (ii) follow from the analogous properties for the minimal path (see (2.3.1)-(2.3.3)). To prove (i), using (2.3.1)-(2.3.3), we have, d ( j ) < d ( i ) + yf cij, then we have, cij~! -- c r j - ( r r ( i ) - d ( i ) ) + ( : r ( j ) - d ( j ) ) = c ~ j - d ( j ) + d ( i ) > O. For (ii) simply repeat the discussion after replacing the inequality by an equality. [] Now that we have a method for constructing potentials, it is necessary to demonstrate that this construction produces an optimal flow.
Theorem (reduced cost optimality) A feasible solution, x*, is an optimal solution of the minimum-cost flow problem if and only if there exists a set of node Jr >__0 V(i, j) in ~ r (x*). potentials, :r(i), such that Cij Proof: For the implication "r suppose that cTj > 0 V(i, j). Since G(x*) is optimal, it contains no negative cycles. Thus by property (ii) above, an arbitrary potential difference may be added to the costs on each arc on each cycle W. For the other direction "=~" suppose that G(x*) contains no negative cycles. Denote with d(.) the shortest path distances from node 1 to all other nodes. Hence d ( j ) < d ( i ) + c i j V(i, j) 6 G(x*). Now define re -- - d then cTj = cij + d ( i ) - d ( j ) > O. Hence we have constructed a set of node potentials associated with the optimal flow. [] The reduced cost optimality theorem proves that each optimal flow has an associated set of potentials, while the potential lemma shows how to construct these potentials. The final step is to demonstrate how to augment the flow using the potentials. To demonstrate this, suppose that we have an optimal flow x and its associated potential zr(i) which produces reduced costs c~j which satisfy the reduced cost optimality condition. Suppose that we want to add one unit of flow to the system, injecting at a source at site, k, and extracting at a site I. Find a minimal path, Pkl, (using the reduced costs c~j) from a excess site k to a deficit site I. Now augment the flow by one unit for all arcs (i, j ) ~ Pkl. We call this flow augmentation 3. The following augmentation lemma ensures that this procedure maintains optimality.
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(flow a u g m e n t a t i o n ) The flow x' : x + 6 is optimal and it satisfies the reduced cost optimality conditions.
Lemma
Proof:
Take re and ~' as in the potential Lemma and let P be the shortest path from node 7T t s to node k. Part (ii) of the potential lemma implies that u (i, j ) ~ P 9cij - O. Therefore r
2-gt
7/-t
= --Cij
-- O. Thus a flow augmentation on (i, j ) 6 P might add ~t
(j, i) to the residual network, but Cji -- O, which means that the reduced cost 7T t
optimality condition Cji > 0 is still fulfilled.
[]
The strategy for the successive-shortest-path algorithm is now clear. Given a set ofexcess nodes E = {ile(i) > 0} and a set of deficit nodes D = {ile(i) < 0}, we iteratively find minimal paths from a node i 6 E to a node j 6 D until no excess or deficit remains. The minimal path for flow augmentation is found using Dijkstra's method on the residual network with the reduced costs given by (3.3.9). After each flow augmentation, the node potentials are recalculated using the potential lemma.
algorithm
Successive
shortest
paths
(convex
costs)
begin
x:0,
Jr(i) = 0 is
while t h e r e begin
compute find the all
a node s with
the r e d u c e d c o s t s c~(x) (Eq. 3.3.9) s h o r t e s t p a t h s d ( i ) f r o m s to o t h e r n o d e s in ~(x) w 9 r " to the r e d u c e d
choose a n o d e t w i t h
augment unit
e ( s ) > 0 do
f l o w on
the
compute 7r(i) = 7r(i) -
costs
c.I~]
e(t)< 0 shortest
path
from
s
to
t
b y one
d(i)
end end
Since we worked hard to construct a system with positive reduced costs, the "find" operation above can be carried out using Dijkstra's algorithm. If we denote the sum of all the sources to be v - Z i l b ( i ) > o b(i), then the number of flow augmentations needed to find the optimal flow is simply v. Each flow augmentation requires a search for a minimum-cost path from a node k c E to a node 1 E D which for sparse graphs and integer flows can be efficiently accomplished with Dijkstra's method, which is O(IAI). Thus for integer flows on sparse graphs with positive costs (as is typical of the physics applications) the successive-shortestpath algorithm is O(vlAI).
2 Exactcombinatorial algorithms 4
4.1
175
Matching algorithms
Introduction and definitions
Given a graph G(V, E) with node set V and edge set E, a matching M C E is a subset of edges, such that no two are incident to the same node (Lov~isz and Plummer, 1986; Graham et al., 1995). An edge contained in a given matching is called matched, other edges are free. A node incident to an edge e 6 M is covered (or matched) others are exposed (or free). A matching is perfect if it leaves no exposed nodes. If e = (u, v) is matched, then u and v are called mates. An alternating path is a path along which the edges are alternately matched and unmatched (for example the highlighted path in Fig. 4.1a). A bipartite graph is a graph which can be subdivided into two sets of vertices, say X and Y, such that the arcs in the graph (i, j) only connect vertices in X to vertices in Y, with no arcs internal to X or Y. Nearest-neighbor hypercubic lattices are bipartite, while the triangular and face-centered-cubic lattices are not bipartite. The more general weighted-matching problems assign a non-negative weight (= cost), w, to each edge e = (i, j). M is a maximum-weight matching if the total weight of the edges in M is maximal with respect to all possible matchings. There is a simple mapping between maximum-weight matchings and minimumweight matchings, namely: let Cij = W - wij, where Wij is the weight of arc (i, j) and W > m a x ( i , j ) ( t o i j ). A maximum matching o n toij is then a minimum matching o n cij. Later in this section we show that minimum (and hence maximum) matchings on bipartite graphs can be mapped to a minimumcost-flow problem. An historical introduction to matching problems, whose origins may be traced to the beginnings of combinatorics, may be found in Lov~isz and Plummer (1986). Matching is also related to thermal statistical mechanics because the partition function for the two-dimensional Ising model on the square lattice can be found by counting dimer coverings (= perfect matchings) (Lov~isz and Plummer, 1986). This is a graph enumeration problem rather than the optimization problems we consider here. As a general rule, graph enumeration problems are harder than graph-optimization problems. Due to the fact that all cycles on bipartite graphs have an even number of edges, matching on bipartite graphs is considerably easier than matching on general graphs. In addition, maximum-cardinality matching and maximumweight matching on bipartite graphs can be easily related to the maximum-flow and minimum-cost-flow (respectively) problems discussed in Section 3. Further, rigidity percolation (even on non-bipartite graphs like the triangular lattice) is related to maximum-cardinality matching on bipartite graphs (see Section 9). We thus treat bipartite matching separately (Section 4.3.1). The more complex problem of matching on general graphs, which are used to solve spin-glass models, is discussed in Section 4.3.2.
176 (
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)
Fig. 4.1 a) A matching (thick) is a subset of edges with no common end. An augmenting path (shaded) starts and ends at exposed nodes, and alternates between unmatched (thin) and matched (thick) edges, b) Interchanging matched and unmatched edges along an augmenting path increases the cardinality of the matching by one. This is called augmentation and is the basic tool for maximum-matching problems.
4.2 Augmenting paths The algorithms for maximum matchings are based on the idea of successive augmentation, which is analogous to the augmenting-path methods for flow problems (see Section 3.2.2). An augmenting path A p with respect to M is an alternating path between two exposed nodes. An augmenting path is necessarily of odd length, and, if G is bipartite, connects a node in one sublattice, U, with a node in the other sublattice, V. Clearly, if matched and free edges are interchanged along Ap, the number of matched edges increases by one. Therefore if M admits an augmenting path it cannot be of maximum cardinality. Similarly a matching M cannot be of maximum weight if it has an alternating path of positive weight, since interchanging matched and free edges would produce a "heavier" matching. The non-existence of augmenting paths is a necessary condition for maximality of a matching. It is also a sufficient condition. A central result in matching theory states that repeated augmentation must result in a maximum matching (Berge, 1957; Norman and Rabin, 1959). Theorem (augmenting path) (i) A matching M has maximum cardinality if and only if it admits no augmenting path. (ii) A matching M has maximum weight if and only if it has no alternating path or cycle of positive weight. Proof: (i) =~ is trivial. To prove r assume M is not maximum. Then some matching M' must exist with IM~I > IMI. Consider now the graph G' whose edge set is E ~ = M A M ~ (the symmetric difference of M and M~). Clearly each node of G~ is
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incident to at most one edge of M and at most one edge of M t. Therefore nodes in Gt have at most two incident edges and the connected components must be either simple paths or cycles of even length, and all paths are alternating paths. In all cycles we have the same number of edges from M as from M ~ so we can forget them. But since IM'I > IMI there must be at least one path in G~ with more edges from M ~than from M. This path must necessarily be an augmenting path. (ii) =, is again trivial. Assume M is not maximum. Some matching M t must therefore exist with w(M ~) > w(M). Consider again ~ = (V, MAMa). By the same reasoning as before, we conclude that ~ must contain at least one alternating path or cycle of positive weight. [3
4.3 Matching problems 4.3.1
Matching on bipartite graphs
Consider a bipartite graph, B(U, V, E), where U is the set of nodes on one sublattice and V is the set of nodes on the other sublattice. It is conventional to draw bipartite graphs as shown in Fig. 4.2, with the two sublattices joined by arcs, which can only go from one sublattice to the other. Now assume that an initial matching M is given (which can be the empty set), as in Fig. 4.2a. It is natural to look for alternating paths starting from exposed nodes (If there are no exposed nodes, M is maximum. Stop). An efficient way to do this is to consider all alternating paths from a given exposed node simultaneously in the following way: build a breadth-first-search (BFS) (see Section 2.3.1) starting from an exposed node, for example node vs, as described in Fig. 4.3a. In the BFS tree, node v5 corresponds to level 0. All its adjacent edges are free. They lead to nodes u3 and u5 at level 1, which are covered. Now since we must build alternating paths, it doesn't make sense to continue the search along free edges. Therefore we proceed along matched edges, respectively, to nodes Vl and v2. From these we follow free edges to u l and u2, and then matched edges to nodes v3 and v4. In the last step, node u4 is found exposed. Therefore (u4, v4, u2, vl, u3, v5) is an augmenting path. After inverting it, the augmented matching shown in Fig. 4.2b is obtained. If no exposed node were found when the B FS ends, then node v5 will never be matched and can be forgotten because of the following result (van Leeuwen, 1990), which is valid for general graphs: Theorem If there is no augmenting path from node uo at some stage, then there never will be an augmenting path from Uo. In practice, there is a difference between the search technique described here and the usual BFS since the searches from odd-numbered levels are trivial. They always lead to the mate of that node, if the node itself is not exposed. Therefore
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\
a)
b) ~
'
\
Fig. 4.2 a) An initial matching is given for a bipartite graph, b) The enlarged matching obtained after inverting the augmenting path discovered from node v5 (see Fig. 4.3).
v+3
Fig. 4.3 a) The BFS tree built from exposed node v5 in Fig. 4.2a. Dashed lines represent non-tree edges to already visited nodes. The search finishes when an exposed node u4 (double circle) is found, b) The auxiliary tree obtained after removing oddlevel nodes and identifying them with their mates. After inverting the augmenting path {v5, u3, Vl, u2, v4, u4}, the enlarged matching in Fig. 4.2b is obtained.
the search can, in practice, be simplified by ignoring odd-numbered nodes and going directly to their mates, as shown in Fig. 4.3b. The search for alternating paths can be seen as a usual BFS on an auxiliary graph from which odd-level nodes
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have been removed by identifying them with their mates. The basic augmenting path algorithm is then as follows (Papadimitriou and Steiglitz, 1982). algorithm ( M a x i m u m - c a r d i n a l i t y begin
bipartite
matching)
e s t a b l i s h a i n i t i a l f e a s i b l e m a t c h M on B c o n t a i n s an e x p o s e d n o d e u E U do
B(U, V,E)
while begin
Find an a l t e r n a t i n g
path,
to an e x p o s e d n o d e invert the a l t e r n a t i n g
update M
P,
v E V path,
from u E U P
end end The best known implementation for this algorithm is due to Hopcroft and Karp (1973). It runs in time O(IEI 9[ , / ~ ) and is based on doing more than one augmentation in one step. There is also a simple way in which to map the maximum-cardinality-matching problem to the maximum-flow problem. Let /3 -- (V, U, E), and define B' by adding a source node s and a target node t, and connecting all nodes in V to s, and all nodes in U to t, by edges of capacity 1. Now let all edges e E E have capacity 1. Because of the integer flow theorem, maximum flows in/3' are integral. Every flow of size f thus identifies f matched edges in/3, and vice-versa. Since maximum flows in 0 - 1 networks are computable in O(IEI 9~/IV[) time, so are maximum-cardinality matchings on /3. The mapping of matching problems to flow problems also applies to the maximum-weight-matching problem on bipartite graphs. This is also known as the assignment problem, because it can be identified with optimal assignment, e.g. of workers to machines, if worker i E U produces a value tOij working at machine j E V. In a similar manner to that described in the previous paragraph, this problem can easily be formulated as a flow problem. Again add a source node s, a sink node t; connecting s to all nodes in U by unit-capacity, zero-cost arcs; all nodes in V to t by unit-capacity, zero-cost arcs. Also interpret edge eij as a directed arc with unit capacity and c o s t w ( e i j ) = W - e i j , where W > max(i,j)(tOij ). The solution to the minimum-cost-flow problem from s to t is then equivalent to the maximum-weight matching which we seek.
4.3.2 Matchingon general graphs Maximum matching on general graphs is considerably more difficult because of the presence of odd-length cycles, which are absent on bipartite graphs. Consider starting a BFS for alternating paths from an exposed node a at level 0 (Fig. 4.4).
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h 9
d/////
Y
e A
a
b
9
v
I
w
..
a)
9
i
j
h 9
a 9
b)
v
k
x
I p n
b _-
'~
B
Fig. 4.4 a) A blossom is an odd-cycle which is as heavy as possible in matched edges, i.e. contains 2k + 1 edges among which k are matched, b) The reduced graph obtained by shrinking a blossom to a single node contains the same augmenting paths as the original graph.
And consider what happens when searching an even-level node x (necessarily covered). Let (x, y) be an unexplored edge incident to x. If y is exposed, we have found an augmenting path, and the augmentation proceeds as usual. If y is covered there are two possibilities: if y is odd-level, nothing special happens. If y is marked as even, there is a special situation: there are two even-length alternating paths, one from a to x and one from a to y, and therefore (x, y) closes a cycle of odd-length. Let c be the last node common to both alternating paths (necessarily at an even level). The odd cycle including c is called a blossom B, and c its base. A blossom is essentially an odd-length alternating path from c to itself, as depicted in Fig. 4.4a. Its presence may conceal an existing augmenting path, like for example {a, b, c, i, j, k, x, y, f, e, d, h, n, p} in Fig. 4.4, which would not be discovered by the BFS since edge (d, h) would never be explored. A blossom might also make us "find" an augmenting path where none actually exists, like for example {a, b, c, d, e, f, y, x, k, j, i, c, b, a} in Fig. 4.4a. The first polynomial-time algorithm to handle blossoms is due to Edmonds (1965). Edmonds' idea was to shrink blossoms, that is, replace them by a single
2 Exactcombinatorial algorithms
181
node/3 obtaining a modified graph ~ as shown in Fig 4.4b. The possibility of shrinking is justified by the following theorem due to Edmonds.
Theorem (Edmonds) There is an augmenting path in G if and only if there is an augmenting path in ~. The existence of a blossom is discovered when edge (x, y) between two evenlevel nodes is first found, and its nodes and edges are identified by backtracking from x and y until the first common node is found (c in our example), which is the blossom's base. Once identified, the blossom is shrunk, replacing all its nodes (among which there might be previously shrunk blossoms) by a single node, and reconnecting all edges incident to nodes in the blossom (necessarily uncovered edges) to this one. The search proceeds as usual, until an augmenting path is found, or none, in which case a is abandoned and no search will ever be started from it again. If an augmenting path is found, which does not involve any shrunk blossom, it is inverted as usual. If it contains blossom-nodes, they must be expanded first and one has to identify which way around the blossom the augmenting path goes. This may need to be repeated several times if blossoms are nested. After inverting the resulting augmenting path, a new search is started from a different node. A simple implementation of these ideas runs in O(IVI 4) time (Papadimitriou and Steiglitz, 1982). The fastest known algorithm for non-bipartite matching is also O(IE[ 9~/Igl) time (Micali and Vazirani, 1980).
5
5.1
Mathematical programming
Introduction
There are many books devoted to linear programming (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Wright, 1997) and its applications. The problems described in Sections 2-4 can be cast as linear or, more generally, convexprogramming problems. In this chapter we give a brief introduction to mathematical programming and its relation to minimum-cost-flow problems and to matching on non-bipartite graphs. Linear programming is the problem of optimizing (minimizing or maximizing) a linear cost function while satisfying a set of linear equality and/or inequality constraints. Linear programming was discussed by George B. Dantzig around 1947, although L.V. Kantorovich had introduced and solved a related problem in 1939. The simplex method for linear programming was published by Dantzig in 1949. From the outset it was realized that flow problems are a special case of mathematical programming with flow conservation at each node being a set of
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linear constraints. Special implementations of the simplex method for network flow can be much more efficient than the general simplex method and are called network-simplex methods. Since 1949, generations of workers in the fields of operations research, economics, finance and engineering have been trained to formulate and solve linear- and non-linear-programming models, particularly in the framework of simplex methods. Mathematical-programming methods have had a much smaller impact on modeling in the physical sciences. If the variables over which we optimize can have continuous variation, then linear programming (LP) is known to be in P, that is, it is solvable in polynomial time. Polynomial bounds on continuous linear programming were first demonstrated for ellipsoid algorithms. However, ellipsoid algorithms are inefficient in practice, in contrast to the N P simplex method which is efficient in practice. In 1984, Karmarkar introduced a second class of P algorithms for linear programming and from this work the interior-point methods have developed. Interior point methods are efficient, especially for large sparse LP and convex-cost problems (Wright, 1997). In many problems, the variables over which we wish to optimize are restricted to be integers. This class of linear-programming problem is called integer linear programming or ILE In contrast to continuous LP, ILP does not, in general, have polynomial bounds. In addition the simplex method is not designed for this general class of problem. However, all of the P problems which we have discussed in Sections 2-4 can be mapped to an ILP problem, and it is only these network ILP problems which are known to have polynomial bounds. The general ILP class includes many of the outstanding "hard" combinatorial problems such as the traveling-salesman problem.
5.2 5.2.1
Linear- and convex-cost programming Problem definition
The standard form (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Wright, 1997) of the primal LP problem is: minimize the cost function,
cost - ~
Ci X i ,
(5.2.1)
and xi >_0.
(5.2.2)
i
subject to the linear constraint equations, AY-b,
If we seek an integer linear-programming solution, we have the additional requirement that xi be an integer. Convex-cost programming is the generalization
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to the case where the cost is a convex function ie. Cost ~ hi (xi), where hi is a convex function. For the problem defined by (5.2.1) and (5.2.2) to have a solution, we need the length (n) of the vector of variables s to be larger than the number of linear constraints (m), otherwise the problem is overconstrained or infeasible. The matrix A is an m • n array which may couple the variables xi to each other. Linear-programming problems may be written in a variety of different forms, and various tricks are needed to reduce them to the standard form. An important alternative form of linear programming is the dual-linearprogramming form. The dual problem is stated as follows: maximize the cost function, cost = Z bi 1Bi , (5.2.3) i
subject to the linear-constraint equations, t ~ A - Y, with - C o _< 1/3 i < CO
(5.2.4)
There are a variety of relations between the primal and dual problems, with one of the most important being,
Lemma (strong duality) If either the primal or dual problem has an optimal solution, then both possess optimal solutions and the two optimal costs are equal.
5.2.2
Minimum-cost-flow as mathematical programming
Define the flow in each bond Xij > O, to be the variable over which we are to optimize; the sources at each site to be co > bi >_ - c o ; the capacities on each bond to be Uij > 0 and; a linear cost function, Z i j CijXij then the minimum-costflow problem can be written as: (Ahuja et al., 1993b)" Minimize the cost function
cost -- Z
CijXij,
(5.2.5)
ij
subject to the flow constraints, MY - b, with 0 _< xij < blij.
(5.2.6)
Other than the capacity restriction on the flow Xij, this is in primal linearprogramming form (compare with (5.2.1) and (5.2.2). It is simple to remove the upper constraint o n xij. Just introduce a set of slack variables, ~ij, ~ij - - U i j - - X i j .
(5.2.7)
This is a set of linear equations, which is added to the set Ms - b which impose conservation of flow at each vertex. This minimum-cost problem is now in primal linear-programming form, with the variables ~ij >~ 0 and Xij > O.
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5.2.3
Weighted matching as linear programming
All weighted-matching problems can be thought of as matching problems on complete graphs (Lov~isz and Plummer, 1986), by just adding the missing edges with zero weight. Furthermore, the number of nodes can always be assumed to be even, since an extra node can always be added which has zero-weight connections to all others. Therefore, optimal solutions will always be perfect matchings. Thus we can formulate the maximum-weight matching problem (over weights 113ij ) a s a minimization problem (over costs C i j ) over the set of perfect matchings on a complete graph, after defining the costs Cij = W -- l13ij, where W is greater than or equal to the largest weight. In order to see that maximum-weight matching can be formulated as a minimization problem on a complete graph with n nodes, we associate an integer variable Xij t o each edge eij, w i t h xij = 1 meaning that edge (i, j) is in the matching and Xij - - 0 meaning that it is not. By convention X i j "-- X j i and Xii --- O. We are then left with the following linear minimization problem. minimize Z
CijXij
(5.2.8)
i,j
subject to n
Z
X i j "--
1
i = 1,---, n
(5.2.9)
j=l
xij > 0
1 < i <_ j < n
(5.2.10)
However, on general (non-bipartite) graphs, optimal solutions do appear for which s o m e Xij are fractional (Fig. 5.1 shows an example of this) and thus do not correspond to a matching. Here, again, the reason for the complication is the existence of odd cycles (Edmonds, 1965). Thus we need to explicitly impose the condition
x i j = O, 1.
However, in the bipartite case this never happens, i.e. the optimal solutions are always integer. This fortunate phenomenon also occurs in the maximum-flow problem, and allows one to solve both the integer maximum-flow and maximum matching on bipartite graphs by means of simplex or any other technique of linear programming. A solution to this problem for the case of general graphs has been found by Edmonds, and consists in adding new constraints, which impose X i j - - O, 1 indirectly. For each odd subset S C V, we impose the additional set of constraints
(
(i,j)6E(S)
xij
) ~ ~(ISkl-'
I)
for every odd-subset Sk of V
(5.2.11)
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1/2
| ~ 1
1~//
1
~1/2
1/2
a)
b)
c)
Fig. 5.1 Example of fractional optimal solution on graphs with odd cycles, a) The graph and its weights, b) An optimal solution of P1 is fractional, c) The desired solution. (From Bieche et al., 1980)
where E(S) denotes the set of edges with both ends in S. Notice that these constraints would disallow the fractional solution in Fig. 5.1, but not the true one. They are automatically satisfied in bipartite graphs, and therefore are not necessary in that case.
5.2.4
Traveling salesman as mixed-integer programming
Finally, to relate integer linear programming to a problem which is known to belong to the NP-complete class, consider the traveling-salesman problem (Papadimitriou and Steiglitz, 1982). This problem seeks to find a loop which visits each of n cities, and which has minimum loop length. Thus we seek to minimize cost -- Z
CijXij,
(5.2.12)
ij
with constraints n
n
Zxij-1,
Zx/j-- 1
i=1
j=l
(5.2.13)
Cij is the distance between cities i and j, and Xij = 0, 1. The two constraints
ensure that each city has one incoming edge and one outgoing edge. However, the problem defined above includes the possibility of subtours. That is, instead of one tour visiting all cities in one circuit, it allows the possibility of two or more
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separate subtours, which all together visit all cities, but not in one continuous loop (Papadimitriou and Steiglitz, 1982). Subtours may be eliminated by imposing the additional constraints that for any partition of the set of cities (S, S), we must have, Z
xij ~> 1.
(5.2.14)
icS,jES
That is, it is not possible to separate the cities without cutting at least one edge in the tour. An alternative to this rather cumbersome set of constraints is the following, Ui --Uj + nxij < n -
1,
1< i r j < n
(5.2.15)
That is, for a set Xij corresponding to a valid tour, it is possible to find a set -cx~ < ui < cxz. For a set Xij which do not correspond to a valid tour, it is not possible to find a set ui which satisfies this constraint. This is because summing the difference u i - u j around a tour gives zero, so for any set {ui } in a tour the condition is satisfied. However, if there are two or more subtours, it is possible to choose a set {ui } which is not a tour, in which case the inequality (5.2.15) can be violated. Due to the fact that ui are real, while the remainder of the problem is integer, this programming problem is called a mixed-integer linear p r o g r a m m i n g (MILP) problem, which are, in general, also not polynomial. It is tantalizing that the problems in the N P - c o m p l e t e class are so similar in structure to linear programming (except for the restriction to integers) and yet have not yielded to polynomial solution. Naturally there have been attempts to adapt continuous-linear-programming methods to the ILP and MILP classes but, at present, there has been no significant progress toward a general polynomial algorithm, and it is generally believed that none will ever be found.
6 6.1
Percolation and minimal path Introduction
In this section we discuss the relationships between search, minimal-spanningtree algorithms and connectivity percolation (Section 6.2) and; between minimalpath algorithms and the problem of a directed polymer in a random media (Section 6.3). We also note that the greedy algorithms (e.g. the Dijkstra and Prim algorithms) used to solve the minimal-spanning-tree and minimal-path problems are examples of extremal d y n a m i c s which lead to self-organized critical behavior (Section 6.4).
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The (connectivity) percolation problem has developed into an important interdisciplinary field of research, and several recent books are devoted to its physics (Stauffer and Aharony, 1994; Havlin and Bunde, 1996), its applications (Sahimi, 1994) and its mathematical foundations (Kesten, 1982; Grimmett, 1999). Here, we summarize (Section 6.2.2) the relationships between the algorithms described in Section 2.3 and the algorithms which have been used to provide high-precision results for percolation problems. This discussion is preceded by a very brief introduction to the connectivity-percolation problem (Section 6.2.1). Although search and spanning-tree algorithms are essentially equivalent to forest-fire (Stauffer and Aharony, 1994) and invasion algorithms (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984), these correspondences have only been pointed out recently (Barabasi, 1996; Duxbury and Dobrin, 1999). Algorithms for the backbone (Herrmann et al., 1984) in percolation are more complex than those for the cluster statistics (Hoshen and Kopelman, 1976), however, uses of depth-first search (Grassberger, 1992a) and matching algorithms (Moukarzel, 1998a) from computer science have lead to significant improvements in the analysis of the percolation backbone. Discussion of the matching algorithm will be delayed to Section 9, where it appears as a special case of the matching methods for the general rigidity-percolation problem. The minimal-path problem also has many physical applications and has been heavily studied (Halpin-Healy and Zhang, 1995; L~issig, 1998). Most work in this area has focused on the directed polymer in random media (DPRM). The DPRM provides a solid-on-solid approximation to a domain wall in a randombond magnet in two dimensions (Huse and Henley, 1985) and is a simple model for a flux line in a dirty superconductor (Blatter et al., 1994). The transfer-matrix method (Huse and Henley, 1985) has been the standard numerical method for studies of the DPRM, and we show (Section 6.3.2), that it is equivalent to a constrained form of Dijkstra's method (see Section 2.3.2) for minimum path. The constraint is to enforce a layer-by-layer growth in the invasion surface and hence a minimal path which has no overhangs. However, Dijkstra's method usually produces a rough invasion surface, and allows the possibility of overhangs in the optimal path. It has been shown that overhangs in the minimal path do not effect the roughness (Cieplak et al., 1994, 1995; Schwartz et al., 1998) in the self-affine-growth regime as is essential for the validity of the analytic methods. However, as has been suggested recently (Duxbury and Dobrin, 1999), the energy fluctuations at the rough Dijkstra growth surface are constant. Instead there is a KPZ (Kardar-ParisiZhang (Kardar et al., 1986)) scaling of the roughness of the Dijkstra growth front. We close this section (Section 6.4) with a discussion of some analogies between extremal dynamics and greedy algorithms (such as Prim and Dijkstra).
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6.2 Connectivity percolation 6.2.1
The connectivity-percolation problem
The percolation problem is defined as follows (Stauffer and Aharony, 1994; Grimmett, 1999). Consider a graph ~(V,A), which is initially connected. Now consider removing, with (uniform) probability 1 - p each arc (i, j ) 6 A. It has been proven that in the limit IVI ~ c~, there exists a well defined percolation threshold Pc, such that for p > Pc a cluster of infinite extent (the "infinite cluster") exists while for p < pc no infinite cluster exists. Exactly at the critical point, Pc, at least one infinite cluster exists and in many cases, it is fractal. There is a beautiful mapping between the percolation transition on nearest-neighbor lattices, and thermal phase transitions in a Potts (spin) model (as q ~ 0). The behavior near Pc is related to the behavior near the critical temperature in the spin model. The terminology and ideas of scaling theory and critical exponents have thus been taken over to the study of behavior near the percolation threshold Pc. Three important quantities in the study of percolation are: the infinite-cluster probability, P~ ~ (p - pc) #, which is the probability that a bond is on the infinite cluster; Pspanning, which is the probability that an infinite cluster exists
and; Pbackbone ~ ( P -- Pc) fl', the probability that a bond is part of the currentcarrying part of the graph. The backbone is a subset of the infinite cluster, and is found from the infinite cluster by trimming off "dangling ends" which are unable to transport an electrical current. 6.2.2
Algorithms and results for percolation
There are a large number of efficient algorithms for connectivity percolation namely; The Leath algorithm (Leath, 1976b); the Hoshen-Kopelman algorithm (Hoshen and Kopelman, 1976); the forest-fire and burning algorithms (Herrmann et al., 1984) and; the invasion algorithm (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984). We classify these algorithms for percolation into three classes and discuss their relation to optimization methods. (i) Search algorithms at fixed p First we consider algorithms where a connected graph ~(V, A) is generated, it is then diluted (i.e. arcs are removed with probability 1 - p). Search algorithms are then used to study the cluster structure of the diluted graph. A breadth-first search (Section 2.3.1) from a source site s finds all sites which are connected to the source and labels them with their "chemical distance" (Stauffer and Aharony, 1994). This is equivalent to the "forest fire" or "burning" method for finding the chemical distance. The latter methods were developed independently, and quite a bit later, than the corresponding developments in the computer-science
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community. A new breadth-first-search is needed for each cluster, so iterative application of breadth-first search identifies the cluster structure. A depth-first search uses singly connected bonds in its first probe to maximum depth. Since depth-first search excludes crossing a bond more than once, the node at the end of a singly connected bond(or sequence of singly connected bonds) which is closest to the source of the search, defines an articulation point in a depth-first-tree. The articulation points divide the depth-first-search tree into its biconnected parts. This procedure was well known in computer science in the early 1970s (Tarjan, 1972), and has recently provided efficient algorithms for the backbone in connectivity percolation (Grassberger, 1992a, 1999). The original burning algorithm for backbone identification uses forward and reverse breadth-first searches to find the backbone (Herrmann et al., 1984). This is inefficient compared to the depth-first search procedure, though high accuracy results have been found by applying this method at Pc (Rintoul and Nakanishi, 1992, 1994). (ii) Growth algorithms at fixed p It is storage inefficient to generate the entire graph and then to classify its cluster structure. Instead growth algorithms keep track of information only at a growth or "invasion" front (these methods are also called "epidemic" methods). As discussed in Section 2.3, breadth-first search, Dijkstra's algorithm and Prim's algorithm are also best implemented in this way. The Leath algorithm (Leath, 1976b) for site percolation grows from a seed site. Each site adjacent the growth front is assigned a uniform random number rj. At each step, choose a site j which has not yet been tested and which is a nearest neighbor to a site which has already been invaded. If rj < p then add site j to the growing cluster. If rj > p, mark site j as tested and do not test it again. For p > Pc, the Leath algorithm grows a cluster of infinite size, while for p < Pc, the growth of the cluster ceases at a finite distance from the source. Very high precision results for percolation have been found using this method, for example, lattices of size 20483 have been studied (Lorenz and Ziff, 1998). The Hoshen-Kopelman algorithm (Hoshen and Kopelman, 1976) grows many clusters simultaneously by assigning cluster labels to each new cluster that is nucleated during growth. This is done "row-by-row" (or "layer-by-layer" in three dimensions) and must take into account the merging of different growing clusters. This merging is accounted for by defining "equivalence classes" which are sets of cluster labels that are set to be equivalent due to merging. Simulations on lattices of up to 4 • 1011 sites have been carried out using this method (Rapaport, 1992). (iii) Invasion or "greedy" algorithms When calculations are carried out at fixed concentration p as above, it is necessary to "tune" the value of p to find the critical concentration pc. The threshold value, Pc, must be known very accurately in order to find the critical exponents to high precision. However, it is possible to define a dynamics which automatically
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stays at the critical threshold. Though this behavior has been known for some time (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) it has recently become a prototype for the more general class of dynamical phenomena now known as "self-organized criticality" (SOC) (Bak et al., 1987; Furuberg et al., 1988; Cieplak et al., 1996). The invasion algorithm for percolation (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) is very similar to the Leath algorithm (Leath, 1976a, b). A growth front evolves outward from a seed site s. Each site adjacent to the growth front is again assigned a uniform random number rj. However, the new feature is that growth is always to the site which has smallest rj. That is, from all the uninvaded sites adjacent to the growth front, choose the one which has the minimum value of its random number, rj. A similar algorithm can be defined for bond percolation. It has been shown that at long times the value of rj adjacent to the growth front converges to pc (Wilkinson and Barsony, 1984). By construction, the greedy algorithm grows outward indefinitely. It has also been shown that the cluster grown in this way leads to the same scaling behavior as the infinite cluster at the percolation threshold (Wilkinson and Barsony, 1984). Since this algorithm "self-organizes" itself to be always at the percolation threshold, it is a pedagogical example of a SOC produced by extremal dynamics (Zaitsev, 1992; Miller et al., 1993; Paczuski et al., 1995; Sneppen, 1995). The invasion algorithm just described is equivalent to Prim's algorithm (see Section 2.3.1) (Barabasi, 1996) for minimal spanning tree, provided we start Prim's algorithm at the source site s (rather than at the cheapest bond r in the graph). Prim's algorithms is a classic "greedy" algorithm in computer science, and it is likely that many other things that are known in computer science about greedy algorithms will prove useful in the study of extremal dynamics (Papadimitriou and Steiglitz, 1982; Ahuja et al., 1993b). From the point of view of numerical calculations, the algorithms for connectivity percolation are quite satisfactory. For example, in three dimensions, the Fisher exponent r = 2.189(2), the finite-size correction exponent is S2 = 0.64(2), and the scaling-function exponent is cr = 0.445(10) (Lorenz and Ziff, 1998). Even the backbone exponents are quite well known (see Section 9.4).
6.3 6.3.1
Minimal path Introduction and scaling theory
The minimal path problem on a graph, G(V,A), where each edge has a cost Cij, was discussed in Section 2.3.2. As demonstrated there, Dijkstra's algorithm is a invasion (growth) algorithm which solves this problem exactly. The physics community has focused on the special case of a directed path
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or a DPRM (Huse and Henley, 1985; Kardar et al., 1986; Kardar, 1987; Kardar and Zhang, 1987; Derrida and Spohn, 1988; Nattermann and Lipowski, 1988; M6zard, 1990; M6zard and Parisi, 1990; Parisi, 1990; Fisher and Huse, 1991; Halpin-Healy and Zhang, 1995; L~issig, 1998). In the continuum limit the DPRM is described by,
S l u } --
dz
-~
+ 6pin(U(Z), Z), ].
Oz
(6.3.1)
The random pinning potential 6pin, which is usually chosen to have mean zero, is characterized by a correlation function,
(F-,pin(U,Z)~pin(U', Zt)) -- K ( z
- z', u - u').
(6.3.2)
Frequently, the random potential is taken to be delta-correlated K (z - z', u(z) u'(z')) = y 8 (z - z')~i(u(z) - u'(z')), with y describing the strength of the pining potential. More complicated situations, like extended defects or columnar defects, can easily be modeled by correlating the disorder in some or all directions appropriately. A generalized lattice version of the Hamiltonian (6.3.1) is,
d-1 H -- Z Z (uz,j - Uz-l,J)P -[- ,~pin(Uz), z j=l
(6.3.3)
where p = 1 if often taken for convenience. The Hamiltonian (6.3.1) is also a standard model of a flux-line (Blatter et al., 1994) in a disordered environment and of a polyelectrolyte in a frozen gel matrix (Kardar et al., 1986; HalpinHealy and Zhang, 1995). The DPRM, by its construction of being stretched in the longitudinal (z) direction, has no self-interaction (i.e. is self-avoiding), but it can fluctuate in the transverse (u(z)) direction. One of the attractive features of the DPRM model (6.3.1) is that some beautiful analytic results may be derived for it (Forster et al., 1977; Kardar et al., 1986; Derrida and Lebowitz, 1998). Firstly, a DPRM configuration u(z) can also be regarded as the world line of a particle in d - 1 (transverse dimensions) moving in a time dependent potential 7. To see this, consider the restricted partition function Z(u, z) -
Du I exp
-
T
dz
2
-+-epin(Ut(Zt),Zt)l } (6.3.4)
which is proportional to the probability for the DPRM's end point at internal coordinate z being located at u. The right-hand side of (6.3.4) is the path-integral formulation of the imaginary-time Green's function of a particle in a time dependent potential g pin (U(Z), Z), where u(z) is the coordinate of the particle, z is
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the imaginary time, and T plays the role of h. Thus it satisfies the Schr6dinger equation
Z
OZ = {
} -~ffV2 + Epin(U(Z),Z) Z.
(6.3.5)
The free energy (of the flux-line/DPRM) f (u, z) = - T In Z (u, z) then obeys the KPZ non-linear diffusion equation
Of 0---z = DV2 f + 2 IVu(z)fl 2 + r/(u(z), z),
(6.3.6)
where D = T/2cr can be interpreted as a diffusion constant, ~. = - T / c r describes Epin(U(Z),z)/T is the strength of the coupling to the non-linearity and 17 a Gaussian noise. The correspondence f --+ h, demonstrates that the height fluctuations during KPZ growth are analogous to the free-energy fluctuations in the minimal path (Kardar et al., 1986). The roughness w(L) and energy fluctuations AE2(L) of a DPRM of length L, obey the scaling relations =
w2(L) = [ll2]av -- [1112 ~ ' L 2~',
(6.3.7)
and its energy fluctuations
AE2(L)-- [E2]av-- [Ej2v ~, L 20.
(6.3.8)
The exponents ~" and 0 obey the exact scaling relation 0 = 2~" - 1. For the DPRM in 1 + 1 dimensions, ~" = 2/3, and 0 = 1/3 are known exactly (Kardar et al., 1986). During growth, surfaces can become "kinetically" rough and their roughness is often described by the KPZ equation (Kardar et al., 1986), which, as shown above, is related to the DPRM problem. The roughness of the KPZ growth front scales as, w(t, l) .~ tt3 ffJ(1/tr (6.3.9) where l is the sample size parallel to the growth front, t is the growth time, and /5 and ~ are critical exponents. At long times, the surface fluctuations saturate to a value w ~ --+ U. For KPZ growth c~ = 1/2. It is then evident that ~ =/~/ot. The scaling function may be found by a mapping to the asymmetric exclusion process (Derrida and Lebowitz, 1998), and generalizations to broad distributions of "waiting times" lead to continuously varying exponents (Tang et al., 1991).
6.3.2
Algorithms for minimal path
As stated above, Dijkstra's method solves the minimum-path problem. For the detailed description below, we specialize to the case p = 1 in the discrete solidon-solid energy (6.3.3). The DPRM is a directed optimal path on the links of a
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lattice (see Fig. 6.1). For a path along the {10} orientation, the path is allowed to step forward, to the left or to the right, with an increased energy cost for motion to the left or right, which models the elasticity of the DPRM (Kardar, 1985; Kardar et al., 1986). An even simpler model is a path in the {11 } orientation (see Fig. 6.1). In this case, there is no explicit elasticity to the DPRM, but the motion is restricted to the transverse direction, and it is believed that this constraint is sufficient to maintain the DPRM universality class. The random potential 6pin(U(Z), Z) is included as a random energy on the bonds of the lattice (see Fig. 6.1). Thus the lattice Hamiltonian is simply
H -- ~
ei
(6.3.10)
9x i
i where the sum is over all bonds i = (u(z), z) of the lattice and xi represents the DPRM configuration starting at 0 (at u = 0 and z = 0, see Fig. 6.1): It is xi = 1 if the DPRM passes through the bond i and xi - 0 otherwise. The ground state of (6.3.10) is the minimum energy path from 0 to some point (u(z), z) at the lower boundary plane of the lattice (either fixed or free). Interpreting the energies as distances (after making them all positive by adding a sufficiently large constant to all energies), and the lattice as a directed graph, this becomes a shortest path problem that can be solved by using, for instance, Dijkstra's algorithm (see Section 2.3.2). In addition, due to the directed structure of the lattice one can compute the minimum energies of DP configurations ending at (or shortest paths leading to) coordinates (u, z) recursively (this is the way in which Dijkstra's algorithm would proceed for this particular case) (Huse and Henley, 1985). This is the same as the transfer-matrix algorithm, which follows from the group property of the partition function (6.3.4): Z(u, z + 1) -- ~
Z(u, u', 1) Z(u', z)
(6.3.11)
Ut
which, in the zero temperature limit, reduces to d-1
E(u(z + 1)) = Minu,(z){E(u'(z)) + e(u(z + 1)) + J ~
luj(z + 1) - u)(z)[}
j=l
(6.3.12) In Fig. 6.2 we show a collection of such optimal paths in the 1 + 1 dimensional case. The transfer-matrix method has been successful in studies of the DPRM in arbitrary dimensions. Simulations in 1 + 1 dimensions confirm the exact exponents (e.g. ( = 2/3) (Huse and Henley, 1985) and have yielded accurate results in 2 + 1 dimensions and in 3 + 1 dimensions (0 = 0.248(4) and 0 - 0.20(1) respectively) (Kim et al., 1991), and have clearly disproved the early conjecture that ( is
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~. _ , ~ m , ,
i~__:
I
e, i
a)
b)
Fig. 6.1 Models for a DPRM. a) In the {10} orientation, b) In the {11} orientation. (From Rieger, 1998a)
Fig. 6.2 A collection of polymers of lowest energy directed along the diagonals of a square lattice with random bonds. Each polymer (crossing 500 bonds) has one end fixed to the apex of the triangle, the other to various points on its base, and finds the optimal path in between. (From Kardar and Zhang, 1987)
superuniversal (Kardar and Zhang, 1987). Questions remain about whether the KPZ equation has a finite upper critical dimension, with recent work suggesting no upper critical dimension (Castellano et al., 1998) or a critical dimension which is less than or equal to four (L~ssig and Kinzelbach, 1997), though the latter result is inconsistent with the majority of the numerical evidence (Kim et al., 1991; Tang et al., 1992; Ala-Nissila et al., 1993; Castellano et al., 1998). Optimization methods have recently been applied to the minimal-pathproblem in (1 + 1) and have confirmed that overhangs are not important at weak disorder (Cieplak et al., 1995; Marsili and Zhang, 1998; Schwartz et al., 1998).
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However, there is an important difference between the minimal-path problem with overhangs as compared to the DPRM, namely that the growth front in the minimal-path problem with overhangs is rough (Duxbury and Dobrin, 1999). By construction, the growth surface in the transfer-matrix method is fiat, because the energy (distance) labels are incremented row by row. The Dijkstra growth front generated by invading a random network obeys the same scaling behavior as (6.3.9), with time replaced by L. Invasion starts at L -- 0 (which is analogous to t = 0 in the growth analogy) and occurs on a strip of transverse size I d-1 . Preliminary work indicates that the Dijkstra invasion front is in the KPZ class, so that the unrestricted Dijkstra algorithm finds minimal paths and generates KPZ growth fronts simultaneously (Dobrin and Duxbury, 1999).
6.4 Greedy algorithms and extremal processes In "greedy" algorithms one chooses and updates locally optimal sites (Papadimitriou and Steiglitz, 1982; Lov~isz and Plummer, 1986; Cormen et al., 1990; West, 1996). As discussed in Section 2.3, the greedy strategy is useful in solving the minimum-spanning-tree problem and the minimal-path problem. There are a variety of other g r a p h i c - m a t r o i d problems that are solved by greedy methods. In computer-science applications, these methods are interesting because they solve certain optimization problems efficiently and exactly. However in physics they are also interesting because they generate physically interesting and relevant growth morphologies. The greedy strategy is analogous to extremal dynamics (Duxbury and Dobrin, 1999). Extremal dynamics occurs in systems which are metastable and are driven weakly (well below the activation barriers) and so are evolving slowly (de Arcangelis et al., 1985; Duxbury et al., 1987; Zaitsev, 1992; Bak and Sneppen, 1993; Miller et al., 1993; Paczuski et al., 1995; Sneppen, 1995). A well-known example is the Bak-Sneppen evolution model which optimizes the "fitness" of a population by the extremal selection of the weakest species (Bak and Sneppen, 1993; Paczuski et al., 1996; Bak, 1996). The hottest-bond algorithm for fuse networks (de Arcangelis et al., 1985; Duxbury et al., 1987) and the invasionpercolation algorithms (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) are two other important examples of extremal dynamics. Notice that in the fracture of fuse networks load enhancement plays a crucial role so that, in spite of the extremal dynamics, there is no "self-organizing" critical steady-state. The connection between an extremal dynamics and critical behavior is related to the question of how to map annealed disorder to quenched and vice versa. In the case of the KPZ equation with thermal noise, as discussed in Section 6.3, the arrival-time mapping connects the quenched disorder of the directed polymer to the noise in the interface equation. And in both limits the system shows non-
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trivial scaling. The mapping can sometimes work in the opposite direction, as in the so-called "run-time statistics", where it was used to find the fractal dimension and dynamics of invasion percolation (see Marsili, 1994). Each greedy algorithm described in Section 2 solves exactly an optimization problem, that is, it minimizes a cost function. This begs the question as to whether other extremal processes have an associated cost function which is minimized by the dynamics. Such a cost function would act as a non-equilibrium free energy which would be useful in the theoretical development of the problem.
7
Random Ising magnets
7.1
Introduction
Random Ising magnets (Young, 1998) provide a key pedagogical basis for understanding the interplay between ordering (e.g. ferromagnetism) and disorder induced by quenched impurities. Many studies indicate that the ordered phase in random magnets and random manifolds is dominated by a zero-temperature fixed point which means that the zero-temperature behavior is characteristic of the entire ordered phase. In this chapter we demonstrate that the ground state of many random Ising magnets can be found exactly using optimization algorithms. Use of these methods is starting to provide detailed tests of functional-renormalizationgroup and replica-symmetry-breaking theories of random magnets. This is especially important at low temperatures, where Monte-Carlo methods are inefficient due to strong metastability effects. We consider the "spin-half" Ising magnets described by 7-[ -- -- ~ Jij~ricrj - ~--](hi + n)~ri (i,j) i
(7.1.1)
The first sum is over nearest-neighbor spins and cri = 4-1. We identify the following special cases: (i) Interfaces in random-bond magnets. Jij >__ 0, hi -- 0. In this case the bulk phase is ferromagnetic, but an interface is enforced by fixing the spins on two opposite faces of a lattice to have opposite orientations. The roughness and energy fluctuations of such domain walls have interesting scaling behavior. This minimal-energy-interface problem maps directly to the maximum-flow problem (with the domain wall being the minimum cut) (Section 7.2.2). (ii) Random-field magnets. Jij >__O. Random-field magnets are ferromagnetic at low values of the random field for dimensions d > 2 (Imry and Ma, 1975;
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Imbrie, 1984), but as the random-field strength increases, they become
frozen paramagnets. Even at zero temperature, there is then the possibility of a bulk phase transition, at which the order parameter (the magnetization) is singular. This problem also maps to maximum flow (the minimum cut defines the domain structure) (Section 7.3.2). (iii) Diluted antiferromagnets in a field (DAFF). Jij -- - - ] J I E i E j , where Ei -- 0, 1. This models experimental systems such as FecZnl-cF2 (Belanger, 1997), which are superexchange antiferromagnets doped with non-magnetic impurities (Zn). A key observation is that (nearest-neighbor) DAFFs on bipartite graphs map to diluted ferromagnets in an alternating field. The latter problem is solved by maximum flow (Section 7.3.2). Many of the experimental systems have bipartite structure, and are dominated by nearest-neighbor superexchange, so the DAFF model is an excellent first approximation. (iv) Frustrated magnets and spin glasses. Frustrated magnets are those in which the exchange interactions themselves cannot all be satisfied. The most famous example of a frustrated magnet is a spin glass (Binder and Young, 1986). A second famous example is the axial next-nearest-neighbor Ising model (ANNNI) (Selke, 1988). However spin glasses combine disorder and frustration (Toulouse, 1977) while there are a large number of magnets that are only frustrated (e.g. ANNNI). Although the two-dimensional Ising spin glass with nearest-neighbor interactions is solvable in polynomial time, the three-dimensional problem has been proven to be N P-complete. Nevertheless NP optimization methods are starting to make some progress even in that problem (Section 7.4). In this section we also present some interesting recent results for the Euclidean-matching problem (M6zard and Parisi, 1988; Houdayer and Martin, 1998), which is solvable in polynomial time in arbitrary dimensions and which shows some of the features of a spin glass.
7.2
Interfaces in random-bond magnets
7.2.1 Introduction and scaling theory We have already discussed an interface in a random-bond magnet in two dimensions (Section 6.3) as it is equivalent to the minimal-path problem. Nevertheless we discuss some further aspects of interfaces in two-dimensional randombond magnets as well as discussing results in three and four dimensions. After describing the continuum and lattice models used to study random-bond interfaces, we summarize some of the scaling predictions. In Section 7.2.2, we show
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how random-bond interfaces may be found, in arbitrary dimensions, by mapping to maximum flow. This is an important development as the transfer matrix is very inefficient for this problem in three and higher dimensions 9 Maximum-flow methods have provided the first precise tests (see Section 7 9149 of the functionalrenormalization-group predictions for this problem, as well as providing detailed information about the dependence of interface roughness on the strength of the disorder and the lattice orientation. The section on random-bond interfaces closes with a discussion of ground-state degeneracy in diluted networks, and some observations on the sensitivity of the ground state to a small drive field (Section 7.2.4). The continuum solid-on-solid(SOS) model for random-bond interfaces is, 7-/({u})--
f
dY
~[Vu(?)
+V(u(Y))+Hu(~)
)
,
(7.2.1)
where the single valued function u(?) measures the height (displacement) of the elastic manifold above a flat reference surface. For a d-dimensional interface in a (d + 1)-dimensional Ising magnet the vector ~ is d-dimensional. V (u (7)) models the random fluctuations in the exchange constants, cy is the interface stiffness and is related to J for Ising domain walls. H is an applied drive field. For many types of disorder, interfaces in random magnets are self-affine, that is they are fractals with different scaling dimensions perpendicular and transverse to the average orientation of the manifold. The equilibrium "width" and "energy fluctuations" of the manifold are defined as usual, i.e w2(L l) -- [uZ]av - [u] 2 and AE2(L, 1) = [E2]av - [E]2v . As a function of the system size parallel to the interface (L) and perpendicular to the interface (1), we have, 9
'
av,
w(L,1) ~ Lr fo(1/L ~)
(7.2.2)
A E ( L , l) ~ L ~ A ~ ( 1 / / ~ ) .
(7.2.3)
The energy fluctuation exponent 0 is related to the roughness exponent ~', (Huse and Henley, 1985; Barabasi and Stanley, 1995; Halpin-Healy and Zhang, 1995), 0 -- 2~" + d - 2.
(7.2.4)
From the minimal-path problem, we know that in (1 + 1) dimensions ~" = 2/3 (Huse and Henley, 1985; Kardar et al., 1986). In (d + 1) dimensions, a functionalrenormalization-group calculation yields ~" = 0.2083(4 - d) (Fisher, 1986). Lattice models of random-bond interfaces begin with the Ising magnet with Hamiltonian,
7-[ = -- Z JijcricrJ - Z (i,j)
Hcyi
(7.2.5)
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where Jij >__O. An interface is enforced by fixing the spins on two opposite faces of the lattice in opposite orientations (i.e. on one side fix o'i -- 1 and on the other O"i " ~ - - 1 ) . As we now demonstrate, the energy and configuration of interfaces in this random-bond magnet can be found exactly in an arbitrary dimension using a mapping to the maximum-flow problem.
7.2.2
Mapping to maximum flow
Although the mapping between random-bond interfaces and the maximum-flow problem is simpler than either the spin-glass (Bieche et al., 1980) or randomfield mappings (Picard and Ratliff, 1975; Angl6s d'Auriac et al., 1985; Barahona, 1985) to optimization problems, it was noticed and used much later. Early connections between minimal-cost interfaces (minimum cuts) were in the context of limiting paths in granular superconductors (Rhyner and Blatter, 1989; Riedinger, 1990, 1992). Rhyner and Blatter (1989) noted that the problem was a linearprogramming problem and solved small samples using the simplex method, while Riedinger (1990) made the connection with the maximum-flow problem and applied the augmenting-path-algorithm to its solution. More recently, Middleton (1995) used the mapping to maximum flow, and the preflow method, to find highprecision values for random-bond exponents in (2 + 1) and (3 + 1) dimensions (see Section 7.2.3). The mapping of random-bond interfaces to maximum flow/minimum cut is quite simple, it only requires using the correct variables. First, for any spin configuration {cri}, we define
S
--
{i E Vl~ri - -+-1}
S
--
{i E W l c r i - - 1 } = V \ S
(7.2.6)
Given a spin configuration described by S, its energy (taking (7.2.5) with H -- 0) is given by,
~(S)
=
-
~ iES,jES
--
--
Jij-
~
Jij+
iE-S,jES
~ Jij + 2 Z Jij iEV, jEV iES,jES
~
Jij
(7.2.7)
iES,jES -- Ebulk -]- Edw
The first term in the second of these equations is the bulk ferromagnetic energy (Ebulk), while the second term is the interface energy (Edw). The interface (or domain-wall) energy, Edw, is positive and we seek to minimize it. Now we show that the domain-wall energy, Eaw, is equivalent to the maximum flow in a related capacitated network, and that the domain-wall configuration is exactly the same as the minimum cut in that network.
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To define the analogous maximum-flow problem, first connect the sites on one face of the lattice to a new source node s, and the sites on the opposite face to a second new node, the target t (these two extra nodes are sometimes called "ghost" sites). Assign to the bonds making these connections infinite capacity. For each existing bond in the lattice make the association Uij - - U j i - - J i j , so that the exchange constants map to flow capacities. This construction leads to a network which is identical to that shown in Fig. 3.1. Note that we have forward and backward arcs for each pair of interacting sites in the lattice. Now we seek the maximum flow in this network, given the capacities U i j . From the minimum cut/maximum flow theorem, the maximum flow is equal to the sum of the capacities on the minimum cut (Section 3.2). The sites on one side of the cut are S, while the sites on the other are S. The capacity of a cut (S, S) is, w
--
u[S, S] =
Z
uij =
Edw
2 '
(7.2.8)
i6S,j6S
which is equal to the domain wall energy, as seen by comparing (7.2.8) with (7.2.7) and recalling that Uij --- J i j .
7.2.3
Roughness exponent and orientation dependence
Early attempts at numerically testing the prediction for the roughness exponent in higher dimensions (~ = 0.2083(4 - d) (Fisher, 1986)) used a transfer-matrix algorithm (Kardar and Zhang, 1989). However, although the transfer matrix is efficient for the DPRM in arbitrary dimensions, it is very inefficient for higherdimensional manifolds. Hence the analysis based on the transfer matrix was restricted to small sizes and the exponents found were not indicative of the infinitelattice behavior. Middleton (1995) used the maximum-flow mapping to obtain high-precision results. Middleton used {111 }-oriented Ising systems with a domain wall pinned (at one point) in the center of the system. His data for the (2+ 1)-dimensional problem is reproduced in Fig. 7.1. The best data collapse (that shown in the figures) using systems with varying lateral size (L) and height (l) yielded exponents: ~ = 0.41 (1), 0 = 0.82(2) in (2 + 1); which are consistent with the predictions of the functional-renormalization-group calculations (~ -- 0.416 in (2 + 1)), and with the scaling relation (7.2.4). In (3 + 1) a similar analysis yielded ~ -- 0.22(1) and 0 - 1.45(5) (1 up to 20 and L up to 30), which again confirm the RG prediction and the scaling relation (7.2.4). Using the mapping to maximum flow, Alava and Duxbury (1996) and R~iis/inen et al. (1998) have extended the analysis of random manifolds to include the dependence of roughness on: the type and strength of the disorder and; on lattice orientation. The dependence of the interface roughness on dilution, 1 - p,
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.
.
.
.
201
.
i
d=2
a)
F
'A<] ~/k E]~
,~, / k ~ '~
L=5 [] L = 1 0 L=20 /k L = 4 0 <] L = 8 0 V L=120
VZ~ /k
0.1 1
I
I
I
. . . .
I
1
O [] O /~
[]
L=5 L=10 L=20 L=40 = 120
b) ,
,
, ,
I
,
,
1
,
,
,
,
, ,
I
lO
1/L~ Fig. 7.1 Scaling plots" a) Roughness. b) Energy fluctuations, of random-bond interfaces in the {111 } orientation with uniform disorder. The calculations are for slabs of width I in the {111 } direction and area L2 parallel to the interface. (From Middleton, 1995)
for {10} orientation interfaces is presented in Fig. 7.2. Also plotted in this figure are results for the fracture surfaces in fuse networks (Duxbury et al., 1987; Kahng et al., 1988; Hansen et al., 1991a) with the same disorder. It is surprising that minimal-energy interfaces and scalar fracture surfaces have the same statistical properties in (1 + 1) dimensions (Roux and Hansen, 1992; R~is~nen et al., 1998). This is not true for (2 + 1)-dimensional fracture surfaces (Bouchaud, 1997), where scalar fracture surfaces are rougher, at least on length scales accessible to these simulations (R~isfinen et al., 1998). The behavior on approach to p -- 1 is reminiscent of the square-root behavior of the angle-dependent surface energy (Krug and Halpin-Healy, 1998). Results for the dependence of roughness on disorder for {100} and {111} orientation interfaces in cubic lattices are presented in Fig. 7.3. These results confirm the random bond exponent ~" = 0.41 +0.01 in (2 + 1) found by Middleton
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~3 0
~2
O O 0
O 9
0
080
090
0.9
9
,00
Bond fraction (p)
Fig. 7.2 Roughness of interfaces (lattice units) as a function of disorder. Solid dots are for domain walls, while the open circles are for fracture surfaces in fuse networks (see text). (From R~is~nen et al., 1998)
(see Fig. 7.1). This is an affirmation of the universality of this exponent, as the simulations in Fig. 7.3 were for dilution disorder, they allowed overhangs, and both the {100} and {111 } orientations were considered. In contrast, Middleton (Fig. 7.1) considered uniform disorder, directed (SOS) interfaces in the {111} orientation. The dependence of the roughness on disorder in the {10} (Fig. 7.2) and {100} (Fig. 7.3) orientations are very different. In the {10} orientation, the roughness grows sharply as disorder is increased from p = 1. However, in three dimensions, {100} interfaces are essentially flat until p* ~ 0.89, after which their roughness grows linearly. There is a (rigorous) claim (Bovier and Kulske, 1996) that interfaces are always roughened by disorder in (2+ 1), so even in the regime p > p . , at long length scales these interfaces are expected to be rough. However this does not exclude the possibility that a weakly rough to strongly rough transition could be occurring at p* = 0.89. This is consistent with recent very large scale simulations at p = 0.90 (Rfiis~inen et al., 1998), which indicate a very weak roughness at large length scales. In contrast a variational calculation indicates that there is an exponentially growing length scale, ld "~ e x p ( k / ( 1 - p ) ) (k an unknown constant), below which interfaces may be flat but above which they have the random-bond exponent (Bouchaud and Georges, 1992). However, some aspects of the latter calculation are argued to be incorrect for d > 2 (Emig and Nattermann, 1998, 1999), although the same conclusion is reached for the d -- 2 case.
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Fig. 7.3 Scaled plots of the interface roughness as a function of bond concentration p, for {100} and {111 } orientation cubic lattices. (From Alava and Duxbury, 1996)
There is also an interesting effect at weak (dilution) disorder for {111 } interfaces (Fig. 7.3), where the roughness is anomalously large. This is a discreteness effect due to the high degeneracy of interfaces in the {111 } orientation at p = 1. Removing a couple of bonds from the graph leads to an average roughness which is ~ L instead of ~ L~ + 1), In(L)(2 + 1) as occurs at p = 1 (Nienhuis et al., 1984). Thus diluted {11 } and {111 } interfaces have a singular behavior on approach to p -- 1. In Section 8.5.3 we discuss a random-surface model which is equivalent to a random-bond Ising problem in which there is periodic (with period 2) disorder in the z-direction (perpendicular to the interface). This model arises as an approximation to disordered elastic media, and we derive it in that context in Section 8.2.5. The maximum-flow mapping described in Section 7.2.2 is applicable for any random or regular potential and so it also solves the problem of periodic elastic media.
7.2.4
Degeneracy and sensitivity
A variety of other calculations concerning the sensitivity of interfaces (Zhang, 1987; Shapir, 1991) to disorder are possible with the maximum-flow algorithms. In addition, it is possible to find the exact degeneracy of the ground state using a detailed analysis of the degenerate-cut structure as we now describe.
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Fig. 7.4 The degenerate interface structure for the BDIM at p = 0.64. Different shades indicate different spin clusters. Note that the ground state can be one of three independent interfaces, each with its own excitations. (From Bastea, 1998)
Consider the bond-diluted Ising model in the {10} orientation of a square lattice. An example of the ground state structure is shown in Fig. 7.4, with neighboring subclusters having different shades of grey. In this example, there are clearly several degenerate states which have no overlap (in the terminology of replica symmetry breaking they are in different "valleys"). In addition, there are a larger number of small degenerate clusters. A detailed study shows that the probability that in the ground state of large samples there are at least two non-overlapping ground states P2(L) ~ constant < 1. That is, there is a finite probability of finding degenerate non-overlapping interfaces in the lattice, but it is not assured. The constant is dependent on the bond concentration p, and must approach one as p --+ 1. There is also a power-law distribution of small degenerate clusters which dress the interface. This power law is observed to have exponent -2.81 (5) (Bastea and Duxbury, 1999). The degeneracy of random-bond interfaces is calculated from the structure of the minimum cut. However, it is not obvious which interface clusters can be flipped independently. This dependency can be encapsulated in a reduced graph which has a node for each degenerate cluster, as illustrated in Fig. 7.5. To enumerate the independent cuts (Bastea, 1998), we count all possible directed partitions of the reduced graph into two parts, one of which contains the source s, and the other which contains the target t. Although this graph partitioning problem is computationally hard, the reduced graph is quite small and hence the algorithm is quite efficient in practice. There has also been a recent analysis of the sensitivity of the ground state of the DPRM to the presence of a small drive field H in the Hamiltonians (7.1.1)
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T
ut
\
s
Fig. 7.5 Typical supergraph obtained using the degeneracy algorithm. S is the set of sites fixed to the source and T the set of sites fixed to the sink. C 1, C2 and C3 are independent clusters; they are made of subclusters that are not independent of each other. The S - c u t is the solution with all the independent clusters on the side of the sink, and the T - c u t is the one with all of them on the side of the source. We also show a directed cut (ground state) in which a part of each independent cluster is on the sink/source side. (From Bastea and Duxbury, 1998) and (7.2.1). This problem has been extensively studied in the context of wetting in random systems (Lipowsky and Fisher, 1986; Forgacs e t a l . , 1991). T h e r e the interface describes a boundary between phases ot and/3, in the presence of an inert spectator phase y. The external force amounts to a chemical-potential difference between the phases separated by the interface, and one may consider random-bond or random-field disorder (Huang e t a l . , 1989; Forgacs e t a l . , 1991). Comparisons of numerical calculations in (1 + 1) with scaling theories, which predict that the thickness of the wetting layer [U]av diverges as [U]av ~
H -0,
(7.2.9)
indicate agreement for the depinning exponent in the "weak fluctuation regime" (i.e. interface fluctuations comparable to the typical wall-interface separation, 1 < < [U]av < w). The prediction for ~p (Lipowsky and Fisher, 1986) is 7t = ~ ' / ( 2 - ~'), and in (1 + 1), numerical simulations give 7t -~ 0.5, in agreement with the scaling arguments (Huang e t a l . , 1989; Wuttke and Lipowsky, 1989). Attempts at testing the exponents in higher dimension using flow algorithms are now possible, though difficult due to strong finite-size effects.
206
M . J . Alava e t al. 10 2
L 1.62+/0.04
10 3
~10
4
.....1
4O0 3OO
g 200 10 5
I00 0
0
200
400
600
800
1000
L 10 6
10j
10 2
10
L Fig. 7.6 Scaling of the jump field H1 for (1 + 1) directed polymers (with l = L). The numerical exponent agrees with the scaling estimate ot = 5/3 (see text). (E. Sepp~il~i and M. J. Alava, 2000)
In the regime [U]av > > w, the interface is in the bulk and its behavior is not controlled by the overlap of its fluctuations with the surface. In that case, we can still ask what happens to the ground state of an interface as the external field H is slowly switched on. This is one way of testing the sensitivity of the ground state to small perturbations. It is known that the DPRM is sensitive to small perturbations (Zhang, 1987; Mdzard, 1990; Shapir, 1991; Hwa and Fisher, 1994a), and this sensitivity is often call "chaos" in analogy with the sensitivity of dynamical systems to small perturbations in their initial conditions. Numerical results using the maximum-flow method indicate that the interface (both the DPRM and the (2 + 1)-dimensional interface) may frequently have a few excitations of finite extent, but at a field H1, [U]av typically jumps over a distance which is proportional to the system size. This is indicated in Fig. 7.6 for the DPRM where it is seen that the typical size of the first large jump in the interface location is 0.4/, where 1 is the system size transverse to the interface (DPRM). The scaling behavior of H1, the point at which the interface "jumps" an extensive distance, can be derived as follows. As the interface moves through the lattice, it sees energy fluctuations E n - EO ~ L ~ where E n is the energy of an arbitrary local minimum and E0 is the ground-state energy. This energy difference should be equated with the jump energy H L d Au. If the interface jumps a distance proportional to 1 (as seen in Fig. 7.6), then we have, H1 ~ L ~
]/1 9
(7.2 10)
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Thus, if l "-~ L, then H1 "~ L -a, where ct = d + 1 - 0. The jump field exponents are then, ct = 5/3(1 + 1) and ot ~ 2.18(2 + 1) respectively. The (1 + 1)dimensional result is supported by simulations using maximum flow (see Fig. 7.6).
7.3 7.3.1
Random-field magnets and DAFF Introduction and overview
In this section we review optimization calculations which have been carried out for the spin-half random-field Ising magnet, ~'~ -- -- ~ Jijo'itYj -- ~ hitTi, (i,j) i
(7.3.1)
with the restriction that Jij >__ 0, which ensures that the exchange constants favor ferromagnetic order. The fields hi can be chosen to suit the problem. We shall consider Gaussian, uniform and -l-h distributions. As we show in Section 7.3.2 there is a non-trivial mapping of this problem to the maximum-flow problem (Picard and Ratliff, 1975; Angl6s d'Auriac et al., 1985; Barahona, 1985; Angl6s d'Auriac et al., 1997). In this mapping the cut structure defines the domain structure in the random-field magnet. In Section 7.3.2, we also demonstrate the mapping of a diluted antiferromagnet in a field (DAFF) on a bipartite graph, to a diluted ferromagnet in an alternating field. The latter problem is of considerable practical interest as it is presumed to be in the RFIM class (Fishman and Aharony, 1979; Cardy, 1984), and it has some beautiful experimental realizations (Belanger, 1997). As the strength of the random field is increased, there is an increasing tendency to form domains which favor the direction of the local random fields. In one and two dimensions any amount of random-field disorder destroys long-range order (Aizenman and Wehr, 1989) and the ground state is in a "domain phase" consisting of a disordered aggregate of domains of up and down spins. However, in three dimensions there is a phase transition from a ferromagnetic state at small random fields (Imry and Ma, 1975; Imbrie, 1984; Bricmont and Kupiainen, 1987) to a domain phase at large random fields. The nature of the ground-state transition, however, remains unclear (Nattermann and Villain, 1988; Belanger, 1997; Nattermann, 1997). The mean-field theory of RFIM magnets exhibits two scenarios: if the random fields hi have a symmetric, binary distribution hi = +h then there is a first-order phase transition from the ferromagnetic state to the domain phase at large enough random fields (Aharony, 1978; Swift et al., 1994). In the case of a Gaussian distribution of random fields this transition is second order. The upper critical dimension of the RFIM is six, above which this mean-field picture
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should apply (Grinstein, 1976; Grinstein and Ma, 1983). Between two and six dimensions, scaling theories (Grinstein, 1976; Grinstein and Ma, 1983; Bray and Moore, 1985b; Gofman et al., 1996) which assume a zero-temperature fixed point lead to a "modified hyperscaling relation" 2 - ~ = (d - y)v. Unfortunately there is no convincing analytical theory to distinguish between this possibility and the possibility of a first-order transition at zero temperature, although there has been recent progress in understanding the origin of the theoretical difficulty of the RFIM (Nattermann, 1997; Brrzin and de Dominicis, 1998). In Section 7.3.3, we review studies, using maximum flow, of the ground-state phase transition of three- and four-dimensional RFIMs and DAFFs. In Section 7.3.4 we discuss domains walls in the ferromagnetic phase of the RFIM, generated by imposing opposite spin orientations on opposite faces of a hypercube. We also make some remarks about the fractal dimension of domain walls in the domain phase of the RFIM in two dimensions, and about domain-size distributions in one and two dimensions. In Section 7.3.5, we describe an analysis of the sensitivity of the random-field domain state to small random-field perturbations. We close with a discussion of the ground-state degeneracy of a RFIM with +h fields and the ground-state degeneracy of the DAFF, again in the domain phase.
7.3.2
Mapping to maximum flow
The correspondence between maximum flow and the RFIM came to the attention of the physics community (Anglrs d'Auriac et al., 1985; Barahona, 1985) soon after the discovery of mappings between two-dimensional spin glasses and optimization problems (Bieche et al., 1980). A very similar mapping was available somewhat earlier in the operations-research literature (Picard and Ratliff, 1975). To map the problem of finding RFIM ground-states to maximum flow, we construct an extended graph using a "ghost-site" technique which is very similar to that used for random-bond interfaces (see Section 7.2.2). Two new "ghost" nodes are connected to the old nodes as follows. All sites with positive random field are connected to a new node, s, and all sites with negative random field to a new node, t: Jsi
J/t
=
hi 0 [hi[ 0
if hi >__0 ifhi <0 if hi < 0 ifhi >0.
(7.3.2)
This construction is illustrated in Fig. 7.7. This construction is general as it applies for arbitrary range exchange c o n s t a n t s Jij >__0, including infinite range. The field terms may be regular and/or random so it also includes an alternating field which
2
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arises from the mapping to the DAFF (see below). With this construction, we can now write the energy (7.3.1), or "cost function", as
E---
Z
Jij~
(7.3.3)
(i,j)EA where the spin orientation of the source and target nodes are fixed: as = 1 and at = - 1 . This is now a random-bond Ising model on the enlarged graph which includes s and t. We proceed in a manner very similar to that given for random-bond interfaces (see Section 7.2.2). The new feature is that the spins of the ghost sites now impose the "antiperiodic" boundary conditions, and enforce an interface. We define, as in Section 7.2.2, S to be the set of up spins and S to be the set of down spins. The energy of the RFIM ground state can then be written as (this is the same as in Section 7.2.2),
E(S) -- -
~ (i,j)Ea
Jij + 2
~
(7.3.4)
Jij.
(i,j)E(S,S)
Now the first term (the negative term) is the energy one obtains by presuming that all of the exchange terms and all of the field terms are satisfied. The second term is the degree to which the field and exchange terms are unable to be simultaneously satisfied. Clearly the ground-state energy is minimized when the positive term in (7.3.4) is smallest. As in Section 7.2.2, making the association Uij -- Jij makes it clear that the positive term in (7.3.4) is equivalent to the cut capacity of a flow problem, where flow is injected at s and removed at t. Hence, from the mincut/max-flow theorem (Section 3.2) the cut energy is equal to the maximum flow. Once the minimum cut has been identified, it defines a partition (S, S) and hence it also defines the domain structure of the RFIM ground state. It is worth stating again that this mapping applies for arbitrary range ferromagnetic interactions and arbitrary fields (for spin 1/2 magnets). However, a generalization of this mapping to Potts models, leads to a k-cut or a k-terminal problem, which has been proven to be in the NP-class (Dahlhaus et al., 1994). For planar graphs, including the k-terminal vertices, there are complicated but polynomial algorithms. Diluted antiferromagnets in a field (DAFF) are expected, in the weak field limit, to be in the same class as the RFIM (Fishman and Aharony, 1979; Cardy, 1984). Examples of diluted antiferromagnets are materials such as, FecZnl-cF2 (Belanger, 1997). c is the concentration of magnetic atoms (Fe) and can be varied. The DAFF Hamiltonian reads m
~-~ -- Z JijEiEj~iO'j -- Z OEi~ (i,j) i
(7.3.5)
where O'i -- '1--1, Jij >_ O, (i, j) are nearest-neighbor arcs on a lattice, and ei E {0, 1} with ei -- 1 with probability c, representing the site concentration
210
M . J . Alava et aL
UP
__db,
IF
I I
IF
I~'
M i n i m u m Cut -h 4 DOWN Source
Fig. 7.7 An example of how to map the one-dimensional RFIM chain into a augmented graph with two extra sites. The sites of the original chain are connected to each other with bonds of capacity J, and depending on the sign of the random field to either of the "ghost sites" with bonds of capacity Ih i I. We need to find the maximum flow from s(source) to t(target or "sink"), and its associated minimum cut.
of spins. Usually one takes Jij --" J, as is relevant to experiment. The antiferromagnetic exchange competes with the external field which tries to produce a finite magnetization. For bipartite networks, with nearest-neighbour interactions, it is trivial to map this problem to a ferromagnetic exchange problem in an alternating field (Mattis, 1976). To see this, divide the lattice into two bipartite sublattices A and B. Every site in A has all its nearest neighbors in B, and vice versa. Now use the "gauge" transformation (here we allow for random fields hi at each site, while for the "usual" DAFF case hi = H.),
t
{ +~ri
for for
i 6 A i EB '
t { -+-sihi hi = -8ihi
for for
i 6 A
i ~ B ' (7.3.6) which flips the spins on one sublattice and leaves the other sublattice unaltered. The Hamiltonian which results is, O'i - -
--O" i
" -- -- Z (i,j)
Jgisjo'/o'J -- Z "8i0"/ + Z "giO'/" i~A iEB
(7.3.7)
(7.3.7) is the Hamiltonian for a diluted ferromagnet in a staggered field, with one sublattice being subjected to an up field, while the other sublattice experiences a
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down field. This model is solvable by mapping to the maximum-flow problem, as are more general problems in which the nearest-neighbour exchange is random (though still positive), and there is a local random field. However, it is restricted to nearest neighbours as further neighbour antiferromagnet interactions lead to couplings within the sublattices.
7.3.3
The phase transition in three dimensions
The analysis of the phase transition in the random-field Ising model has a history filled with controversy, and in some cases, acrimonious differences of opinion. An early perturbation analysis (Parisi and Sourlas, 1979) suggested that "dimensional reduction" should be valid, so that the behavior in a pure system in dimension d should correspond to random-field behavior in d + 2. This implies that there should be no phase transition in the d -- 3 RFIM. However, it was then proven that the d = 3 RFIM is ordered at weak fields (Imbrie, 1984), demonstrating that dimensional reduction is incorrect. There is still no reliable ~-expansion for the RFIM (Br6zin and de Dominicis, 1998). Concurrently there was a bitter conflict between experimentalists who claimed that DAFFs in three dimensions exhibited a phase transition and those who claimed that they did not. This was also resolved in favor of a phase transition (Belanger, 1997). Two dimensions is the lower critical dimension for the RFIM and there is a rigorous proof that a unique Gibb's state exists for arbitrary weak fields in two dimensions (Aizenman and Wehr, 1989), implying that there is no finite magnetization for any finite random field in two dimensions. Using the mapping to maximum flow (see Section 7.3.2), Ogielski (1986) studied the ground-state phase transition in the RFIM in three dimensions for Gaussian and bimodal disorder and clearly confirmed that there is indeed a phase transition. He concluded that the phase transition is second order (at zero temperature) for the Gaussian-random-field case and for the bimodal case the order parameter had a tendency toward a first-order behavior. For the Gaussian-disorder case, he found the magnetization exponent r = 0.05 and the correlation-length exponent v -- 1.0. The mean-field exponents for the Gaussian case (which is second order) are v = 1/2, /3 = 1/2. Several recent calculations of a similar type to Ogielski's (Swift et al., 1997; Nowak et al., 1998; Hartmann and Nowak, 1999), as well as Monte-Carlo simulations (Rieger and Young, 1993; Rieger, 1995a) have confirmed many of his general conclusions. In all cases, the magnetization exponent is very small indicating, perhaps, a first-order behavior. However, there is no indication of a latent heat. The correlation-length exponent also seems to be non-universal, with values near v - 1.1 reported for Gaussian RFIM and DAFF, while values closer to v = 1.6 are typical of the (+h) RFIM. Compare these with the latest experiments on a high-quality DAFF sample, that indicate a correlation
212
M . J . Alava e t al.
length exponent of v ~ 0.87 (Slanic et al., 1999). A typical scaling analysis for the DAFF is presented in Fig. 7.8, where the staggered magnetization is M -[Iml]av, and the corresponding "disconnected" susceptibility is Xdis • L3[m2]av. Here m is the sublattice magnetization for a given realization of the disorder, and the average is over disorder. The scaling behaviors M -- L - M v l f I ( ( A - A c ) L l/v) and Xais = L - y / v X ( ( A - A c ) L l/v) are assumed. As remarked above the values of v and fl found in these simulations are quite close to those reported by Ogielski (1986). However, the small size of fl suggests a first-order behavior in that quantity, and which motivates a detailed look at the behavior of the magnetization as a function of field. An example of the field dependence of the magnetization, for a single realization of a DAFF, is presented in Fig. 7.9. This figure suggests that the magnetization decays via a sequence of jumps, rather than in a continuous manner. These jumps are "averaged out" in the data of Fig. 7.8. There is then a question about how the size of the magnetization jumps change as the system size is increased. Angl6s d'Auriac and Sourlas (1997) and Sourlas (1999) have done an interesting analysis in which they show that the size of the magnetization jumps ji (e.g. the five largest ones) appearing in data such as Fig. 7.9 should be plotted according to Ci
ft"
ji = j ~ + L--i~(1 + - ~ ) 6 j i .
(7.3.8)
From this analysis it is concluded that the magnetization jumps do not disappear in the infinite-lattice limit. Instead, there is a finite, and large, jump in the magnetization in the infinite-lattice limit, for example for the DAFF, they find that the magnetization jump j ~ ~ 0.55. In addition, the correlation-length exponents v found from the finite-size-scaling analysis of (7.3.8) are non-universal and are significantly larger than the results found using conventional analysis (Ogielski, 1986; Swift et al., 1997; Nowak et al., 1998; Hartmann and Nowak, 1999). This clearly raises questions about the nature of the RFIM phase transition in three dimensions, and it is likely that optimization methods will greatly assist in their resolution. Analysis of the transition in four dimensions agrees with both the mean-field limit, and recent high-temperature series expansions. For binary disorder the transition is discontinuous, and for Gaussian disorder it is continuous (Swift et al., 1997).
7.3.4
Domains and domain walls in the R F I M
At sufficiently high random fields in three dimensions, and at all finite fields in one and two dimensions, random fields destroy ferromagnetic long-range order in the ground state. We define the "breakup lengthscale", Lb, to be the size scale at
2
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Exact combinatorial algorithms
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-
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[]av a n d
15
the c o r r e s p o n d i n g
s u s c e p t i b i l i t y gdis -- L3[m2]av for the D A F F on a c u b i c lattice w i t h site c o n c e n t r a t i o n c = 0.50. T h e f o l l o w i n g v a l u e s w e r e f o u n d Ac = ( H / J ) c
-- 0.62, v = 1.14,/3 = 0.02,
-- 3.4. ( F r o m H a r t m a n n and N o w a k , 1999)
I
I
0.8
0.6 0.4 ---t
0.2
rl
I _
o
~--~.,'-
I _I- -- -- -
I _
---L__
_
_
_
_ i, . . . . .
I
I
I
I
I
I
1
2
3
4
5
6
7
A
Fig. 7.9 T h e g r o u n d - s t a t e s t a g g e r e d m a g n e t i z a t i o n , m, o f a D A F F w i t h c = 0 . 5 0 and L = 15 on a c u b i c lattice, as a f u n c t i o n o f the strength o f the a p p l i e d field A. T h e g r o u n d state is d e g e n e r a t e , the m i n i m u m (solid) and m a x i m u m ( d a s h e d ) c u r v e s i n d i c a t e the variation o f m o v e r the set o f d e g e n e r a t e states. ( F r o m H a r t m a n n and N o w a k , 1999)
214
M . J . Alava et aL
which ferromagnetic domains cease to exist. In one dimension, Lb diverges algebraically on approach to A -- 0. In three dimensions, L b diverges algebraically on appoach to Ac, if the transition is continuous. If the transition is first order, the domain size should remain finite up to the critical point. In two dimensions, which is the lower critical dimension for the RFIM, the behavior is different. There, as suggested by Binder (1983) and Grinstein and Ma (1983), the data is consistent with, Lb ~ exp (A[1/A]2),
(7.3.9)
where A is a disorder-dependent constant. This form has been confirmed, with A = 1.9 q- 0.2 and 2.1 -+- 0.1 for bimodal and Gaussian disorder, respectively. This result is different from that obtained with finite-temperature Monte-Carlo simulations for small L (Fernandez and Pytte, 1985), which demonstrates the advantages of ground-state calculations in pushing to large enough system sizes. The surfaces of the magnetic domains scale with non-trivial exponents (Esser et al., 1997). We note that for the Gaussian disorder case, Vives et al. claim that there is a phase transition to a ferromagnetic state (Frontera and Vives, 1999), which is inconsistent with the Binder prediction and is also inconsistent with the Aizenman-Wehr theorem (Aizenman and Wehr, 1989). Nevertheless an interesting conflict between the conventional analysis of the RFIM and the results of ground-state simulations is beginning to emerge (Sepp~il~i et al., 1998). Specifically the role of the percolation of a macroscopic domain. Preliminary work indicates that percolation of ferromagnetic order occurs in the RFIM at a fairly large value of A ~ 2. This is not too surprising since even in the strong-field limit, the 50/50 concentration of up and down fields is close to the site percolation threshold, pc -- 0.59. As the field strength is reduced domains grow so percolation of a macroscopic domain is expected. If this percolating domain remains fractal all the way down to A -- 0, there is no violation of the AizenmanWehr theorem. Nevertheless, there is a "phase transition" at the percolation point to a quasi-long-range-ordered phase. This appealing picture warrants an intensive numerical analysis. The renormalization calculations (Fisher, 1986; Halpin-Healy, 1989) and an Imry-Ma argument indicate that the roughness of domain walls in (d § 1)dimensional RFIMs should have the exponent ( = (4 - d ) / 3 . Early studies of RFIM domain walls in (1 § 1) used the transfer-matrix method (Fernandez et al., 1983). The domain-wall energy is calculated, as usual, from Edw -- Eap - Ep
(7.3.10)
where Eap denotes the energy of the system with antiperiodic boundary conditions and E p the energy with periodic boundary conditions. Fig. 7.10 shows an example of a (1 § 1) RFIM domain wall.
2
Exact combinatorial algorithms
215
Fig. 7.10 An RFIM interface. The calculation is for an L = 100 square lattice with weak bimodal disorder, A = 10/17. Note the large jumps on the interface and the lack of overhangs. (From Sepp~l~ et al., 1998)
The idea that domain walls in the RFIM are self-affine has been supported by the work by Fernandez et al. (1983) and by Halpin-Healy and Herbert (1993), both using the transfer-matrix technique. Moreover, later studies with combinatorial optimization further supported this claim (Moore et al., 1996; Jost et al., 1997). If RFIM interfaces are self-affine, the variations in the domain-wall energy would scale as, A E ( L ) ~ L ~ as in the random-bond case discussed in Section 7.2, and the energy should be linear in L. Notice that this is in direct contrast to the argument as to why the two-dimensional ground state should break up into domains: the domain-wall energy has a logarithmic correction which can be measured in ground state calculations (Seppfilfi et al., 1998). The data in Fig. 7.11 shows that (1 + 1) RFIM interfaces scale, at weak disorder, with a roughness exponent close to 6/5. Weak disorder implies that the system size is smaller than the "breakup" length, L < < Lb, so that the bulk state is ferromagnetic. Once beyond the breakup length scale, the roughness exponent returns to 1 (see inset to Fig. 7.11). The origin of the super-rough behavior seen in Fig. 7.11, is a broad distribution of jump sizes which is visually evident in Fig. 7.10. A quantitative analysis indicates that the distribution of jump sizes P ( 3 h ) is a stretched exponential, but also that large jumps occur quite frequently at large sample sizes as in some models of kinetic growth (Krug, 1994). For the weakdisorder regime one obtains an effective energy-fluctuation exponent 0 ~ 0.9 (Sepp~il~ et al., 1998). One should finally note that the roughness exponent as determined for ground-state domain walls has nothing to do with the similar one for driven interfaces with random-field disorder (quenched Edwards-Wilkinson equation) (Kessler et al., 1991; Leschhorn, 1993).
216
M . J . Alava et aL
102 m m
101
9
/ oo~
9
10 ~
A
10~ 10~
.
10 ~ .
.
.
.
.
.
.
i
102 L
10 2 .
.
.
.
.
i0 a .
.
.
103
Fig. 7.11 Scaling of the interface width for an RFIM with bimodal disorder. A = 2/3 (empty triangles) and 3/2 (filled squares). The line indicates a least-squares fit with a roughness exponent ~ = 1.20 4- 0.05. The inset shows the cross-over in interface properties with increasing system size (A = 10/9). (From Sepp~il~iet al., 1998) 7.3.5
Sensitivity and degeneracy
In this section we describe studies of the properties of the paramagnetic state of the RFIM and the DAFE In this regime, the minimum cut gives the complete domain structure of these complex ground states. We study the stability of these ground states to small perturbations (sensitivity) (Alava and Rieger, 1998), and in the case of the DAFF and the (+h) RFIM, the ground-state degeneracy (Bastea and Duxbury, 1998; Hartmann, 1998c). In spin glasses (Binder and Young, 1986; Rieger, 1995c; Young, 1998) it is well known that small changes of parameters like temperature or external field cause a complete rearrangement of the equilibrium configuration (Fisher and Huse, 1986; Bray and Moore, 1987). This is sometimes called "chaos" in analogy with the sensitivity to initial conditions exhibited by chaotic dynamical systems. This chaos has experimentally observable consequences like reinitialization of aging in temperature cycling experiments (Refregier et al., 1987; Lefloch et al., 1992; Mattsson et al., 1993; Rieger, 1994) and has also been investigated in numerous theoretical works (Ritort, 1994; Kisker et al., 1996; Kondor, 1989; Kondor and Vdgs6, 1993; Franz and Ney-Nifle, 1995). A slight random variation of the quenched disorder has the same effect on the ground-state configurations. In theoretical calculations, chaos with respect to temperature changes is harder to observe than chaos with respect to disorder changes (Ney-Nifle and Young, 1997), and the latter phenomena has been used to quantify spin-glass chaos in numerical investigations (Bray and Moore, 1987; Rieger et al., 1996).
2
Exact combinatorial algorithms
217
Fig. 7.12 A two-dimensional RFIM ground state plus the perturbation-induced changes. The original spin-orientations are indicated in grey for oi = + 1 and white for cri -- -1, the flipped spins are indicated in black. L = 320, A -- 2 and 6 -- 0.1. (From Alava and Rieger, 1998)
This type of chaos has also been demonstrated in the directed polymer in a random (bond) medium (Zhang, 1987; Feigel'man and Vinokur, 1988; Nattermann and Lyksutov, 1992), where it is quantified using a "displacement exponent". The displacement exponent is related to the roughness exponent ~" (Feigel'man and Vinokur, 1988; Nattermann and Lyksutov, 1992) and is also related to the susceptibility of such interfaces (Shapir, 1991). The extent to which the ground state ({cri}) and the perturbed state ({a/(3)}) of a magnetic system differ, is quantified by the overlap,
q - ~
1
!
O'i~ri ( 5 ) .
Z
(7.3.11)
i
In glassy systems, thermodynamic states are typically chaotic, which means that on length scales L > L* the two spin configurations {or} and {or~} lose all correlations. In contrast, the RFIM ground state remains correlated (see Fig. 7.12) at long length scales, although there are considerable domain excitations at small length scales. The "sensitive" regions in the RFIM model concentrate on the cluster boundaries of the ground state. Fig. 7.13 presents the value of the overlap q as a function of the random-field amplitude A in two dimensions. This figure has been obtained with a random field distribution and a perturbation distribution that have a constant probability density between --A and A and - 5 / 2 and 5/2, respectively, and Jij = 1. In any
218
M.J.
Alava e t al.
1.00
0.95
0.90
0.85
[] A V O 9 . . . .
~ 1
.
.
.
.
.
oL=40 &L=80 V L = 160 9L = 240 o L = 320 .
.
.
10
Fig. 7.13 Scaling of the chaos or overlap, q, of the two-dimensional RFIM model with Gaussian disorder, for system sizes L = 40, 80, 160, 240 and 320 and for 6 = 0.1. (From Alava and Rieger, 1998)
dimension, the limit A --~ cx~ goes over to a site-percolation problem, i.e. the local RF-orientation determines the spin state at a site. In that limit the overlap q is determined by the probability that the applied perturbation 6 will change the local field orientation. At small enough fields, the ground state (on finite lattices) is ferromagnetic and there are no "sensitive" interface regions in the ground state. The overlap then approaches one. As seen in Fig. 7.14, the ground-state degeneracy of (• RFIM also occurs at the interfaces between up and down domains in the RFIM ground state. This sort of degeneracy also exists in the DAFF (see below) and so may be of experimental relevance. These degenerate configurations are found from the degenerate cut structure using the same method as that used for degenerate random-bond interfaces (see Section 7.2.4) (Bastea, 1998) (see also (Hartmann and Usadel, 1995)). The degenerate domains evident in Fig. 7.14 produce a finite ground-state entropy for a wide range of A = h / J , as demonstrated in Fig. 7.15. Prior work in one dimension (Derrida et al., 1978; Bruinsma and Aeppli, 1989; Igloi, 1994), on Cayley trees (Bruinsma, 1984), and in two dimensions (Morgenstern et al., 1981), had already indicated the singular nature of the degeneracy as a function of the field. In particular, the large degeneracies at fields (h/J) which are small rationals. However, the presence of ground-state entropy even for irrational h / J was a surprise (Bastea and Duxbury, 1998). The mechanism for this is easy to construct. The degenerate clusters at irrational A have zero field energy and the same domain-wall energy in both the up and down states of the cluster. For the
2
Exact combinatorial algorithms
219
Fig. 7.14 Ground-state structure of the bimodal RFIM for A = h / J = 3/2. The spins frozen "up" are the dominant grey color, the spins frozen "down" are white, while the other shades of grey represent the spins making up the degenerate clusters. A dot indicates a field "up" and the absence of it indicates a field "down". (From Bastea, 1998)
RFIM on the square lattice, the lowest order degenerate clusters of this sort are indicated in the inset to Fig. 7.15. The number of these clusters (or two-level systems (TLS)), nTLS, can be estimated from the expression, L2 nTLS O~ P T L S ~ Lb
(7.3.12)
where L is the linear size of the system, and Lb is the domain size. From (7.3.9), we have Lb o~ e x p [ ( 1 / A ) 2 ] . L Z / L b is the total length of interface between up and down spin domains in the system. PTLS is the probability of occurrence of a TLS at a given interface site. PTLS -- pn/4, where 1/4 is the probability of occurrence of the up-down pair of fields and Pn is the probability that this pair is surrounded in the ground state by frozen spins with the appropriate configurations. The entropy density is then s o~ p n e x p [ - ( 1 / A 2 ) ] for A < 4 and 0 f o r A > 4. Pn is discontinuous at A -- 2, because the dominant TLSs for A < 2 are different than those for A > 2 (for example there are twice as many spin configurations that lead to a TLS below than above A -- 2). If we take the observed jump ~ 3.3, then the above argument leads to the curve given in the inset to Fig. 7.15, which is very close in form to the continuous part of the entropy presented in Fig. 7.15. In the regime Ac < A _< A., we can consider the ground state to be composed of a frozen background in which is embedded a set of largely non-interacting free superspins (for each independent cluster). The ground state thus contains a large number of magnetic two-level systems (Coppersmith, 1991). This gives rise to a
220
M . J . Alava et aL
q)
0.06
5
@ i
2
0.03
0
o
I
2
hlJ
3
hlJ
4
Fig. 7.15 The ground-state entropy of the two-dimensional RFIM as a function of h / J = A. The inset shows the smallest two-level systems (TLSs) at irrational A and an estimate of the entropy (from (7.3.12)) produced by them. A dot indicates where the local random field favors the "up" spin direction. The system sizes used were from 10 x 10 to 130 x 130 and the entropy was found as the slope of the line (ln D).vs.N, where D is the degeneracy, N is the system size (total number of spins) and the average is over the disorder (1000 samples were used). (From Bastea and Duxbury, 1998)
paramagnetic response at low temperatures for both the RFIM and DAFF in this regime. The natural ground-state order parameter for the paramagnetic response in the regime Ac < A < A , is the magnetization mpferro for the RFIM and the staggered magnetization mpstaggered for the DAFE These quantities can be calculated by applying an appropriate infinitesimal field or by polarizing all of the degenerate domains in a given orientation. The basic features of the ground-state degeneracy are then reflected in the ground-state paramagnetic magnetization. In three dimensions, the entropy remains zero at low A, reflecting the ferromagnetic state for A < Ac. While for large fields A > A, = 6 in three dimensions, the field polarizes all of the spins and there is no degeneracy. Calculations of the paramagnetic order parameter for the DAFF, mpstaggered, for square and cubic lattices are shown in Fig. 7.16. (The situation is similar for the RFIM.) There is a strong sublattice paramagnetic response for all Ac < A < A,, with spikes at certain rational values. These figures are for a DAFF with dilution p = 0.9, but only the details change as p is varied, at least when p > Pc,site. In the regime A < Ac there is a spontaneous staggered magnetization, and low temperature measurements (such as neutron
2
Exact combinatorial algorithms
221
0
0.2
d E ffl
E
0
0.1
2
4
6
H/J
i
0
,
J
i,-,
1
i
2
H/J
I
3
i
,
4
5
Fig. 7.16 The order parameter for the ground-state paramagnetism (mpstaggered) of the DAFF on square and cubic (inset) lattices. (From Bastea and Duxbury, 1998)
scattering and NMR) should be influenced by both the "staggered paramagnetic" response and the spontaneous staggered magnetization. The existence of the additional order parameter mpferro in the case of the +h RFIM, and rnpstaggered in the case of the DAFF may complicate the analysis of the phase transition in three dimensions, particularly because it has been shown that these degenerate clusters are power-law distributed close to the phase transition (Bastea and Duxbury, 1999).
7.4 7.4.1
Ising spin glasses and Euclidean matching Introduction and overview
The spin-glass problem has been a central problem in theoretical physics since the introduction of the Edwards-Anderson model (Edwards and Anderson, 1975).
222
M. J, Alava et aL
The Edwards-Anderson model is "]-[ -- -- Z Jij~ (i,j)
(7.4.1)
where (i, j) runs over nearest-neighbor pairs on a d-dimensional lattice, and ai are Ising spins. The exchange couplings Jij a r e independent random variables. In particular the + J model is defined by P ( J i j ) = p6(Jij - J) + (1 - p)3(Jij + J)
(7.4.2)
where p is the density of ferromagnetic bonds. Continuum distributions of exchange constants, for example Gaussian or uniform distributions, are also of interest. This model captures the essential features of disorder and frustration (Toulouse, 1977; Vannimenus and Toulouse, 1977; Mrzard et al., 1987; Mrzard and Monasson, 1994), which are believed to promote spin-glass behavior. Frustration is illustrated in Figure 7.17 for a plaquette on a square lattice. If the number of antiferromagnet interactions in a square loop is odd, no spin configuration can simultaneously satisfy all of the exchange interactions. Plaquettes with an odd number of negative interactions are therefore frustrated. A similar conclusion is valid for any closed loop. It is easy to see that a frustrated loop always encloses an odd number of frustrated plaquettes, i.e. there is a "topological charge" associated with isolated (not paired) frustrated plaquettes, the effects of which can be measured by performing a "loop integral" in the same way that electric charge (but not pairs of opposite sign) can be detected by measuring the electric field on a surface enclosing it. There are several similarities (Kirkpatrick, 1977; Fradkin et al., 1978) between short-range spin-glass systems and gauge theories, which justifies making these analogies. The solution of the infinite-range version of the Edwards-Anderson model (Kirkpatrick and Sherrington, 1978; Parisi, 1980) has provided a compelling vision of the nature of the spin-glass ordered state. In this model there are an infinite set of ultrametrically related ground states. It has been argued that this structure is destroyed in finite dimensions and instead there are a finite number of degenerate ground states (in spin glasses with continuous distributions of exchange constants) and that there are power-law distributions of droplet excitations which dominate the low temperature behavior of spin glasses (Fisher and Huse, 1986). Despite an enormous amount of computational work (mostly Monte Carlo) and analytical analyses (Binder and Young, 1986; Young, 1998), the spin-glass ordered phase is not well understood. The two-dimensional spin-glass ground-state problem was the first magnetism problem to be mapped to a polynomial optimization (matching) problem (Bieche et al., 1980), however, it was soon realized that three-dimensional spin glasses
2
4
,,,,,
Exact combinatorial algorithms
-o3
4
223
+
o3
+
,.I,.I1
o
+ 2
a)
1
2
b)
Fig. 7.17 a) If the number of negative interactions around a plaquette is even, it is always possible to satisfy all four interactions, b) If the number of negative couplings is odd, this is not possible and the plaquette is said to be frustrated.
and two-dimensional spin glasses in a field are in the N P-complete class (Barahona, 1982). In Section 7.4.2 we discuss the mapping of spin-glass problems to matching (planar graphs) and to maximum cut (general case). In Section 7.4.3 we review calculations of the exact ground states of spin glasses in two dimensions. In Section 7.4.4, we describe the Eulcidean matching problem, which is a toy model for spin-glass behavior, and which is solvable in polynomial time using matching algorithms.
7. 4.2
Mapping to optimization problems
(i) Mapping to a matching problem We first show how a nearest-neighbour spin-glass problem on a square lattice, with free boundaries, is mapped to a matching problem on a general graph (Bieche et al., 1980; Barahona, 1982; Bendisch et al., 1994; Palmer and Adler, 1999). Associate with each unsatisfied bond an "energy string" joining the centers of neighboring plaquettes sharing this bond, and assign to each energy string a "length" equal to I Jij I. Clearly the energy of the system equals the total length of these energy lines, up to a constant. Let us for example put all spins pointing upwards, so that each negative bond will be unsatisfied and thus cut by an energy string (Fig. 7.18a). Unfrustrated plaquettes have by definition an even number of strings crossing their boundary, therefore energy lines always enter and leave unfrustrated plaquettes. Frustrated plaquettes on the other hand have an odd number of strings, and therefore one string must begin or end in each frustrated plaquette. These observations hold for any spin configuration. If boundary conditions are open or fixed, some of the energy strings can end at the boundary, whereas for periodic boundary conditions frustrated plaquettes occur in pairs and are paired by energy lines.
224
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b)
c)
Fig. 7.18 Thick lines represent negative couplings. Energy strings (dotted) are drawn perpendicular to each unsatisfied coupling. Frustrated plaquettes (odd number of negative couplings) are marked by a dot. a) All spins up configuration, b) A ground state, c) Another ground state. Clearly, finding a ground state is equivalent to finding a minimum length perfect matching (Section 4) in the graph of frustrated plaquettes (Bieche et al., 1980; Barahona, 1982; Barahona et al., 1982). If IJij] = J, i.e. all interactions have the same strength, Fig. 7.18b shows one possible ground state. An equivalent ground state is obtained by flipping all spins inside the gray area in Fig. 7.18c, since the numbers of satisfied and unsatisfied bonds along its contour are equal. Degenerate ground states are related to each other by the flipping of irregularly shaped clusters, which have an equal number of satisfied and unsatisfied bonds on their boundary. If the amplitudes of the interactions are also random, this large degeneracy of the ground state will, in general, be lost, but it is easy to see that there will be a large number of low-lying excited states, i.e. spin configurations which differ from the ground state in the flipping of a cluster such that the "length" of unsatisfied bonds on its boundary almost cancels the length of satisfied ones. Similar ideas apply to three-dimensional systems (Fradkin et al., 1978), although the problem is not a matching problem. In this case, the unit-length segments through the centers of frustrated plaquettes always form closed rings. One isolated negative bond produces four frustrated plaquettes. Fig. 7.19 shows some simple examples. If one associates a unit square perpendicular to each unsatisfied bond, it is easy to see that an open surface is formed whose boundary is the ring of segments through the centers of frustrated plaquettes. Thus minimizing the energy is equivalent to minimizing the extension of this spanning surface, very much like in the foam problem. Barahona (1982) has shown that finding a threedimensional ground-state is A/P-hard, so it is unlikely that a polynomial-time algorithm will ever be found for this problem.
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Fig. 7.19 Energy surfaces in a frustrated three-dimensional Ising system. Finding a minimum energy configuration is equivalent to finding a minimum spanning surface with fixed boundaries. Thick black lines represent negative interactions. Frustrated plaquettes are indicated with a dot, and it is always possible to draw a closed ring (black line) through centers of frustrated plaquettes. Energy surfaces (gray) are perpendicular to unsatisfied bonds, and span these rings, which are not planar in the general case.
(ii) Mapping to a cut problem In analogy to (7.2.7) for the interface in random-bond ferromagnets and to (7.3.3) for the RFIM, the problem of finding the ground state of the spin-glass Hamiltonian H defined in (7.4.1) is again equivalent to finding a minimal cut (S, S), see (7.2.6), in a network m
min~_ H'(o') -- min(s,g )
Z
Jij - min(s,g)Ecut,
(7.4.3)
(i,j)E(S,S)
where H: -- ( H + C ) / 2 with C -- Z(ij) Jij a constant. However, now the capacities U ij --- Jij of the underlying network are no longer non-negative, so the minimal cut we seek cannot be identified with a minimum-cut/maximum flow problem. Note that for the DAFF (7.3.5) it was possible to gauge away the sign of antiferromagnetic couplings Jij. This is not possible when positive and negative couplings are mixed randomly. The problem of minimizing Ecut is the same as the problem of maximizing - E c u t , which is the well known maximum-cut problem in combinatorial optimization. This formulation is obviously more flexible than the matching formulation, but the general maximum-cut problem is not solvable in polynomial time. If the field is zero and the graph is planar, minimum-cut can be mapped into a "Chinese postman" problem (Barahona et al., 1982), which can be solved in polynomial time by a method due to Edmonds. For the case of periodic boundary conditions without field, Barahona (unpublished) has found a polynomial-time algorithm although it is reported (De Simone et al., 1996a) to be too slow to be useful.
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But even in the non-polynomial cases, it is still possible to find exact ground states in reasonable times (Barahona, 1982; Kawashima and Suzuki, 1992; De Simone et al., 1996a, b; Klotz and Kobe, 1994), and in fact it seems to be a nonpolynomial branch-and-cut algorithm (De Simone et al., 1996a, b) which gives efficient running times for the spin-glass problem, even in two dimensions. In what follows we would like to sketch the idea of an efficient but nonpolynomial algorithm (De Simone et al., 1996a) for the maximum-cut problem formulated in (7.4.3). Let us consider the vector space R A. For each cut [S, S] define X (s'~) E R A, the incidence vector of the cut, by X~es's) -
1 for each
edge e - (i, j) E (S, S) and X~es's) - 0 otherwise. Thus there is a one-to-one correspondence between cuts in G and their {0, 1}-incidence vectors in R A. The cut-polytope Pc(G) of G is the convex hull of all incidence vectors of cuts in G" Pc(G) - conv{x (S'S) E R A IS c_ A}. Then the max-cut problem can be written as a linear program max {u~x Ix E Pc(G)} (7.4.4) since the vertices of Pc(G) are cuts of G and vice versa. Linear programs (see Section 5.1) usually consist of a linear cost function u r x that has to be maximized(or minimized) under the constraint of various inequalities defining a polytope in R n (i.e. the convex hull of finite subsets of R n) and can be solved for example by the simplex method, which proceeds from comer to comer of that polytope to find the maximum (see (Lawler, 1976" Chv~ital, 1983; Derigs, 1988)). The crucial problem in the present case is that it is A/'79-hard to write down all inequalities that represent the cut polytope Pc(G). However, it turns out that partial systems are also useful, and this is the essential idea for an efficient algorithm to solve the general spin-glass problem as well as the traveling-salesman problem and other "mixed-integer" problems (i.e. linear programs where some of the variables x are only allowed to take on some integer values, like 0 and 1 as in our case) (Lawler et al., 1985; Thienel, 1995). One chooses a system of linear inequalities L whose solution set P (L) contains Pc(G) and for which Pc(G) - convex hull {x E P(L)Ix integer}. In the present case these are 0 < x < 1, which is trivial, and the so-called cycle inequalities, which are based on the observation that all cycles in G have to intersect a cut an even number of times (have a look at the cut in Fig. 7.18 and choose as cycles for instance the paths around elementary plaquettes). The most remarkable feature of this set L of inequalities is that the separation problem for them can be solved in polynomial time. The separation problem for a set of inequalities L consists in either proving that a vector x satisfies all inequlaities of this class or to find an inequality that is violated by x. A linear program can be solved in polynomial time if and only if the separation problem is solvable in polynomial time (Gr/3tschel et al., 1988). For those that can be solved in polynomial time" the cutting-plane algorithm, starting from some small initial set of inequalities,
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iteratively generates new inequalities until the optimal solution for the full subset of inequalities is feasible. Note that one does not solve this linear program by the simplex method since the cycle inequalities are still too numerous for this to work efficiently. Due to insufficient knowledge of the inequalities that are necessary to describe P c ( G ) completely, one may end up with a non-integral solution x*. In this case one branches on some fractional variable Xe (i.e. a variable with x e ~_ {0, 1}), creating two subproblems: one having Xe = 0 and the other having Xe = 1. Then one applies the cutting plane algorithm to both subproblems, hence the name branch-and-cut. Note that in principle this algorithm works for any dimension, boundary conditions, and in a finite field. However, there are realizations of it that run fast (e.g. in two dimensions) and others that run slow (e.g. in three dimensions) and there is ongoing research which attempts to improve on the latter. A detailed description of this rather complex algorithm can be found in Thienel (1995).
7. 4.3
Ground-state calculations in two dimensions
In their pioneering work, B ieche et al. (1980) studied the ground-state behavior of the ( + J ) spin glass as a function of the fraction x of anti-ferromagnetic bonds. From simulations of systems of 22 x 22 spins they deduced that ferromagnetic was destroyed at x* - 0.145. This zero-temperature transition is detected by the appearance of fracture lines which span the system, i.e. paths along which the number of satisfied and dissatisfied bonds is equal, and which can thus be inverted without any cost in energy. The authors also looked at the fraction of spins in connected components, defined as sets of plaquettes which are matched together in any ground-state. Later investigations by Barahona et al. (1982) located the loss of ferromagnetism at a somewhat lower density of antiferromagnetic bonds, x* ~ 0.10, suggesting that in the regime 0.10 _< x _< 0.15 a random antiphase state exists which has zero magnetization but long-range order. This state is, according to the authors, characterized by the existence of magnetic walls (which are different from fracture lines) across which the magnetization changes sign. Thus the system is composed in this regime of "chunks" of opposite magnetization, so < M >-- 0 although the spin-spin correlation does not go to zero with distance. At x = 0.15 a second transition occurs, this time due to the proliferation of fracture lines, and rigidity (long-range order) is lost since the system is now broken into finite pieces which can be flipped without energy cost. Their conclusions were supported by later work using zero-temperature transfer-matrix methods (Ozeki, 1990). Freund and Grassberger (1989), using an approximate algorithm to find low-energy states on systems up to size 210 • 210, located the ferromagnetic transition at x* = 0.105 but found no evidence of the random antiphase state. The
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largest two-dimensional systems studied to date using exact matching algorithms appear to be 1800 • 1800 square arrays (Palmer and Adler, 1999). Bendisch et al. (1994) examined the ground state magnetization as a function of the density p of negative bonds on square lattices, and concluded that 0.096 < x* < 0.108, but their finite-size scaling analysis is not the best one could think of. They did not analyze the morphology of the states found, so no conclusion can be reached regarding the existence of the random antiphase state. More recently Kawashima and Rieger (1997), compared previous analyses of the ground state of the two-dimensional ( i J) spin glass, in addition to performing new simulations. They summarized the results in this area in the phase diagram given in Fig. 7.20. Kawashima and Rieger have found that the "spinglass phase" is absent and that there is only one value of Pc. They thus argue for a direct transition from the ferromagnetic state to a paramagnetic state, for both site- and bond-random spin-glass models. Their analysis is based on the energy difference A E = E p - E a , where E p is the ground-state energy with periodic boundary conditions and Ea is the ground-state energy with antiperiodic boundary conditions. The scaling behavior (Bray and Moore, 1985a), [AE]av ~ L p ,
[ A E 2 ] a v ~ L 20
(7.4.5)
was assumed. In a ferromagnetic state p = 1 and 0 = 2, while in a paramagnetic state p < 0 and 0 < 0. However, in an ordered spin-glass state we have p < 0, and 0 > 0. Although the conclusion of this analysis was the absence of a spinglass phase, the exponent they found, 0 -- -0.056(6) for p < Pc, is small. Although, in our view, the numerical evidence that there is no finite-temperature spin-glass transition in the two-dimensional Edwards-Anderson model with a binary bond distribution is compelling, one should note that a different view has been advocated (Shirakura and Matsubara, 1995). A defect-energy calculation similar to the one described above has been presented (Matsubara et al., 1998) to support this view. It was shown that the probability distribution of IAEI does not shrink to a delta-function centered at A E for L --+ ~ , instead it maintains a finite width. However, even if l i m L _ ~ [AEI = A E ~ > 0 a f i n i t e value of this limit indicates that the spin-glass state will be unstable with respect to thermal fluctuations since arbitrarily large clusters will be flipped via activated processes with probability e x p ( - A E ~ / T ) at temperature T. The correct conclusion is then that there is no finite-T spin-glass transition in the two-dimensional EdwardsAnderson model with binary couplings. The two-dimensional Ising spin glass with a binary ( + J ) bond distribution is in a different universality class than the model with a continuous bond distribution. The degeneracies, which are typical for a discrete bond distribution, are absent for the continuous case for which the ground state is unique (up to a global spin flip). Even in the continuous case, the ground state is found using a
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P T
...................
SG(?)/
0 5
..,(2) --(1) Pc Pc
"
1
P Fig. 7.20 Phase diagram of a two-dimensional (+J) Ising model, with fraction p -- 1 - x of ferromagnetic bonds. (From Kawashima and Rieger, 1997)
minimal weighted matching algorithm (with the modification that now not only the length of a path between two matched plaquettes counts for the weight, but also the strength of the bonds laying on this path). The latest estimate for the stiffness exponent of the two-dimensional Isingspin-glass model with a uniform bond distribution between 0 and 1 obtained via exact ground-state calculations (Rieger et al., 1996) is [AE2]av cx L 20
with
0 = -0.281 + 0.002,
(7.4.6)
which implies that in the infinite system arbitrarily large clusters can be flipped with vanishingly small excitation energy. Therefore the spin-glass order is unstable with respect to thermal fluctuations and one does not have a spin-glass transition at finite temperature. Nevertheless, the spin-glass correlation length ~ (defining the length scale over which spatial correlations like [(Si Si+r)Z]av decay) will diverge at zero temperature as ~ ~ T -1/u, where v is the thermal exponent. A scaling theory (for a zero-temperature fixed-point scenario as is given here) predicts that v = 1/101 which, using the most accurate Monte-Carlo (Liang, 1992) and transfermatrix (Kawashima et al., 1992) calculations gives v = 2.0 4- 0.2, is inconsistent with (7.4.6). This is certainly an important unsolved puzzle, which might be rooted in some conceptional problems concerning the use of periodic/antiperiodic boundary conditions to calculate large-scale low-energy excitations in a spin glass (these problems have first been discussed in the context of the XY spin glass
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(Kosterlitz and Simkin, 1997) and the gauge glass (Kosterlitz and Akino, 1998)). Further work in this direction will be rewarding. Next we would like to focus our attention on the concept of chaos in spin glasses (Bray and Moore, 1987). This notion implies an extreme sensitivity of the spin glass state with respect to small parameter changes like temperature or field variations. There is a length scale ~.-the so-called overlap length-beyond which the spin configurations within the same sample become completely decorrelated if compared for instance at two different temperatures
CAT --[((riCri+r)T(Cricri+r)T+AT]av ~ e x p ( - r / ~ . ( A T ) ) .
(7.4.7)
This should also hold for the ground states if one slightly varies the interaction strengths Jij in a random manner with amplitude 6. Let {or} be the ground state of a sample with couplings Jij and let {ort} be the ground state of a sample with couplings Jij 4- SKi j, where the Kij a r e random (with zero mean and variance one) and 3 is a small amplitude. Now define the overlap correlation function as !
!
C~(r) = [o'io'i_Fr (Tio'i+r]av ~ ?(r31/ff),
(7.4.8)
where the last relation indicates the scaling behavior we would expect (the overlap length being ~. ,~ 6-1/C) and ~" is the chaos exponent. Rieger et al. (1996) confirmed this scaling prediction with 1/~" -- 1.2 4- 0.1 by exact ground-state calculations of the two-dimensional Ising spin glass with a uniform coupling distribution, and the corresponding scaling plot for C~(r) is shown in Fig. 7.21. The two-dimensional Ising spin glass in a homogeneous field (i.e. with an additional term -h Z i Si in the Hamiltonian) is already a much harder problem that cannot be solved in polynomial time. Nevertheless efficient methods exist for this case (see Section 7.4.2) so that large systems can still be solved exactly in a reasonable computational time (Barahona, 1994; De Simone et al., 1996a, b; Rieger et al., 1996). A non-zero external field h induces a non-vanishing magnetization m = N -1 y-~N=I SO in a system with ground state {sO}. The relation between magnetization and field strength is highly non-trivial in general and motivates the introduction of a new exponent 3h characterizing this relation in the infinite system (L --+ oo) for small fields (h << J),
moo(h) ~ h 1/3h.
(7.4.9)
Rieger et al. (1996) found, using a branch-and-cut algorithm for the calculation of the ground states, that 6h -- 1.48 -4- 0.01. (7.4.10) According to the scaling theory already mentioned ~h is not an independent exponent but is related to the thermal exponent via 3h -- 1 4- 1/v, which is indeed
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,,,, ~+
0.8 o~
5 = 0.0025 5 = 0.005 5 =0.01 5 = 0.02 5 = 0.05 5=0.1
0.6 r
0.4
+ • 9 o
~_. *. %
9
o o
%
0.2
o
. . . . . . . .
0.001
i
. . . . . . . .
0.01 r
i
0.1 ~1.05
1
Fig. 7.21 Scaling plot of the overlap correlation function C~(r) versus r/L* with L* = 3-1/~'. The best data collapse (for data confined to r < L/4) is obtained for 1/r = 1.05. The system size is L = 50 and the data are averaged over 400 samples. These were obtained by creating > 80 reference instances and creating five random perturbations of strength 6 for each. (From Rieger et al., 1996).
fulfilled. Note that due to the apparent violation of the scaling relation v = 1/i0i the otherwise equivalent relation 3h -- 1 -- 0 is also violated. Again we stress that in our view this is due to some misconception in the way in which 0 is determined in spin glasses. Besides chaos and overlap length there is another very important concept in spin glasses, which is the notion of " P ( q ) " , i.e. the probability that two equi-
Y]i (S~S/fl)
of value librium states ct and fl of a spin glass have an overlap N - ] q. Obviously in simple systems like a ferromagnet P (q) is simply a single delta function at q -- 0 in the paramagnetic phase and at q = + m 2 (m being the spontaneous magnetization) in the ferromagnetic phase. For a "real" spin glass, P (q) should be a non-trivial function, but care has to be taken with the boundary conditions. Here we do not want to enter into a detailed discussion of this issue (for details see Young (1998)) but we do want to mention what can be done using ground-state computations. The problem is that in the system of interest the ground state is unique and a naive application of the above concept would always give q -- 4-1. The method that has been devised to study the overlap function is to calculate P(q) for spins in a smaller block of spins in a larger system and then increase the system size, keeping the block size fixed. This procedure, which can be used straightforwardly using any ground-state algorithm,
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L.
r
L
d
-1 | ! I
!
b
" L'
Fig. 7.22 The ground state for L ~ -- 120 and L = 80 subsystems of a single infinite sample are compared. The solid lines inside the L = 80 region (dashed box) indicate the difference (the "relative domain walls") in the two ground states in their common area. The solid box indicates a window of size w = 40. In this example, expansion of the boundary conditions changes the ground state inside the window by the introduction of a domain wall that crosses the window. As can be seen, domain walls near the edge of the L = 80 subsystem are induced by the boundary condition change; most do not propagate into the middle of the region. (From Palassini and Young, 1999) gives important information about the ground-state structure (Middleton, 1999; Palassini and Young, 1999), which turns out to be trivial for the two-dimensional Ising spin glass. As indicated in Fig. 7.22 as the system size L increases, it is less likely that a domain wall of size w occurs in the middle of the sample. Equivalently one can say that the above mentioned overlap P (q = 0) vanishes in the thermodynamic limit, and it does so in accordance with a simple scaling theory (Palassini and Young, 1999). A consensus has been reached that in three dimensions there is an Ising spin-glass phase transition at a finite temperature, which means that the lower critical dimension for short-range Ising spin glasses is below three. Unfortunately the only reliable evidence comes from large-scale Monte-Carlo simulations (Kawashima and Young, 1995; Hukushima and Nemoto, 1996; Marinari et al., 1997), as the domain-wall renormalization group (DWRG) based on exact ground-state calculations are confined to very small system sizes (L _< 4). Transfer-matrix calculations (Bray and Moore, 1985a) and more recent optimization calculations (Hartmann, 1998b) using heuristic methods (Hartmann, 1997, 1998a, 1999a, b, c); nevertheless, indicate that the stiffness exponent is positive, 0 = 0.19 4- 0.02, which is consistent with the existence of a finite-temperature spin-glass transition as found using Monte-Carlo simulations.
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~s~
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S
9
I I
-oI
Fig. 7.23 Example of a dimer state in the Euclidean matching problem. Solid lines are the ground-state matching, while dotted lines indicate "loop" excitations. (From Houdayer and Martin, 1998)
7. 4.4
Glass-like excitations: Euclidean matching
Euclidean matching consists of pairing a set of N points, such that the sum of the distances between matched pairs is minimal. This has been proposed as a simple model which illustrates a range of spin-glass-like behavior (M6zard and Parisi, 1988). This problem is a minimum-weight perfect matching on a general graph. The solid lines in Fig. 7.23 indicate such a matching on a set of N -- 12 points. In general this problem is solvable in polynomial time using the Edmonds' algorithm (see Section 4.3.2). In this case, the graph is complete because, in general, one should allow any site to pair with any other site. In practice, one can limit the possibilities to close neighbours. In order to study the excitations from the minimal-cost matching, Houdayer and Martin (1998) suggest taking a pair in the minimal-cost matching and then setting its "distance" to infinity. This excludes that pair from the minimum-cost matching. A re-calculation with this new cost in place leads to a new matching. This new matching can be overlaid on the old matching, and what results is a "loop excitation" like those shown in Fig. 7.23. It is argued that these loop excitations are analogous to the excitations in spin-glasses. It was found that although at short length scales there is a scaling like that assumed in the "droplet" phenomenology of spin glasses, there are also largelength low-energy excitations as in the mean-field theory. The latter "large-loop" excitations obey the scaling law Prob([1 - x) ~ G ( x ) / ~ / - N ,
(7.4.11)
where the scaling variable i1 - l l/~/-N. The physical picture that can be extracted from this model is that there are well-separated almost-degenerate states as in the
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infinite-range spin-glass model. However, each of these states has "droplet-like" excitations, which should dominate the physics on short length and time scales. The Euclidean-matching problem thus has a behavior intermediate between the "droplet picture" and the "hierarchical picture" of the spin-glass ordered phase.
8
8.1
Line, vortex and elastic glasses
Introduction and overview
A large number of models for various "glassy" states have arisen in recent years. Several of these models were motivated by the discovery of high Tc materials and the wide variety of vortex states which are caused by their combination of small coherence lengths, large penetration depths and complex microstructures (B latter et al., 1994). In this chapter we discuss three broad classes of glassy system which are of interest both in the context of disordered flux states and because they arise in other contexts. In Section 8.2, we derive them in the context of disordered flux states. The following three types of models have been mapped to polynomial optimization problems and are discussed in detail in Sections 8.3-8.5. (i) Arrays of DPRMs with f-function interactions. It is quite remarkable that the ground state of hundreds of self-avoiding polymers which have f-function interactions can be found in a couple of minutes (Rieger, 1998b). This miracle is performed by mapping to the minimum-cost-flow algorithm. In fact the mapping is quite trivial (see Section 8.3.1). This problem can be used to model flux lines in the dilute limit (note that it ignores long-range interactions so is rather poor for real flux lines). It is more appropriate for entangled polymer arrays, but it has not yet been applied to that problem. This mapping has just been discovered, so there are only a few results available. However, an enormous number of different problems can be tackled with this mapping. (ii) Vortex glass in the strong-screening limit. The vortex-glass Hamiltonian is derived in Section 8.2.4. It is a model for Josephson-coupled arrays (Ebner and Stroud, 1985) and also for granular superconductors (John and Lubensky, 1986). Unfortunately, we do not know of a mapping of the full vortex-glass Hamiltonian to a polynomial optimization method. However, in the limit of strong screening, it does map to the minimum-cost-flow problem (Section 8.4.1). There it is found that the ground state is disordered (based on domain-wall scaling calculations) (Kisker and Rieger, 1998). (iii) Periodic elastic media. This area has been very active recently, and is turning out to be very rich. It is related to weakly disordered flux lattices, as demonstrated in Section 8.2.5. However, similar models apply to pinned chargedensity waves (Griiner, 1988) and random surfaces (Shapir, 1991). Only a limiting case of these problems has been mapped to polynomial optimization prob-
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lems, namely, the random-surface model (see Section 8.5.1). This, in turn, has several limiting cases which map to different optimization problems. The most general random-surface problem maps to minimum-cost flow (see Section 8.5.2) (Blasum et al., 1996; Rieger and Blasum, 1997). The random-surface model with energy Y~(i,j)Ihi -- hjl maps to maximum flow (Zeng et al., 1996), and corresponds to the random-bond interface problem treated in Section 7.2, but now with a periodic potential in the height direction. Finally the even-morespecial case of a restricted solid-on-solid model in the {111 } orientation of a cubic lattice, maps to a maximum-weight-matching problem (Zeng et al., 1996) (see Section 8.5.4). All of the random-elastic-media simulations indicate that the ground state (the "Bragg glass") of (2 + 1)-dimensional elastic media is superrough, so that w ~ In L (Cardy and Ostlund, 1982; Toner and Di Vincenzo, 1990; Tsai and Shapir, 1992; Blasum et al., 1996; Zeng et al., 1996) (Section 8.5.5), instead of w ~ (InL)1/2 as occurs at high temperatures. After working hard to reach a consensus that this is correct, it has been recently demonstrated that when dislocations are included in the model, they proliferate and destroy the Braggglass state (Zeng et al., 1999). The nature of the ground state in the presence of these dislocations remains unresolved (see Section 8.5.6).
8.2 8.2.1
Disordered flux arrays Introduction
In this section we show how the limiting models discussed in Sections 8.3-8.5 arise in the context of disordered vortex matter. In contrast to conventional (type-I) superconductors which have a direct phase transition from the superconducting Meissner phase to the normal phase, type-II superconductors exhibit the so-called mixed (or Schubnikov) phase. Whereas the Meissner phase is characterized by a complete expulsion of magnetic flux from the superconductor at low magnetic fields H < Hc~ (T), at large fields the magnetic field penetrates the superconductor in the form of vortices or flux-lines (FL) each carrying a magnetic flux quantum ~0 = h c / 2 e ~ 10-7Gcm 2. Thus, within the mixed phase, the number of flux-lines per unit area is simply given by n = B~ ~ o and, in a pure superconductor, they arrange in a triangular lattice (the Abrikosov flux-line lattice). When the density of flux-lines increases (with increasing applied field H) the vortex cores begin to overlap and superconductivity is destroyed above the upper critical field Hc2. Since the high-Tc superconductors are strongly type-II the theoretical and experimental interest in the properties of the vortex lattice has increased significantly in the last decade, see Blatter et al. (1994) for a review. The technological importance of these materials is obvious since they provide the possibility of dissipation-free currents at relatively high temperatures. However, for large ap-
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plied fields as well as for large currents (creating large self-fields) these materials operate in the mixed phase, and a current can cause motion of the vortices via the Lorentz force, which leads to dissipation. In order to produce a technologically useful high-Tc superconductor, disorder in the form of defects that pin the fluxlines and hinder them from moving, has to be introduced. Thus, there is immense interest in the analysis of pinning effects on flux lines and flux lattices.
8.2.2
Ginzburg-Landau theory
The starting point of a phenomenological description of superconductors is the Ginzburg-Landau theory, starting from an expansion of the free-energy functional in terms of the complex order parameter, the macroscopic wave function of the condensate of Cooper pairs qJ (x) = (c t (x) (c+ (x)) coupled to the electromagnetic vector potential A:
f
d3x
h ~mm
2 e A ) qJ
V - ih---~
b B2 -+-alqjl2 -+- 2 Iqjl4 -~ 8zr
-.} 4rr
(8.2.1) where B - V x A is the microscopic magnetic field, H is the applied magnetic field, a and b are the Ginzburg-Landau (GL) expansion parameters, and a -- a ( 0 ) ( 1 - T/Tc) changes sign at the transition temperature Tc whereas b stays constant. The homogeneous solutions (minima) of (8.2.1) are given by qJ0 -- 0 (normal state) and I%1 - lal/b (superconducting state). The inhomogeneous (spatially varying) solutions, which are obtained via variation with respect to qJ* and A are characterized by two important length scales: the coherence length being the correlation length of the Cooper pairs and setting the scale of spatial variation in the order parameter qJ, and the magnetic penetration depth )~, determining the scale of variations of the magnetic field A. The ratio tc - ~./~ characterizes the sensitivity of the superconductor to fluctuations allowing for a local penetration of the magnetic field and reduction of the order parameter: For x > 1/v/2 the superconductor becomes type-II, and magnetic flux penetrates above a lower critical field Hcl (T), determined by )~, in the form of flux-lines. Within the mixed phase, the number of flux-lines per area is simply given by n - B / ~ 0 , and they arrange in a triangular lattice (the Abrikosov flux-line lattice) with a lattice constant a~ - (2/x/~)I/2(OPo/B) 1/2. When the density of flux-lines increases (with increasing applied field H) the vortex cores begin to overlap and superconductivity is destroyed above the upper critical field Hc2, which is determined by ~. The ratio of the upper and lower critical field depend on the ratio of the two basic length scales )~ and ~" Oc2 lOci 0(. (~./~)2. High-Tc materials have extremely large penetration depths )~ and short coherence
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lengths ~, with x ,~ 100. Therefore it is evident that nearly the whole region of their phase diagram that is relevant for applications is dominated by the presence of flux-lines.
8.2.3
The low-field limit
In the ordered (superconducting) phase the GL-parameter a is large and negative and the amplitude of the pair wavefunction qJ(x) approximately uniform IqJ(x)l = ~0, only the phase varies in space qJ(x) = ~oexp(iO(x)) with 0 [0, 2rr ]. Up to a constant the qJ-independent part of f c L becomes 2e
Fqj m, f d3X-~m 1 VO
m
2 (8.2.2)
~ A
tic
Configurations of the phase field in which 0 winds 27r around a closed loop enclosing a singularity are called vortices. In two dimensions these singularities are point-like objects, in three dimensions they are lines. At low temperatures minimization of (8.2.2) requires V0 = A, which integrated around the loop implies 2re = 2 e ~ / h c , where 9 = f d f . B is the magnetic flux enclosed by the loop. Thus 9 -- ~0 and each vortex line carries a quantum of magnetic flux and the density of vortices is determined by the field via n - B / ~ 0 . At low fields the first approximation is to treat the vortices as independent, which leads naturally to a directed polymer description. A vortex can be considered as an elastic string that has a line tension cr that is (in the long wave length limit) equal to its line energy e0 ln~./~, with e0 = (~0/47r~.) 2 the energy scale determining the self-energy of the vortex. A gradient expansion then yields the random DPRM elastic energy
Fel --
dz a
1+
~z
-1
~
dz 2
-~z
'
(8.2.3)
where the d-component vector u(z) denotes the displacement field of the vortex which we suppose to be aligned with the z-axis. A vortex at rest with respect to the laboratory frame of reference which is exposed to a current j experiences a Lorentz force fL -- ( ~ o / c ) j x n, where n is a unit vector pointing in the direction of the vortex. In an inhomogeneous material the vortex can be pinned by macroscopic or atomic defects, the latter being typically the oxygen vacancies in the high-Tc superconductors (doped cuprates). The DPRM model has been a standard starting point in the analysis of the effect of columnar pins, and their competition with point pins, on the morphology of flux lines in random media (Blatter et al., 1994). With optimization methods it is possible to treat many DPRMs with a f-function interaction (Section 8.3).
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8.2.4
Gauge (vortex)-glass limit
A new aspect in disordered superconductors is the appearance of the so-called vortex-glass state (Fisher, 1989; Fisher et al., 1991), in which the off-diagonal long-range order of the pair condensate ~ (x) has a phase that is random in space but frozen in time, reminiscent of spin-glass order (see Section 7.4). (Note that the Abrikosov flux lattice of a pure sample, is destroyed in less than d - 4 (Larkin, 1970; Larkin and Ovchinikov, 1979) beyond the so-called Larkin length lLarkin, where the energy gained by the pinning centers overcomes the elastic energy). There is experimental evidence for a transition to this vortex-glass state (Koch et al., 1989; Gammel et al., 1991; Olsson et al., 1991). The existence of this state is important for superconductivity, as it implies that the resistance can vanish without the presence of an ordered Abrikosov lattice (Fisher et al., 1991). Deep inside the superconducting phase, the superconducting order parameter (x) attains a non-vanishing modulus ~0 and the dominant low-energy excitations are phase fluctuations i.e. qJ(x) = ~0exp[i0(x)]. A careful RG-treatment shows that density fluctuations (i.e. spatial variations in the modulus of qJ (x)) are also irrelevant at the critical point. Thus, besides the quadratic part in magnetic field, the relevant part of (8.2.1) is then (dropping all constants for simplicity)
f d3x(V
-
-
iA)ei~
>Z
Iei[O(x)-O(x+e~)-A(x'x+e~)]-- 1 12.
(8.2.4)
X,O/
On the right-hand side we have put the discrete version, the coordinates x now denoting discrete lattice points, ot -- 1, 2, 3 the space directions and A(x, x + e~) --
f x+e~ ds- A
(8.2.5)
dX
are the phase shifts induced by the vector potential A. The latter can be split into an external (quenched part) A ext, induced by the applied magnetic field, and a fluctuating part a, that is responsible for screening effects. Changing the notation from coordinates x to lattice indices i for a simple cubic lattice one thus obtains from (8.2.4) the well-known XY-Hamiltonian nxy
-- - J
L~ (i,j)
cos(0/ -
Oj - A i j - ~ . - l a i j )
1
+ ~L ~ [ V x a] 2
(8.2.6)
[]
where J is the interaction strength and )~ the screening length or penetration depth and the sum is over nearest-neighbour sites. The last term describes the magnetic energy (which is f d3xB 2 in (8.2.1) with B -- V x A the microscopic magnetic field), and is the sum over all plaquettes of the lattice, where the curl is given as the directed sum of the aij around one plaquette.
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In the limit of infinite screening length (X --~ oo) the fluctuating vector potential is irrelevant, which is a situation that is typically assumed in the context of Josephson-junction arrays and granular superconductors, for which the Hamiltonian (8.2.6) with ~. --+ oo has originally been proposed (Ebner and Stroud, 1985; John and Lubensky, 1986) HX~.---~ Y oo
-- -- Z Jij (i,j)
cos(0/
-
Oj - A i j ) ,
(8.2.7)
Oi n o w representing the superconducting phase in a particular grain with index i of the two- or three-dimensional array and Jij (x 1/Rij describes the strength of the Josephson coupling between grain i and j, with Rij being the corresponding resistance. In the non-homogeneous situation one has two sources of randomness, 1) the phase shifts Aij c a n be random due to the random location and distance of the superconducting grains, and 2) the couplings Jij a r e random due to the random resistance. Two different models arise: (i) The XY spin glass, in which Jij is random and Aij = 0 or constant. The lower-critical dimension for the appearance of an XY spin-glass state is believed to be four (Banavar and Cieplak, 1982; Morris et al., 1986; Jain and Young, 1986). (ii) The gauge glass, in which Jij = J constant and Aij E [0, o'] is distributed randomly over a continuous range (0 < o- < 2Jr) measures the strength of the disorder, here we discuss only the case of strong disorder, i.e. o" -- 27r. It is believed to be in a different universality class from the XY spin glass, since it does not have the global reflection symmetry Oi --+ -Oi, and its lower critical dimension is believed to be three, leading to the presence of a vortex-glass state in superconductors in three dimensions (Fisher, 1989). In the strong-screening limit the vortex-glass state is destroyed as is demonstrated in Section 8.4.
8.2.5
Perturbing the flux lattice
When a homogeneous superconducting material is placed in a magnetic field of strength H > Hcj, a triangular vortex lattice, with lattice constant a = (2/~/~)1/2(~o/B)1/2 , is formed. In the presence of weak disorder, the flux lattice is distorted, which we describe by the d - 1-component displacement field ui(z). (In the continuum description we use R and u(R, z)). Note that when setting u(Ri, z) = u i (Z) one assumes that there are no dislocations present, otherwise such a relabeling is impossible to perform unambiguously. For weak disorder one expects u(Ri, z) to be slowly varying on the scale of the lattice and one can use a continuum elastic energy (Blatter et al., 1994; Giamarchi and Le Doussal, 1995), as a function of the continuous variable u(r), with r -- (r, z):
1 Hel -- -~ ~,
f.
ddq z (2n) d u ~ ( q ) ~ u ~ ( - q ) '
(8.2.8)
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M.J. Alava et al.
where or,/3 = 1. . . . . d - 1 label the lattice coordinates, BZ denotes the Brilloun zone and ~t~ is the elastic matrix. An appropriate (dispersion-free) approximation for the high-Tc superconductors that can be modeled by stacks of coupled planes and therefore described by layers of two-dimensional triangular lattices would then contain the three elastic moduli: the compression modulus cll, the shear modulus C66 and the tilt modulus C44: nel --- 1/2 f d2Rdz((Cll C 6 6 ) ( Z o t O~U~)2 -~- C66 Y~,fl(O~Ufl) 2 -F- C44 Z~(OzU~)2), where or, fl run over the transverse co-ordinates of the flux line. Here, for simplicity, we confine ourselves to a fully isotropic situation in which
c f d d r[vu (r) ]2 Hel -- -~
(8.2.9)
corresponding to O~,t~(q) - cq23a~ Next we include disorder via a Gaussian pinning potential e(u, z) acting on the flux lines. The total Hamiltonian then reads
H -- Hel +
f
d clr ~pin(r) p (r),
(8.2.10)
where the flux-line density at a given point r -- (R, z) is given by p(r) -- Z
6(R - Ri - u ( R i , z))
(8.2.11)
i
Some care has to be taken when going to the continuum description since the discrete translational symmetry u ---> u + Ri of the density has to be preserved in any approximation. Since the coordinates Ri only serve as an internal label of the flux-lines it is more convenient to use a label that is a function of the actual position of the flux-lines. One can introduce the slowly varying field via the implicit equation 4~(R, z) = R - u(4~(R, z), z) that has a unique solution in the absence of dislocations. Using this relabeling field q~(r) the density (8.2.11) can be rewritten (Giamarchi and Le Doussal, 1995) as p(r) -- P0 det[0~p/~] ~
e iKj(p(r),
(8.2.12)
J
where Kj are the vectors of the reciprocal lattice and P0 = 1/[R] 2 = B/Cbo is the flux-line density. When one retains only the smallest reciprocal lattice vector K1 and neglects the term VRU (since fluctuations in the density are slowly varying) one obtains the standard continuum representation for the disorder energy of the flux-line lattice
ndi s ,~
f
d3repin(r) 2 p ~ 1 7 6
u(r)])"
(8.2.13)
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For concreteness, let us consider the two-dimensional case. In this case the elastic energy given in (8.2.8) can be made isotropic by a rescaling of the coordinates and the full Hamiltonian reads,
f
H2d ~" -21
d2 r {K (Vu) 2 -+- W (r, u) }
(8.2.14)
with Gaussian distributed periodic disorder with zero mean and correlation function (W(r, u)W(r', u')) = 2g cos(K](u - u'))32(r - r'). Note that this formulation is indeed equivalent to (8.2.13). The elastic constant is proportional to ~/C11/C66and g is the strength of the disorder. In the literature the Hamiltonian (8.2.14) is often represented as the vortex-free two-dimensional XY model in a random symmetry-breaking field
'fd2
HXy -- -~
r {K(Vu) 2 + 2~/-~cos[u(r) - 0(r)]}
(8.2.15)
where the random phases 0(r) are uniformly distributed at each point and uncorrelated at different points: (0(r)0(ff)) cx 32(r - if). In the large g limit this model reduces to the random-surface model which is solvable by optimization methods. This is what we study in Section 8.5, where the three-dimensional version of this model and the effect of dislocations on it are also considered. Concluding this section we remark that at low temperatures the two-dimensional model (8.2.15) and the three-dimensional version of it are expected to be in a glassy phase with quasi-long-range order (the Bragg glass) (Giamarchi and Le Doussal, 1994, 1995). Many of the properties of this phase, including its stability or instability with respect to dislocations, can be investigated via ground state (T = 0) calculations.
8.3 Arrays of directed polymers 8.3.1
Mapping to minimum-cost flow
In Section 8.2.3 we pointed out that in the low-field limit of high-Tc superconductors a directed-polymer description is appropriate. In the literature usually only the properties of a single line, or at most two lines (Tang, 1994) are studied numerically, generally with the transfer-matrix method, whose computational demands increases exponentially with the number of lines. Here we demonstrate how the complete N-line problem (with local interactions) can be solved numerically in polynomial time using optimization methods. Here we consider N directed polymers. Actually the model also applies to undirected polymers as well, just as Dijkstra's method solves for minimum path
242
M . J . Alava e t al.
including the possibility of overhangs (see Section 6.3). The lattice version of this model is given by the Hamiltonian H(x) = Z
eij 9Xij,
(8.3.1)
(i,j) where
Z(i,j) is a sum over all bonds (i, j) joining sites i and j of a d-dimensional
lattice, e.g. a rectangular (L d-1 x 1) lattice, with periodic boundary conditions (b.c.) in d - 1 directions and free b.c. in one direction. The bond energies eij >>_0 are quenched random variables that indicate how much energy it costs to put a segment of polymer on a specific bond (i, j). The polymer configuration x (xij > 0 ) , is modeled by associating it with an integer flow. Each bond is allowed an integer flow which is either Xij - - 1 when arc (i, j ) is occupied by the polymer and Xij = 0 otherwise. This is imposed in a flow network by setting the arc capacities to be U ij = l. This restriction also ensures that there is a hard-core repulsion between polymers, on the bonds of the lattice. Note that two polymers can visit the same site, though this is improbable due to the bond repulsion. The connectivity of the polymers is ensured by the fact that flow is conserved, that is, on each site of the lattice the incoming flow balances the outgoing flow, V . x = 0,
(8.3.2)
where V. denotes the lattice divergence. Obviously the polymers have to enter, and to leave, the system somewhere. To control the number of polymers in the sample, we attach all sites of one free boundary to an extra "ghost" site (via energetically neutral arcs, e = 0), which we call the source s, and the other side to another extra "ghost" site, the target (as in Fig. 3.2). Now if we want to place N polymers in the sample, we inject a flow of N units at the source s and extract a flow of N units at the sink. (7.X)s = +N
and
(7.x)t = -N,
(8.3.3)
Note that we could consider more general situations in which we put flow in and take flow out of the system at any desired location in the lattice provided the flow entering the graph is equal to that leaving. With the above construction, the problem of N non-intersecting polymers in a random media is solved exactly by solving the minimum-cost-flow problem with: linear cost (8.3.1); with N sources and N targets at locations which can be chosen to suit the physical configuration and, with capacity constraint Uij = 1 to ensure non-intersection. An example for two polymers is presented in Fig. 3.2. As demonstrated in Section 3.3.3, the successive-shortest-path algorithm solves this problem in a time bounded by O(IAIN). Examples are presented in Fig. 8.1.
2
Exact combinatorial algorithms
243
a) 30 25 20 15
25 20 15
25 20 15
10
1
1 302
' ~ 1 0 ~ 00 ~ 5
10 15 .
.
.
30 25 20 15 10
b)
i
.
25 20
o
10 ~
1
0
]~ --
Y '~10 ~
25 20 15 10
o
1
0
l
o ~o~
~--
J
r
~
o
Fig. 8.1 a) Optimal polymer configurations for two-dimensional samples of linear size L = l -- 100. The number of polymers, N, is 1,2,4,8,16,32. b) Optimal polymer configurations for a series of three-dimensional samples (linear size L = 32).
8.3.2
Polymer/flux line fluctuations in d = (1 + 1) and d = (2 + 1)
The one-polymer problem (DPRM) has been discussed in Section 6.3. Whereas two lines (N = 2) are still tractable (Tang, 1994), the transfer-matrix method fails to work efficiently for an increasing number of lines since its complexity grows exponentially with N. Since it is the dense limit N -- p L d-1 with p of order one
244
M.J. Alava et aL
which is expected to contain new physics the minimum-cost-flow method, which solves this problem in polynomial time, is highly desirable. We now describe some calculations which test the effect of polymer density on the roughness of directed polymers. A glance at Figure 8.1 indicates that in (1 + 1)-dimensions, non-intersecting polymers are confined by their inability to pass through each. In contrast, in (2 + 1)-dimensions directed polymers are far less confined by their neighbours and hence their fluctuations may grow more rapidly with increasing system size. Recall that for a single directed polymer, the roughness w 2 = [tt2]av - [tt]2v ~ 12~, where ~" -- 2/3 in d = (1 + 1) (Huse and Henley, 1985; Kardar et al., 1986) and ~" ~ 0.625 in d -- 2 + 1 (Kim et al., 1991). Consider the situation in (1 + 1) dimensions, where N polymers or flux lines are present in a sample of length 1 in the direction of the polymers and L transverse to them. Fig. 8.1 shows a number of optimal line configurations with varying density obtained with the minimum-cost-flow algorithm. As long as the transverse fluctuations are smaller than the mean distance between the lines, i.e. l~ < L / N -- 1/p the line roughness grows simply like l~, as for a single line confined to a strip of width 1/p. However, for 1~ > 1/p the lines start to feel each other (via the hard-core repulsion), and the lines begin to compete collectively for the deepest minima. When one confines the transverse fluctuations by choosing free boundary conditions (at x=l and x = L ) the line roughness saturates for 1 ~ c~ (Knetter et al., 1999) (see Fig. 8.2). Wsat(L) ~ p-1 In L,
(8.3.4)
where p-1 is the average distance between two neighboring lines and plays the role of the microscopic length scale. This is equivalent to saying that the meansquare displacement [uZ]av ( o f the nth line from its average position [Un]av -n 9p-1) is proportional to In 2 L, or that for the displacement-displacement correlation function (of line n and n + r) one has [(Un Un+r)Z]av (X In 2 r. We will encounter this super-rough behavior again further below in the context of flux line arrays and two-dimensional elastic periodic media in Section 8.5. From this we learn that a hard-core repulsion between individual polymers or flux lines is sufficient to put the N-line system defined above into the universality class of the XY-model with random phase shifts (8.2.15) defined in Section 8.2.5. We remark that similar conclusions have been reached for a model of interacting flux lines with an elastic part in the Hamiltonian (Nattermann et al., 1991). The crossover from the single-line roughness w ~ 1f to the collective behavior (8.3.4) occurs at some characteristic longitudinal length ~tl (L, p), which can be extracted by scaling the data obtained according to -
w(1, L, p) ~.. //)sat" g ( I / ~ l l )
(8.3.5)
as shown in Fig. 8.2. For system sizes L < 256 it turns out (Knetter et al., 1999) that ~11(L, p) ~ c(p) 9L 2/3, and c(p) ~, 1/p2/3, a result that is not completely
2 8.0
Exact combinatorial algorithms
o
245
o L=128
.............. L=64 o o L=32 , L=16
6.0
4.0
/
2.0
a)
0.0
10
/
/
100
1000
10000
100000
1000
10000
100000
I
L=256 1.0
[] o
[] L = 1 2 8 o L=64
................. L=32 0.8
0.6
/
//
0.4
0.2
0.0
b)
10
100
Fig. 8.2 Scaling plot of the line roughness in two dimensions for constant density: a) p = 1/32 and b) p -- 1/16. Each data point is averaged over 500 independent samples, the saturation roughness Wsat has been extrapolated for l --+ cc and obeys (8.3.4), and the characteristic crossover length scale ~11(L, p) has been chosen in such a way to achieve the best data collapse for the collective region 1/~11 >> 1. (From Knetter et al., 1999).
understood regarding the fact that the XY-model with random phase shifts, to which the present line system should be asymptotically equivalent, is isotropic in both space directions (implying ~ll cx L). Let us finally mention that in three dimensions the super-rough two-dimensional result (8.3.4) will no longer hold, and one expects the n o r m a l roughness / / ) s a t ( 3 d ) "~ ~/ln L, see Section 8.5.5 and Knetter et al. (1999). As an outlook for further applications we note that besides point disorder (the energies eij all independent), columnar disorder (eij identical along vertical or
246
M.J. Alava et aL
tilted columns) can be treated (Hwa and Nattermann, 1995; Krug and HalpinHealy, 1993; Nelson and Vinokur, 1993; Tang and Lyksutov, 1993; Arsenin et al., 1994). Moreover, if there is no disorder present (e.g. eij -- const.) the N-line ground state is simply given by N straight vertical lines at arbitrary positions. We can lift this degeneracy and force the polymers to arrange themselves in a periodic pattern by a columnar modulation of the local energies eij. Adding a random perturbation to these energies leads to a competition between disorder and the periodic potential, which eventually, for strong enough perturbation, leads to the destruction of the periodic structure. So for instance the disorder induced melting of the triangular (Abrikosov) lattice can be studied. It is also possible to model a driven depinning transition of the polymers (flux-lines) by adding a force term to the local energies eij which increases linearly with the space coordinate in one direction, in order to model a transport current. A less trivial extension is the introduction of a soft-core repulsion, which can be modeled by allowing a multiple occupancy of a bond (Xij = 0, 1, 2 . . . . ) but punish high polymer densities with an energy eij (Xij) increasing faster than linear with the number of flux units Xij o n the bond (i, j). Thus the N-line problem with soft repulsion consists in minimizing (x) - ~ eij (xij), (i,j)
(8.3.6)
under the constraints (8.3.2) and (8.3.3). The local energy functions eij c a n be chosen arbitrarily for each bond (i, j), however, they have to be convex as for instance eij(xij) = kij . x i j with n > 1 arbitrary. The energies eij have now to be replaced by the quantity eij(Xij + 1) - eij (xij), which is the energy needed to increase the flow Xij on arc (i, j) by one unit. Since it depends on the current flow x the convexity of eij is needed to ensure that the reduced costs fulfill the inequality c u > 0 after the flow modification. Whereas with hardcore repulsion it was only possible to put N - L d-1 polymers into the system, the polymer density can now be arbitrarily high and an interplay between the repulsion and the disorder effects lead to a much richer phenomenology (Rieger, 1998b). This problem is also solved by the successive-shortest-path algorithm (see Section 3.3.3).
8.4 8.4.1
Gauge (vortex) glass The strong-screening limit
The model (8.2.7) is believed to describe various aspects of bulk superconductors correctly. When we set all Aij in (8.2.7) to zero, the model is just the XY ferromagnet, which is in the correct universality class to describe the transition to the Meissner phase neglecting screening, since it has the same value for the
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order parameter everywhere. If Aij a r e chosen to correspond to a uniform field in some direction, the model exhibits a transition to a vortex-lattice state, which corresponds to an antiferromagnet in the magnetic analogue. As is well known from the classical XY-model (Jos6 et al., 1977; Kleinert, 1989) the spin-wave degrees of freedoms of the Hamiltonians (8.2.7) and (8.2.6) can be integrated out and one is left with an effective Hamiltonian for the topological defects, the vortices, which are the singularities of the phase field 0 interacting with one another like currents in the Biot-Savat law from classical electrodynamics, i.e. like 1/r, where r is the distance (see also Li et al. (1996)). An additional integration over the fluctuating vector potential sets a cutoff for this long-range interaction beyond which the interaction decays exponentially, and can thus be neglected. To be specific one obtains, in three dimensions, the vortex Hamiltonian (Bokil and Young, 1995) H3d_
1 Z(Ji 2 (i,j)
- b i ) G ( i - j ) ( J j - bj),
(8.4.1)
which is defined on the dual lattice, which again is a simple cubic lattice. The Ji, represent the vortex density of the phase field 0 on bond i, and there are threecomponent integer variables running from - o c to oe living on the links of the dual lattice and satisfying the divergence constraint ( V . J)i =- 0 on every site i. The bi are magnetic fields which are constructed from the quenched vector potentials Aij by a lattice curl, i.e. one obtains bi as 1/(27r) times the directed sum of the vector potentials on the plaquette surrounding the link on the dual lattice which bi lives on. By definition, the magnetic fields satisfy the divergencefree condition (V 9b)i -=- 0 on every site, since they stem from a lattice curl. The vortex interaction is given by the lattice Green's function G ( i , j ) = J (2at)2 ~
..... N
1 - exp(ik 9(ri - rj))
(8.4.2)
2 }-~nd--__iil---- COS'~n-)) -7 7,.O2,
which behaves asymptotically as G ( r ) ~ r -1 exp(-r/~.) in three dimensions, i.e. it is a ( I / r ) interaction screened for distances r > ~., as described above. In two dimensions the singularities of the phase field (0) are points and the corresponding vortex Hamiltonian is actually a two-dimensional Coulomb gas (Fisher et al., 1991) H2 d _~ _ l
Z(ni
_ bi)G2d(i _ j ) ( n j - b j ) ,
(8.4.3)
2 (i,j)
where now ni are integers (the vortex strengths), subject to the charge neutrality c o n s t r a i n t ~"~i rti - - 0 and G 2d (i - j) is the given by (8.4.2) with d = 2 describing
248
M . J . Alava et al.
a logarithmic interaction G(r) ~ In(r) for distances r < ~.. The real numbers bi are given by 1/2zr times the directed sum of the quenched vector potentials on the links of the original lattice which surround site i of the dual lattice. Most of the theoretical work so far has concentrated on establishing numerically the lower critical dimension of the gauge-glass model, both with and without screening of the interactions between vortices. Without screening, there is no finite temperature transition to a vortex-glass phase in two dimensions (Fisher et al., 1991; Gingras, 1992; Bokil and Young, 1995; Kosterlitz and Simkin, 1997), whereas in three dimensions there is evidence for a finite Tc, as has been found by domain-wall renormalization-group analyses (DWRG) (Gingras, 1992; Bokil and Young, 1995; Reger et al., 1991; Kosterlitz and Simkin, 1997; Kosterlitz and Akino, 1998; Maucourt and Grempel, 1998) and finite temperature Monte-Carlo simulations (Huse and Seung, 1990; Wengel and Young, 1997), though due to limited system sizes and insufficient statistics the earlier DWRG studies (Gingras, 1992; Bokil and Young, 1995; Reger et al., 1991) could not fully rule out the possibility that the lower critical dimension is exactly d = 3. Sufficiently close to the critical point (i.e. at the superconductor to normalstate phase transition), screening effects become important, since the correlation length ~ diverges more strongly than the screening length ~. and the two length scales eventually become comparable (Bokil and Young, 1995). The effect of screening was investigated (Bokil and Young, 1995) by a DWRG study and more recently (Wengel and Young, 1996) by means of a finite-temperature Monte-Carlo simulation, and the results indicate that screening is a relevant perturbation, destroying the finite temperature transition in three dimensions, though the DWRG analysis could only be performed for rather small system sizes (L < 4).
8.4.2
Mapping to minimum-cost flow
For any non-vanishing value of the screening length ~. (in particular for the nonscreened (;~ ~ cx~) case (8.2.7) the vortex Hamiltonians are non-local in the vortex variables Ji or ni. The two-dimensional case is then equivalent to the Coulomb-glass problem, for which it is NP-hard to find the ground state, and the three-dimensional case is obviously still harder. These problems can in principle be solved with modem integer-linear-programming techniques (branch and bound). Since the underlying graph is complete due to the long-range interactions, the system sizes that have been feasible up to now are very modest. However, one can argue (with a coarse-graining picture in mind) that for all finite values of the screening length ~. the vortex Hamiltonians fall into the same universality class, in particular the same as the the strong-screening limit, k - 0. In this case the Green's function G ( i - j ) reduces to G(0) = 0 f o r / = j and G(i, j) = J(2rr,k0) 2 for i ~ j with exponentially small corrections (Wengel and Young, 1996). Thus
2 Exact combinatorial algorithms
249
if we subtract J (2rr ~.0) 2 f r o m the interaction and measure the energy in units of J(2zrZ0) 2, one obtains the simpler Hamiltonian t4z__>O __ _1 Z ( j i ~v 2
- bi) 2 .
(8.4.4)
i
We remark that Hv is still highly non-trivial due to the divergence condition ( V . J)i :-- 0. Finding the ground state of the Hamiltonian in (8.4.4) subject to the constraint (V 9J)i = 0 is a minimum-(convex)-cost-flow problem and can be restated as (8.4.5) Minimize z(J) -- Z Ci (Ji) i
subject to the constraint (V 9J)i - 0, where the cost functions ci(Ji) -- (Ji bi)2/2 have been defined. Applying the successive-shortest-path algorithm presented in Section 3.3.3, the determination of exact ground states can be performed in polynomial time.
8.4.3
Domain-wall energy and chaos
As in spin glasses (see Section 7.4) the issue of the existence of a finitetemperature phase transition can be scrutinized via ground-state calculations of the Hamiltonians (8.4.1) and (8.4.3) by studying the scaling behavior of the largescale low-energy excitations, typically done via a DWRG analysis. For the latter one usually calculates the ground-state energy once with periodic and once with anti-periodic boundary conditions in the original phase variables 0, the energy difference being the defect energy A E (L) measuring the energy cost of a domain wall of length L (see Kosterlitz and Simkin (1997) for a more refined recipe). To this end one has to carefully treat the boundary conditions of the vortex Hamiltonian (8.4.1) and (8.4.3) leading to an additional term that is non-local in the vortex variables Ji or ni. Note that the vortex Hamiltonian in (8.4.4) with periodic b.c. and without the additional boundary term corresponds to fluctuating boundary conditions in the gauge-glass model (Olsson, 1995; Gupta et al., 1998). Again such a non-local term makes the determination of exact ground states computationally hard, in particular it is not a pure minimum-cost-flow problem any more. To overcome this difficulty a different procedure (see below) to induce lowenergy excitations in this model has been devised (Kisker and Rieger, 1998). This procedure also avoids some conceptual ambiguities that are present in the DWRG using periodic/antiperiodic boundary conditions for the phase variables (Kosterlitz and Simkin, 1997; Kosterlitz and Akino, 1998). In the model under consideration a low-energy excitation of length scale L is certainly a global vortex loop encircling the three-dimensional torus (i.e. the
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L x L • L lattice with periodic b.c.) once (or several times) with minimum energy cost. For the pure case the global minimum-energy loop is simply a straight line that costs energy A E (L) = J L, which is exactly what one expects for a domain wall of length L in a three-dimensional XY-model, and which is also obtained from the energy difference between ground states with periodic and anti-periodic b.c. To induce a global vortex loop, without manipulating the boundary conditions, we instead manipulate the costs for flow in one particular space direction. Suppose we have found an exact ground state, which also specifies the cost required to increase the flow variables jx,y,z with respect to j0 by one unit, e.g." A c x - - c i ( J Ox -Jr-1)- ci(J? x) -- j?x b x + 1/2. If we smoothly decrease the variables Ac x and apply our min-cost-flow algorithm to this modified problem, at some point a configuration J] that is the original ground state plus a global loop in the x-direction will appear as the new optimal flow configuration for the modified problem. This extra loop, which can be easily identified by comparing the new optimum with the original ground state, is the low energy excitation we are looking for. Its energy A E ( L ) is simply the difference H ( J 1) - H(J~ found by evaluating the vortex Hamiltonian (8.4.4). Note that energy is always positive, since it is definitely an excitation (in contrast to the usual DWRG procedure where the b.c. is modified). Four remarks are in order: (1) small, simply connected loops are not generated by this procedure, since all that can be gained in energy is lost again on the return. (2) In the pure case this procedure would not work, since at some point spontaneously all links in the z-direction would increase their flow value by one. It is only for the disordered case with a continuous distribution for the random variables bi that a unique loop can be expected. (3) Sometimes (in about 5% of the samples) the global flux changes discontinuously by more than one unit, however, we still define these to be elementary excitations of length scale L. (4) In the presence of a homogeneous external field one has to discriminate between different excitation loops. Those parallel and those perpendicular to the external field need not have the same energy (however, it turns out that the disorder averaged defect energy is identical in all directions, see below). Schematically the numerical procedure is the following: (1) Calculate the exact ground-state configuration {j0} of the vortex Hamiltonian (8.4.4). (2) Determine the resulting global flux along, say, the x-axis fx = ~ Z i
j?x.
(3) Find the cost for increasing the flow corresponding to introducing a global vortex loop in the x-direction Ac x -- ci(J~ + 1) - ci(J ~ - b x + 1/2.
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(4) Reduce the Ac x until the optimal flow configuration {j1 } for this minimumcost-flow problem has the global flux ( f x + 1), corresponding to the socalled e l e m e n t a r y low energy excitation on the length scale L. (5) Finally, the defect energy is AE -- H({j1}) - H({J~ Kisker and Rieger (1998) have computed (using the procedure above) the disorder-averaged domain-wall or defect energy A E as a function of the system size L, with L up to L -- 50. It was found that AE ~ L ~
with
0--0.95(3).
(8.4.6)
In this way one re-establishes that Tc -- 0 for the gauge-glass model in the strongscreening limit, as has been found by Bokil and Young (1995) and Wengel and Young (1996). From the stiffness exponent 0 the thermal exponent v, which describes the divergence of the correlation length, can be calculated. For Tc - O, the correlation length behaves as ~ ~ T -v. By equating the thermal energy with the energy of a low-lying excitation on the length scale of the correlation length, it follows that 1
v - ~.
(8.4.7)
101
From this relation one obtains v - 1.05(3), which agrees well with a result from a finite-temperature Monte-Carlo simulation for the same model (Wengel and Young, 1996), where system sizes L _< 12 have been studied and a zerotemperature phase transition with v - 1.05(1) has been found. Another important feature of a glassy systems is their sensitivity with respect to changes of various parameters ("chaos"). In Kisker and Rieger (1998) the change in the ground-state configuration when the random vector potentials Aij are perturbed by a small amount has been studied. To be specific, one defines ! new vector potentials by Aij -- Aij + ~-ij, where 6ij is randomly drawn from the interval [ - 8 , 6] and then one calculates the ground state for both realizations of the disorder {Aij } and {A~j}. The distance D between the resulting ground-state configurations {J} and {J'} by D ( 8 ) -- Y ~ i ( J i Rieger, 1998) that DL(8) -- d(L61/()
j,i)2. It turns out (Kisker and
with ( -- 3.9(2),
(8.4.8)
where d ( x ) is a rapidly increasing function of x. This implies that ground-state configurations are de-correlated on length scales L > L*, where the overlap length behaves like L* ~ 8 -1/~ as a function of the perturbation strength 8. It is straightforward to consider for model (8.4.4) a homogeneous external field B ext via bi-
1 7---[V x Arand]i + Bext AT/"
(8.4.9)
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and specifying the boundary conditions to be periodic in all space directions (corresponding to fluctuating boundary conditions in the phase variables of the original gauge glass Hamiltonian) and choosing B ext - B ez. This models a situation in which the external field points in the z-direction. It turns out (Pfeiffer and Rieger, 1999a) that the disorder averaged domain wall energy A E is independent of the value for B (for a single sample it, however, depends on B), which means that also the stiffness exponent 0 is independent of B. Moreover, one observes that the ground state rearranges chaotically with varying B (which is the manifestation of field chaos) and the overlap length L* diverges like L* ",~ B -1/r with the same chaos exponent ~" as in (8.4.8) for the disorder chaos. Finally let us remark that it would be highly desirable to extend such investigations to the case of a non-vanishing screening length ~.. However, this is not feasible with existing mappings and algorithms.
8.5 8.5.1
Disordered elastic media Elastic glasses and the random-surface problem
As demonstrated in Section 8.2.5, a simple model to describe the effect of disorder on the periodic flux lattice is the random-phase sine-Gordon model (also called the elastic-glass model). This, and the models derived from it are generally refered to as disordered elastic media. The lattice version of the model (8.2.15) is given by H
~_,(Ui (i,j)
-- U j ) 2 -- ~. ~
COS(2Yr(Ui -- Oi))
(8.5.1)
i
where i denote the individual sites of a square lattice, (i, j) denote arcs between neighbor sites, Oi are independent random variables each distributed uniformly over the interval [0, 1], and ~. is the coupling strength to the disorder. In the limit of infinite coupling strength ~ ~ cx~ the cosine-term in the Hamiltonian (8.5.1) forces the displacements to be ui = Oi + ni, where ni are arbitrary integers. Thus, in this limit the above model maps onto a solid-on-solid (SOS) model on a disordered substrate: H -- ~ _ ~ ( n i --}- Oi - n j - Oj) 2,
(8.5.2)
(i,j)
where hi -- ni + O i denotes the total surface height at lattice site i, ni is the integer height above the random surface which is described by heights Oi (see Fig. 8.3). The models (8.2.15), (8.5.1) and (8.5.2) have been investigated intensively. The model (8.5.2) possesses a phase transition at a critical temperature Tc to a low temperature, glassy phase, in which the displacement-displacement correlation
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Fig. 8.3 The SOS model on a disordered substrate. The substrate heights are denoted by di E [0, 1] (d i = 0 i is the substrate roughness), the number of particle on site i by ni E Z, which means that they could also be negative, and the total height on site i by h i = d i + n i .
function G ( r ) = [((u(0) - u(r))2)lav
(8.5.3)
shows an anomalous ("super-rough") behavior. Two different predictions have been made for T < Tc, one proposing a log-linear behavior G ( r ) = B ( T ) ln(r), with a coefficient B ( T ) that approaches a non-vanishing constant for T ~ 0 (in contrast to thermal roughness without disorder, which would simply vanish at T = 0) and another one proposing a stronger, log-square increase: G ( r ) = C ( T ) lnZ(r). In the language of superconductivity the former prediction leads to an algebraic decay of the order parameter correlation function [(exp{-i (u(0) u(r))})]av ~ r - B ( T ) / 2 , however with an enhanced exponent, whereas the latter leads to a faster than algebraic decay. Whereas the existence of this transition is established now, the qualitative and quantitative features of the glassy low-temperature phase are still debated. Predictions of earlier renormalization-group (RG) calculations (Cardy and Ostlund, 1982; Toner and Di Vincenzo, 1990; Tsai and Shapir, 1992; Hwa and Fisher, 1994b) turned out to be incompatible with results of extensive numerical simulations (Batrouni and Hwa, 1994; Cule and Shapir, 1995; Rieger, 1995b). The subsequent discovery of the relevance of replica symmetry breaking (RSB) effects in variational (Bouchaud et al., 1991; Korshunov, 1993; Giamarchi and Le Doussal, 1994) and RG (Kierfeld, 1995; Le Doussal and Giamarchi, 1995) calculations lead to a variety of new results, which are again in disagreement with the most recent numerical studies (Lancaster and Ruiz-Lorenzo, 1995; Marinari et al., 1995). All the works, analytical as well as numerical, cited so far are confined to a region around the critical temperature, though it is generally believed that the behavior at low temperatures is typical. This is the reason why an investigation of the ground states is most useful.
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M a p p i n g to m i n i m u m - c o s t f l o w
To calculate the ground states of the SOS model on a disordered substrate with general interaction function f (x) we can map it onto a minimum-cost-flow model (Blasum et al., 1996; Rieger and Blasum, 1997). We define a network G by the set of nodes N being the sites of the dual lattice of our original problem and the set of directed arcs A connecting nearest neighbor sites (in the dual lattice) (i, j ) and (j, i). If we have a set of height variables ni we define a flow x in the following way: suppose two neighboring sites i and j have a positive height difference ni - n j > 0. Then we assign the flow value Xij = ni -- n j to the directed arc (i, j) in the dual lattice, for which the site i with the larger height value is on the fight-hand side, and assign zero to the opposite arc (j, i), i.e. Xji = O. And also x i j = 0 whenever site i and j are of the same height. See Fig. 8.4 for a visualization of this scheme. Obviously then we have: u E N "
~ Xij -~" ~ Xji. {j I (i,j)EA} {j I (j,i)EA}
(8.5.4)
On the other hand, for an arbitrary set of values for xij the constraint (8.5.4) has to be fulfilled in order to be a flow, i.e. in order to allow a reconstruction of height variables out from the height differences. This observation becomes immediately clear by looking at Fig. 8.4. We can rewrite the energy function as H (x) -- Z cij (Xij), (i,j)
with
cij (x) = f (x - dij),
(8.5.5)
with dij - di - dj and f ( y ) = y2 (actually f ( y ) can be any convex function, for instance f ( y ) = [y in, with integer n, and it does not even need to be identical for all bonds). Thus our task is to minimize H(x) under the constraint (8.5.4), which is (see Section 3.3.3) a m i n i m u m - ( c o n v e x ) - c o s t - f l o w problem with the mass-balance constraints (8.5.4) and convex cost functions Cij (Xij). The most straightforward way to solve this problem is to start with all height variables set to zero (i.e. x = 0) and then to look for regions (or clusters) that can be increased collectively by one unit with a gain in energy. This is essentially what the negative-cycle-canceling algorithm discussed in Section 3.3.2 does: the negative cycles in the dual lattice surround the regions in which the height variables should be increased or decreased by one. However, the successive-shortestpath algorithm is more efficient and solves this problem in polynomial time (see Section 3.3.3). This algorithm starts with an optimal solution for H (x), which is given by Xij = -Jr- 1 for dij > 1/2, xij " - - - 1 for dij < - 1/2 and Xij = 0 f o r dij E [ - 1 / 2 , + 1/2]. Since this set of flow variables violates the mass-balance constraints (8.5.4) (in general there is some imbalance at the nodes) the algorithm iteratively removes the excess/deficit at the nodes by augmenting flow.
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Fig. 8.4 The flow representation of a surface (here a "mountain" of height n i 3). The broken lines represent the original lattice, the open dots are the nodes of the dual lattice. The arrows indicate a flow on the dual lattice, that results from the height differences of the variables n i on the original lattice. Thin arrows indicate a height difference of xij -- 1, medium xij 2 and thick xij 3. According to our convention the larger height values are always on the right of an arrow. Observe that on each node the mass-balance constraint (8.5.4) is fulfilled. =
a r r o w s
--
a r r o w s
--
We would like to mention that this mapping of the original SOS model (8.5.2) onto the flow problem works only for a planar graph (i.e. free or fixed boundary conditions), otherwise it is not always possible to reconstruct the height variables ni from the height differences Xij. AS a counterexample consider flow in a toroidal topology (i.e. periodic boundary conditions) where the flow is zero except on a circle looping the torus, where it is one. Although this flow fulfills the mass balance constraints (which are local) it is, for the height variables, globally inadmissible. To the right of this circle the heights should be one unit larger than on left, but left and right become interchanged by looping the torus in the perpendicular direction, which causes a contradiction.
8.5.3
A m a p p i n g to m a x i m u m f l o w
The random-surface problem maps to the minimum-cost flow problem for general convex-cost functions f ( x ) . In the special case f ( x ) = Ix I, there is a simple mapping to the maximum-flow problem (Zeng et al., 1996). This mapping is appealing physically as well as computationally, as we now discuss. The Hamiltonian is, H - y ~ Ihi - h j l (i,j)
(8.5.6)
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with hi : ni q- di, di E [0, 1]. The ground-state configuration is degenerate with modulation one lattice unit in the vertical(height) direction. A different way to represent the problem is to define the Hamiltonian on a fiat substrate but with bond energies in the vertical direction alternating between di and 1 - di. That is, the disorder is "periodic" in the vertical direction. Note that the exact mapping is actually to a "brickwork" lattice which has these altemating bond energies. However, the universality class is expected to be the same for a cubic lattice with couplings having modulation two. It is obviously possible to define a bond energy having modulation of an arbitrary wavelength ~, in which case = 2 is the random-surface problem, while ~. = oo is the random-bond interface problem discussed in Section 7.2. This construction can be used in any lattice orientation. The random-surface problem is equivalent to finding the ground-state configuration of an interface in the presence of this periodic potential in the height direction (Zeng et al., 1996). It can be treated with the maximum-flow algorithm in the same way as for the random-manifold problem (Section 7.2). Note that the problem is now three-dimensional, though it may be implemented in a "slab" which must be wide enough to avoid the fluctuating interface.
8.5.4
A special case solvable by matching
We now introduce a restricted SOS model which is believed to exhibit some of the important behaviors of the random-surface model. The starting point (Zeng et al., 1996) is the triangular Ising solid-on-solid (TISOS) model (Nienhuis et al., 1984) HTISOS-- Z J i j ( l n i - n j l - 1), (8.5.7)
(i,j) where the sites i are on a triangular lattice, the height variable ni can take on only integer values, the summation is over nearest neighbors, and the coupling c o n s t a n t s Jij are chosen uniformly (and independently for each bond) in the interval [0, 1]. In contrast to the model (8.5.6), further constraints are imposed on the height variables: (1) the height difference Ini - n j[ for every bond must be either 1 or 2; and (2) the total height increment (clockwise or counterclockwise) along any elementary triangle is zero (see Fig. 8.5). One can easily check that the SOS surface so defined describes an interface along the {111 }-direction of a simple-cubic lattice. The TISOS model can now be related to matching or "dimer-covering" on a hexagonal lattice. One identifies each bond which has Ini - njl - 2 on the triangular lattice with a "dimer" on the bond of the dual lattice. The dual lattice is hexagonal (see Fig. 8.5) and any allowed surface configuration on the triangular lattice maps to a complete dimer covering of the hexagonal lattice. The constraint that the sum of heights around a loop is zero, ensures that there is one bond with
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o
0
257
I
i
i .......) 0
o f
L-3 Fig. 8.5 Various representations of the TISOS model. The height variables defined on each vertex of the triangular lattice (dashed lines) are shown in the figure. The equivalent dimer coveting is indicated by the thick bonds on the dual hexagonal lattice. The corresponding two polymers are displayed as contiguous triple lines. The linear size of the system is denoted by L(= 3). Periodic boundary conditions are imposed (twisted from top to bottom). (From Zeng et al., 1996). Ini - njl -- 2 in each triangle. The energy of a perfect matching is simply given by E -- ~ tOe (8.5.8) eeM
where M is a perfect matching and tOe E [0, l] is a random weight for the edge e. One can verify (Zeng et al., 1996) that the two different representations (interface along {111 }-direction dimers) are energetically equivalent, with a transformation of the random bonds that maintains its uniform and independent distribution. To solve the minimum-cost perfect-matching problem on the hexagonal lattice, we divide the lattice into its two sublattices U and V. Any edge (i, j ) in the arcset A of the graph has one end site in U and other end site in V. A source b(i) = I is connected to each of the sites in one sublattice, i 9 U, while a sink of size b ( j ) -- - 1 is connected to each of the sites in the other sublattice, j e V. The task is to find the minimum-cost flow through a hexagonal lattice with costs Cij --- tOgj , given the set of sources and sinks stated above, and with the constraint that each arc has capacity one, Uij l. This problem is efficiently -
-
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i
i
,
9 D i s o r d e r e d substr at 9T I S O S
r.~ 4.o
~'
3.0
r.~ 2.o
~
1.0
0.0
0.0
,
1
1.0
,
i
2.0
log L Fig. 8.6 Scaling of roughness in the random-surface problem. (From Zeng et al., 1996). solved using the successive-shortest-path algorithm. The random-surface problem on large sample sizes (up to L -- 420 with high statistics) have been studied using this matching method (Zeng et al., 1996).
8.5.5
The super-rough phase and loop statistics
All of the optimization methods described in Sections 8.3-8.5 have been used to study the ground state of the the random-surface problem. In (1 + 1), the randomsurface problem is trivial, the surface is an uncorrelated random walk and hence its mean roughness behaves as w ".~ L ]/2. In (2 + 1) dimensions, the functional renormalization group predicted that the roughness should scale as w 2 ~ (lnL) 2, instead of w 2 ~ In L as applies to thermally rough surfaces. High precision data supporting the functional-renormalization-group prediction are presented in Fig. 8.6. A brief summary of the other results obtained recently is as follows: (Blasum et al., 1996; Zeng et al., 1996; Rieger and Blasum, 1997): 9 The height-height correlation function G(r) as defined in (8.5.3) diverges like G(r) c~ log2(r) with the distance r for all non-linearities of the local cost functions f (x) - I x l n. 9 XL
--
L-4
Y~(i,j)[(hi
-
hj)2]av can be fitted to Xc = a + b log(L) +
c log2(L) (see Fig. 8.6), again indicating a log 2 dependence of the heightheight correlation function. Moreover, the coefficients a, b and c depend
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on the power n in f (x) -- Ix In: c increases systematically with increasing g/.
9 By considering a boundary induced step in the ground-state configuration one sees that the step energy increases logarithmically with the system size: AE = a + b In L with b = 0.56(2). This is analogous to a domainwall or defect energy, where the defect is a dislocation line between two point defects at fixed position at a distance L (see Section 8.5.6 for a more detailed study of dislocations, in particular those in optimized positions). Furthermore the step is fractal, it's length increases like/~step (X Ldl with the fractal dimension d f ~ 1.35(2). 9 Upon a small, random variation of the substrate heights di of amplitude the ground-state configuration decorrelates beyond a length scale L* 3-1/C with ~" ~ 1. This illustrates the chaotic nature of the glassy phase in this model, in analogy to spin glasses. Recently (McNamara et al., 1999) the three-dimensional version of the model (8.5.6) has also been investigated, which is a lattice realization of the threedimensional elastic medium defined by the Hamiltonian (c.f. Section 8.2.5) H = Hel -+- Hdi s as given by (8.2.9) and (8.2.13). It is found that 9 The Fourier-transformed S(k) of the displacement-displacement correlation function G(r) as defined in (8.5.3) behaves like S(k) ~ Ak -3 with a universal prefactor A. This implies that the average roughness as defined for the two-dimensional case, XL diverges logarithmically with system size. 9 By considering a twisted boundary condition analogous to the step inducing boundary conditions in two dimensions one finds that the excitation or domain-wall energy increases algebraically with the system size: AE L ~ with 0 ~ 1. Furthermore, as in two dimensions, the domain-walls are fractal, their area increases like Swall cx Ldl with the fractal dimension d f ~ 2.60. 9 Upon a small, random variation of the disorder configuration of amplitude the ground-state configuration decorrelates beyond a length scale L* 6 -1/c with ~" - 0.39(2). The scaling relation ~" - d f / 2 - O, as proposed originally in the context of spin glasses (Bray and Moore, 1987) is fulfilled. The results on the displacement-displacement correlations are in agreement with renormalization group and variational calculations (Giamarchi and Le Doussal, 1995) and the results on the domain-wall energy are consistent with the prediction of the marginal stability of the system with respect to the introduction of dislocations (Fisher, 1997).
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Fig. 8.7 {111 } interface of a simple cubic lattice and its two loop representations. An example of a ground-state interface confined between two flat {111} layers is shown. The level set at mean height consists of closed contour loops due to periodic boundary conditions. Also shown is the fully-packed-loop (FPL) representation of the same interface (see text for details of construction). (From Zeng et al., 1998)
Another interesting representation of an interface in the {111 }-direction of an simple cubic lattice, is in terms of loops. Given the exact shape of the {111 }interface its topography is completely characterized by a contour plot with the level spacing equal to a single step of the discrete height variable. The contour plot consists of contour loops which live along the bonds of the hexagonal lattice s The contours are closed due to periodic boundary conditions which we impose in both lateral directions. For example, in Fig. 8.7, we show all the contour loops (at the mean surface height) for the {111 } interface. The union of all the contour loops for different realizations of disorder is the contour-loop ensemble. As well as this natural contour-loop characterization, there exists yet another interesting loop representation of the {111}-interface, the fully packed loops (FPLs). These loops owe their existence to the one-to-one mapping between a {111 }-interface and a complete dimer coveting of the hexagonal lattice /2. By removing the bonds of E that coincide with the dimers, we are left with a configuration of fully packed loops, as shown in Fig. 8.7, where every site of E belongs to one and only one loop. A physical realization of FPLs is magnetic
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domain walls in the ground state of the Ising antiferromagnet on the triangular lattice (Nienhuis et al., 1984). FPL models of general loop fugacity in the absence of disorder have been studied recently (Batchelor et al., 1994; Blrte and Nienhuis, 1994; Kondev et al., 1996) and were shown to be critical for values of the loop fugacity that does not exceed two. The interface-FPL mapping thus allows one to consider the effect of quenched disorder on the critical FPL model on the honeycomb lattice with fugacity equal to one (Zeng et al., 1998).
8.5.6
Dislocation effects
Various kinds of topological defects can be present in a line lattice: Interstitials and vacancies, edge and screw dislocations (Nabarro and Quintanilha, 1980). The proliferation (via thermal or disorder fluctuations) of such dislocations can induce a topological phase transition that destroys the long-range or quasi-long-range order, i.e. the flux-line lattice or the Bragg-glass phase, respectively. Recently the effect of dislocations on the glassy phase of the random-surface model has been considered (Gingras and Huse, 1996; Middleton, 1998; Zeng et al., 1999). Within the elastic-glass model, dislocations can be included if we treat the phase field q~(x) or the height field h (x) as a multi-valued function which may jump by 4-2zr or 4-1, respectively, at surfaces which are bounded by dislocations lines. Including these effects, it has been argued (Gingras and Huse, 1996) that for weak enough disorder the system is stable with respect to the formation of topological defects. However, for strong disorder the vortex lines were predicted to proliferate and thus to destroy the (quasi)-long-range order. An example of a dislocation, a pair of point defects, in the random surface model (8.5.2) is shown in Fig. 8.8. Here the energy in terms of the flow variables X i j "-- n i - - n j is H - - Z ( i , j ) ( x i j - dij) 2, where d i j - - d i - dj. With the given configuration of substrate heights the absolute minimum-energy configuration of flux variables Xij a r e xij = 0 in every bond of the dual lattice, except for the thick straight line between the two dots, where Xij - - 1. This would actually be the starting configuration of the success-shortest-path-algorithm as discussed in Section 3.3.3. This is not a feasible solution for the surface problem since the flow x is not divergence free, which implies that one cannot find a displacement or height field n i that fulfills X i j = n i - - n j for all pairs (i j). This flow configuration corresponds to a flat surface with one dislocation line between the two point defects shown in the figure. Actually, in its search for a feasible solution the successive-shortest-path-algorithm will remove this dislocation by sending flow from the fight excess node to the left deficit node. Then the optimal surface (without dislocation) is flat, ni = 0. This already demonstrates that the ground state of the random surface (without dislocations) is unstable with respect to the introduction of dislocations. If the latter are allowed the energy is simply the
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Fig. 8.8 Sketch of a dislocation pair for the random-surface model. The substrate heights di are indicated in grey scale (white means di = 0). The optimal surface would be flat (Vi n i = 0), however, the dislocations as indicated would (if allowed) decrease the total energy (see text).
- dij) 2 without the mass-balance constraint global optimum of H -- Y~(i,j)(xij for the variable xij. It is easy to see that half of all nodes have excess or deficit in the global optimum, which means that the global optimum is full of dislocations. Thus if dislocations are allowed, without penalty, Bragg-glass order is certainly destroyed.
One can ask more detailed questions about a single dislocation pair in the random surface model. For instance one can fix the position of this pair by defining a graph with one deficit and one excess node in a distance L, or one can let them choose an optimal position in the distance (Pfeiffer and Rieger, 1999b; Zeng et al., ! 999). In the first case the energy of the dislocation increases (Pfeiffer and Rieger, 1999b; Zeng et al., 1999) like In L, as has already been found (Rieger and Blasum, 1997) for the average step energy with fixed step position at the boundary (see Section 8.5.5), whereas in the latter case it decreases with L like (In L)-~, with ~ = 3/2. In the framwork of matchings, dislocations arise for incomplete (non-perfect) matchings (see Fig. 8.9). In the case of an incomplete matching, there is no longer a uniquely defined height variable corresponding to displacements of the d = 2 elastic medium. An energy for the dimer configuration can then be assigned as follows: (1) a local pinning energy where vertices are covered by dimers in M; and (2) a defect energy for vertices that are not the endpoints of dimers in M. The
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Fig. 8.9 Left: a defect occurring in an otherwise perfect matching is shown. The unmatched vertex on the A (B) sublattice correspond to a negative (positive) vortex. The dashed line indicates a possible path for a defect string that connects two defects. If this were the defect string, then in the ground state with no defects, the unmatched edges along this path would become matched and the matched edges would be removed. Right: the symmetric difference between matchings for a ground state with no defects and a ground state for the same disorder realization, with a defect energy of Ec = 1.2 (384 • 384 unit cells.) Dimers are included if they belong to a matching in one of the ground states, but not both. Vortices are at the ends of the defect lines. The defect lines themselves show where the phase change due to the introduction of defects is localized. (From Middleton, 1998)
energy of a matching is then,
E-
NcEc- ~
We,
(8.5.9)
eEM
where Nc is the number of non-covered vertices and Ec is the core cost of a defect. Minimizing E in (8.5.9) gives the T = 0 configuration for the elastic medium with defects that have an associated core energy. Over local regions where the height is well defined, the energy is still periodic in the height (displacement) variable. It turns out (Middleton, 1998) that the density of defects decreases approximately exponentially with the defect core energy Ec. However, this calculation was for defects placed in specified (non-optimal) positions. By looking at the energy of these dislocations, Middleton (1998) argued that, at weak disorder, the super-rough phase is stable with respect to dislocation formation, in agreement with Gingras and Huse (1996). However, Zeng et al. (1999) considered dislocations placed in optimal positions, and they found that at arbitrarily weak disorder dislocations are favorable. They thus conclude that, in two dimensions, dislocations proliferate and destroy the Bragg-glass state. As shown in Fig. 8.10, for the fully packed loop model, the energy for a fixed dislocation pair, on average, increases with the distance of the two defects, however, in the optimal position their energy decreases with distance as (In L) -3/2 in agreement with the above mentioned results for the SOS model on a disordered substrate (Pfeiffer
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c)
a) 8
20
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(see left-hand side), a) The probability distributions
of the energy of afixed dislocation pair with separation L/2 for sample sizes from L -- 12 to L -- 384. The solid lines are Gaussian fits. b) The corresponding average defect energy
Ea (solid circle) and the root-mean-square (rms) width ~r(Ea) (solid square) are found to scale with system size as In L. The solid lines are linear fits. Energetics of optimized defects (see right-hand side), c) The defect energy probability distributions for sample sizes from
L
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The solid lines are guides to the eyes. d) The average
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and Rieger, 1999b). The state which results due to this proliferation is not yet understood (Zeng et al., 1999).
9 9.1
Rigidity theory and applications Introduction
Connectivity percolation became popular in the late 1960s and its application to the conductivity of practical materials, such as composites and porous media, has
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further emphasized its importance. The simplest model of charge transport in random networks is the random-resistor network in which each present bond is a resistor. It is natural to consider a second simplemodel in which each present bond is a Hooke spring which obeys the force law F -- -k/zr (IF[ - r0), where r0 is the spring's natural length, II means modulus, and #r is the unit vector in the direction of ?, i.e. #r ~/1~1. Note that we have written this in its vector form to emphasize the fact that the restoring force is in the same direction as the radial vector. There is no restoring force for angular distortions or twists. Networks of Hooke springs and other systems in which there is no restoring force for angular deflection or twists are called central-force systems. A connected network of resistors is able to transport a current. A connected network of Hooke springs is not always able to carry a force. The problem of rigidity percolation or network rigidity is to find the configurations of springs which are able to transmit stress. Related questions were raised by James Clerk Maxwell (1864) due to his interest in the stability of truss structures (e.g. truss bridges) and he developed a lower bound on the number of zero energy ("floppy") modes in these networks. However, his bound is poor near the rigidity threshold (this is the critical volume fraction of Hooke springs needed to support a stress). An exact theory relating the rigidity of a structure to its connectivity is elegant and quite sophisticated (Asimov and Roth, 1978, 1979; Sugihara, 1980; Imai, 1985; Tay and Whiteley, 1985; Hendrickson, 1992). This rigidity theory is developed in Section 9.2. Within the framework to be developed here, usual (scalar) connectivity is just a particular case of rigidity. Although we will discuss "rigidity", the concepts and methods apply to connectivity as well. A set of mini-reviews of this area is now available (Thorpe and Duxbury, 1999). The key practical tool in using rigidity theory is a bipartite matching algorithm (Hendrickson, 1992; Moukarzel, 1996; Jacobs and Hendrickson, 1997) (see Section 4.3.1). Rigidity theory compares the number of degrees offreedom which a set of nodes has to the number of constraints which exist in the graph. In the case of Hooke springs connecting sites in d dimensions, each node has d degrees of freedom (d translations), while each Hooke spring is a constraint. Bipartite matching is used to "match" or assign constraints to sites (see Section 9.2.3). This bipartite matching procedure is exact for the rigidity of graphs in the plane. It is also applicable to the special case of connectivity percolation in arbitrary dimensions. The use of bipartite matching has markedly improved understanding of rigidity percolation in two dimensions (Section 9.3) and has also provided high precision results for the connectivity backbone in all dimensions (Section 9.4). Although the general rigidity problem in three and higher dimensions does not have a well-developed rigidity theory, several important subclasses do. In particular if there is only one unconstrained angle in a three-dimensional structure, then the bipartite matching procedure works. This applies to the rigidity of glasses and proteins where the bond-bending forces are typically much larger than the -
-
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azimuthal twist forces (which are treated as the unconstrained degree of freedom). This is discussed briefly in Section 9.5. It is important to note that the bipartite matching methods, and indeed much of rigidity theory, is devoted to generic rigidity (Laman, 1970; Gluck, 1975; Asimov and Roth, 1978, 1979; Crapo, 1979; Whiteley, 1979; Tay, 1985; Tay and Whiteley, 1985; Hendrickson, 1992) (Section 9.2.3). This concept is foreign to the connectivity problem because all graphs are generic when it comes to connectivity. However, in the rigidity case, some constraints may be degenerate and these constraints violate the counting rules that apply in generic networks. This is analogous to the difference between a regular bead pack (which is not genetic) and random packing (which is generic) and genetic-rigidity concepts have been shown to have important consequences for granular media due to this observation (Moukarzel, 1998b).
9.2 Rigidity theory 9.2.1
Rigidity and bar independence
Consider a framework F made of n point-like joints in d-dimensional space, connected by b bars of arbitrary fixed length. A framework can be formally represented by means of a graph G(V,E) plus a set X = {Xl,X2 . . . . . Xn} in which xi is the spatial location of joint i in d-dimensional space. We call X a realization of G. Clearly ~ contains the topological information about F , while X contains the geometric information. We now look for possible transformations of X which leave all bar lengths unchanged. A continuous transformation satisfying this condition is called a flex. Clearly, if the number b of bars is small, it will be possible to continuously deform F while keeping all bar lengths constant. We say in this case that .T" is flexible (Fig. 9.1a). On the other hand if F has a "large enough" (to be defined later) number of bars, the only possible flexes will be isometries, i.e. translations and rotations in Euclidean space. In this case we say that ~ is rigid (Fig. 9. lb). A second important notion is the concept of infinitesimal rigidity, which amounts to the non-existence of non-trivial infinitesimal flexes. More precisely, an infinitesimal flex V is defined by a set of instantaneous velocities {Vl, v2 . . . . . Vn} satisfying (Xi -- X j )
9 (l)i -- V j ) = 0
V
pair (i, j) connected by a bar.
(9.2.1)
That is, the relative velocity of each pair of joints connected by a bar must be normal to it. If the only vectors V satisfying (9.2.1) are infinitesimal isometries (i.e. trivial), F is infinitesimally rigid. Otherwise it is infinitesimally flexible.
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a
a)
b)
c)
Fig. 9.1 a) Flexible; b) Rigid; c) Degenerate frameworks.
Let us now write (9.2.1) in matrix form as MV - - 0 ,
(9.2.2)
where M is a dn x b matrix whose rows are in one-to-one correspondence with the bars of U. We will say that a subset of bars is independent in a given realization, if their associated rows in M are linearly independent. Clearly, infinitesimal motions of U belong to a vector space of dimension dn. Within this space, infinitesimal isometries span a subspace of dimension Gd -- d + d ( d - 1) / 2 (d translations plus d (dL--1)/2rotations), and they are always solutions of (9.2.2). Therefore the rank K (M) of M can be at most dn - Gd. If K (M) < n d - Gd, there are additional, non-trivial, solutions of (9.2.2), and .T" is flexible. Definition A framework is infinitesimally rigid if and only if K (M) -- n d - Gd.
Since K equals the number of independent rows of M we have the following important equivalence Definition A framework is infinitesimally rigid if and only if it has nd - Gd independent bars.
By definition, removing an independent bar reduces K by one. In other words, each independent bar effectively eliminates one degree of freedom from the space of motions. Dependent bars, on the other hand, (also called redundant bars) can be removed without modifying K i.e. they are essentially useless for rigidity. In order to decide whether a given framework .T" is rigid, all we need to do is to identify dependent bars. Once these are removed, if there are n d - G a independent bars remaining, then U is rigid. If the remaining number of bars is smaller (it cannot be larger), 5 is flexible.
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To illustrate the difference between rigidity and infinitesimal rigidity, consider the two-dimensional example shown in Fig. 9.1 c, which is rigid, since no continuous transformation (other than roto-translations) leaves all bar lengths constant. But it is not infinitesimally rigid, since an infinitesimal displacement of joint b as depicted in the figure satisfies (9.2.1). This situation (i.e. a framework which is rigid but not infinitesimally rigid) is rare, and can only happen for particular choices of joint locations, which are called degenerate. In the particular example of Fig. 9.1 c, a degeneracy occurs because a, b, c are on the same line. Otherwise this framework would be both rigid and infinitesimally rigid. Degeneracies have zero probability if the joint locations are randomly chosen, and therefore one can safely ignore them if random locations are assumed. The distinction between rigidity and infinitesimal rigidity leads us to the concept of generic properties, i.e. properties which are valid for most choices of X. In this sense, a graph is generically rigid, if almost all its realizations are infinitesimally rigid, with the exception of a zero-measure subset of degenerate ones. Notice that generic rigidity is a graph property, i.e. it does not depend on geometry but on topology alone. The claim that generic properties are "typical" and degeneracies an exception can be justified by observing that IMI is a polynomial of the joints' locations. Therefore K can be less than its maximum value over all locations, only on a subset of zero measure, which are the zeros of IM]. The complement of this subset (i.e. the generic configurations) is therefore an open dense subset. This is contained in an important result due to Gluck (1975). Theorem (Gluek) If G has a single infinitesimally rigid realization, almost all its realizations are infinitesimally rigid. A statement similar to Gluck's theorem holds for bar (or, in graph terminology, edge) independence: if the edge set E of G is independent in a single realization, then it is independent in almost all realizations. Therefore we will in the following only be concerned with generic properties, which only depend on the topological information contained in G, relying on the fact that they will be valid for most frameworks generated from G. We now seek a graph-theoretic method to identify redundant and independent bars in ~. Because of what we have discussed, such a method would also constitute a graph-theoretic characterization
of generic rigidity.
9.2.2
Graph-theoretic characterizations of rigidity
Because generic properties depend on topology and not on geometry, it should be possible to decide whether or not a certain framework is (generically) rigid just by looking at its graph, i.e. without any knowledge of the nodes' positions. A graph-theoretic characterization of rigidity is a combinatorial method to decide
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g
[ a)
A
]
b)
Fig. 9.2 a) This three-dimensional structure has n = 8 joints and 3 x 8 - 6 = 18 well distributed bars, yet they are not independent and therefore it is flexible, as can be seen by considering relative rotations of its two halves around the dashed axis. b) A body-joint-bar framework.
whether a graph is generically rigid. Because of our previous discussion, deciding whether a graph is rigid is entirely equivalent to being able to identify redundant bars. We now discuss methods to detect bar redundancy. It is easy to give a sufficient condition for bar redundancy (equivalently, a necessary condition for bar independence). Because our considerations about the maximum rank of M apply to any subset of Y, if any subgraph {7' with n t nodes has m o r e than n t d - G d bars, some of them are necessarily redundant. If on the other hand no subgraph has too many bars, i.e. if b' < d n t - G d for all subgraphs, we will say that the bars are w e l l distributed. Correct distribution in the sense above is a n e c e s s a r y condition for bar independence, in all rigidity problems that we will be discussing, but regretfully it is not always a sufficient condition. In dimensions greater than two, it is possible to build j o i n t - b a r structures with welldistributed yet dependent bars. A well-known example of a three-dimensional structure with well-distributed yet dependent bars is shown in Fig. 9.2a. In two dimensions, correct distribution is a necessary and sufficient condition for bar independence (Laman, 1970). The matching algorithm (MA) to be developed in this section only works for "rigidity problems" in which correct distribution is a necessary a n d sufficient condition for bar independence, since it is simply a method to decide (efficiently) whether bars are correctly distributed or not. As already advanced, the rigidity problem is not only meaningful for bar-joint
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frameworks, but also for more generally defined structures, in which the number of degrees of freedom of the objects represented by the nodes have arbitrary values. These objects are still connected by bars, each fixing the distance between two points and thus acting as one constraint. In many of these cases, correct distribution of bars is a sufficient condition for bar independence, and thus the algorithm to be presented here can be applied to it. We next advance some examples. We start with two-dimensional joint-bar frameworks, for which Laman (1970) has shown the following theorem. Theorem (Laman) A graph ~ contains no dependent bars in two dimensions if and only if all nontrivial subgraphs G', with n I joints and b f bars, satisfy b ~ _< 2n t - 3. If no subgraph contains too many bars, we can be sure that all bars are independent. This is the non-trivial part of Laman's theorem, which, as said, does not hold in d > 3. Moreover, Laman's theorem does not immediately suggest a good combinatorial algorithm. A naive implementation would imply testing all subgraphs, of which there are an exponentially large number. But Sugihara (1980) showed that Laman's theorem can be recast in different terms, involving a matching on an associated bipartite graph. This equivalent restatement gave rise to the first polynomial-time algorithm for rigidity. Hendrickson (1992) later used Sugihara's restatement as a base to provide an algorithm for two-dimensional rigidity with an O(n 2) time-complexity. It is on variants (Moukarzel, 1996; Jacobs and Hendrickson, 1997) of this algorithm that most recent large-scale investigations (Moukarzel and Duxbury, 1995; Jacobs and Thorpe, 1995, 1996; Moukarzel et al., 1997a; Jacobs and Thorpe, 1998; Moukarzel and Duxbury, 1999) of rigidity percolation are based. In its original form, Hendrickson's algorithm performs close to its worst-case complexity on typical problems. Moukarzel (1996) proposed a modification of this algorithm with an improved average behavior (almost linear in n). This modified algorithm is based on the idea of successively condensing (shrinking) rigid subsets of the framework as they are identified, and treating them as single rigid objects called bodies. The correctness of the condensation procedure relies on the observation that Laman's theorem also holds for B-frameworks, that is, structural bar-linkages which, in addition to joints, also include bodies (Fig. 9.2b). Bodies are extended rigid objects and thus have three degrees of freedom in two dimensions. Bars can be connected between two arbitrary points on different bodies, and fix the distance between these two points. The topological structure associated with B-frameworks is a multigraph since several bars can join two bodies, but we will ignore this technicality and simply call it a graph. For two-dimensional
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B-frameworks a graph-theoretic characterization of rigidity also exists, and it is similar to Laman's theorem (Moukarzel, 1996).
Theorem (body Laman) A graph G with n joints, m bodies and b bars contains no dependent bars in two dimensions if and only if all nontrivial subgraphs G', with n ~joints m z bodies and b f bars, satisfy b ~ < 3m ~ + 2n ~ - 3 There are still at least two further cases in which a complete characterization of rigidity in graph-theoretic terms can be given. The first is body-bar rigidity in arbitrary dimensions (though without point-like joints). In d-dimensional space, bodies have Gd -- d ( d + 1)/2 degrees of freedom, which correspond to d (d - 1)/2 rotations plus d translations. For these body-bar linkages, Tay (1985) demonstrated that
Theorem (Tay) A graph G with m bodies and b bars contains no dependent bars in d dimensions if and only if all non-trivial subgraphs G', with m ~bodies and b ~ bars, satisfy b ~ <_ Gdm ~ -- Gd. Last but not least, connectivity on arbitrary graphs can also be interpreted as a simple case of rigidity (Moukarzel, 1998a; Moukarzel and Duxbury, 1998, 1999; Duxbury et al., 1999). While rigidity is a vector problem where each point-like joint has d degrees of freedom (its position) in d dimensions, connectivity is a scalar problem and each joint has one degree of freedom. One can think of this degree of freedom as a voltage vi associated with each node, or as fluid pressure in an array of cavities connected by thin pipes. In the electric realization of the connectivity problem, bars represent batteries with a known fixed voltage so that the potential difference between two nodes connected by a bond is fixed. The analog of (9.2.1) contains the fact that the potential differences are fixed, and is written as (1) i - - l ) j ) --- 0
V
pair (i, j) connected by a battery,
(9.2.3)
where 1)i denotes an infinitesimal change in the potentials Vi. In this scalar connectivity problem, geometry plays no role whatsoever. Most of the discussion of Section 9.2.1 can be extended to this problem. Noticing that in connectivity there is just one "isometry" 9a constant shift in all voltages, we conclude that Gd -- 1 in all dimensions. In particular, everything we said about the relation between rigidity and the rank of the associated matrix M holds, and the definition of bar independence goes along the same line. The following theorem holds trivially in any dimension for the scalar connectivity problem.
Theorem (connectivity) A graph G contains no dependent bonds if and only if all non-trivial subgraphs G', with n ~nodes and b ~bonds, satisfy b' _< n ~ - 1.
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Independent bond-sets are easily seen to be collections of trees (a forest in graph-theoretic terminology). A tree is a cycle-free connected graph and therefore the connectivity theorem is true with "dependent bonds" replaced by "cycles", in which case it is self-evident. In other words, in scalar connectivity a bar is redundant if and only if it closes a loop.
9.2.3
The matching algorithm
In order to discuss the matching algorithm we consider a general rigidity problem that contains all the examples discussed in Section 9.2.2 as particular cases. This rigidity problem is defined to have two types of nodes: "joints" with gd degrees of freedom, and "bodies" with G d degrees of freedom. The number of joints is n, and the number of bodies is m. We will furthermore assume that the following graph-theoretic characterization of edge independence holds. Condition (counting I) A graph ~ with n joints, m bodies and b edges contains no dependent edges if and only if for all non-trivial subgraphs ~' with n' joints m ~bodies and b ~edges is b t < Gdm ~ + g d # -- Gal. Hendrickson's matching algorithm (1992) was developed for two-dimensional joint-bar networks (gd = 2, Gd = 3), but is easily extended to a generalized rigidity problem with gd and G d arbitrary, if the counting condition above holds. The counting condition can be trivially restated in the following form. Condition (counting II) If the counting I condition holds, then the following are equivalent. A. The edges of G are independent. B. If Gd copies of any edge e are added to ~ obtaining Ge, there is no subgraph ~e of ~e for which b' > Gdm' + grin' The matching algorithm builds a maximum independent subset of edges El, and this is done by testing all edges in E one at a time. Each time a new edge e is considered, counting condition IIB is used to decide whether e is independent from the existing set El. If e is independent it is added to El, otherwise e is marked redundant and removed from the graph. For simplicity we will refer to the process of adding Gd copies of an edge e to a graph as replicating edge e. In these terms, the independence test involves replicating each edge in a graph, and looking for subgraphs with too many edges. This has to be done after each new edge is added, and is still very time-consuming. The following observation, due to Hendrickson (1992), is key for reducing the complexity of the procedure: if one adds a new edge enew to an independent set
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El, then in order to decide whether EI tO enew is still independent one only needs to replicate enew and look for subgraphs with too many edges. If none exists, then enew is independent.
Theorem (Hendrickson) Let EI be an independent edge set and consider the graph G whose edge set is EI tO enew. The following are equivalent: A. EI to enew is independent.
B. When enew is replicated obtaining ~enew' no subgraph {7etn e l / ) of Genew has more than G d m ~ + gdn ~ edges.
Proof: A =~ B is a consequence of counting condition II. To prove B =~ A assume E1 to enew is dependent. Then because of counting condition II, there must be some subgraph ~ of Gg (obtained by replicating edge g,) with "too many" edges. But ~ must necessarily include enew since El is independent. Therefore {7~ would contain the same number of edges if enew were replicated instead of ~, and the result is proven. [] Therefore, edges are added one at a time in order to build an independent set. Each time a new edge is added and replicated, we have to check whether any subgraph has too many edges. This checking can be done efficiently by mapping Hendrickson's condition B onto a bipartite-matching problem (Sugihara, 1980; Csima and Lovfisz, 1992; Hendrickson, 1992). In order to illustrate this mapping, we first introduce the following notions. Given a graph {7 = (V, E) we define an auxiliary directed graph D(G) by assigning to each edge e 6 E a direction, i.e. transforming it into an arrow. Arrow orientations are arbitrary, so that D({7) can in principle be in one out of many "states", each corresponding to a configuration of arrow orientations. We say that a configuration of arrows, one per edge in E, is satisfying, if no node i 6 V has more than Yi incoming arrows, where Fi is the number of degrees of freedom of node i, that is, Fi = gd if node i is a joint, and Fi = G d if it is a body. If node i has exactly Fi incoming arrows, we will say that it is saturated (See Fig. 9.3). From now on, only satisfying configurations will be allowed on D(~). Consider now a bipartite graph B with node set (V1, V2), where V1 - E and V2 is composed of ?'i copies of each node i 6 V. Nodes Vl 6 V1 and v2 6 V2 are connected by an edge if the corresponding objects (resp. an edge in E and a node in V) are incident to each other in ~. Satisfying arrow configurations on D are in one-to-one correspondence with complete matchings from V1 to V2, that is, matchings that leave no node in V1 exposed. Because of this, we say that an arrow is matched to a node when it points to it in a satisfying arrow configuration. We use the bipartite matching techniques described in Section 4.3.1 in order to
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IB 1
c
I 1 VV~
A a)
b)
Fig. 9.3 a) A B-framework in two dimensions (bodies A and B have Y = 3 and joint c has F -- 2). b) The associated satisfying arrow configuration when one bar is replicated. All nodes are saturated in this example.
construct satisfying arrow configurations on 79. The relevance of the matching problem for our independence characterization (Hendrickson's condition B) is a consequence of the following equivalence. T h e o r e m ( S u g i h a r a - H e n d r i c k s o n ) The following are equivalent. A. A satisfying arrow configuration exists on "D(~e .... )" B. ~enew contains no subgraph with too many edges. Proof:
A ::~ B is a trivial consequence of the definition of ~)(~enew)" To prove that B =:~ A assume that no satisfying arrow configuration exists on ~:)(~enew)' and let C be a maximally satisfying arrow configuration, i.e. one with the maximum possible number of matched arrows and all unmatched arrows removed. Let ~ be one of the arrows that cannot be matched. Both its end nodes must therefore be saturated. Send both nodes to a queue Q and start a BFS from Q, following arrows in reverse direction only. This means, from each node x we only explore arrows which are pointing to x. All nodes visited in this BFS must be saturated, otherwise an augmenting path would have been discovered and g, could be matched, contradicting our maximality hypothesis. Therefore, for each joint-node visited during the BFS, we simultaneously find gd unvisited edges (those matched to it), and for each body-node we find Gd unvisited edges. So when the BFS ends, we have visited a subgraph with a total of m bodies, n joints and exactly gdn + Grim edges. Now taking into account ~, we have a subgraph with at least one excess edge. t3
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We now sketch the matching algorithm for the general rigidity problem. We are given a graph G(V, E) and we want to determine a maximal independent edge set EI c_ E, or equivalently: we want to identify and discard redundant edges. We start with 79 having no edges. Then for each edge enew E E, we add (Gd + 1) arrows between the corresponding two nodes in 79 and: 9 If we are able to find a matching (a configuration of arrow orientations that is satisfying), then enew is independent and is added to EI. In this case, the Gd "extra" copies of enew are removed from 79. 9 If on the other hand no satisfying configuration exists, enew is redundant and all copies of it are removed from 79. In this way, the edge set of 79 is kept, at all times, independent. In practice a matching in 79 is kept from previous stages of the algorithm, and we only have to enlarge it to include the (Gd + 1) copies of each new edge enew. The enlargement is done by BFS from any of the nodes incident to enew, as described in Section 4.3.1. If the matching on 79 cannot be enlarged we learn that enew is redundant and it is discarded. This procedure enables us to build a maximal subset of independent bars. If this subset contains exactly ngd + mGd -- Gd bars, then our B-framework is rigid. If it contains less bars then our B-framework is not rigid. But this knowledge alone is not always very useful since in most physical applications (e.g. on disordered systems), the framework under consideration will almost certainly contain nonrigid parts, so that this naive procedure would only give us an obvious piece of information: that the system as a whole is not rigid. In practical applications we would more generally be interested in decomposing the system into rigid clusters. Fortunately this is also possible within the matching algorithm. When a matching fails, we learn that enew is redundant. In this case we also obtain an important piece of information from the algorithm: a new rigid cluster is identified, as the following two results show. Theorem If only Ga copies of any new edge enew are added to 79 whose edge set EI is independent, a matching is always possible. Proof: Assuming no matching is possible, start a BFS as described in the proof of the Sugihara-Hendrickson theorem and a subgraph will be identified with bI > grin' + Gdm', involving the Gd copies of enew. If now the Gd copies of enew are replaced by Gd copies of any other edge in this subgraph, a matching would also not be possible, but this is in contradiction with the assumed independence of Ei. []
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So the first Gd copies of any new edge can always be matched if the edge-set in 79 is independent. Therefore it is the result of the (Gd + 1)th search for an augmenting path that decides whether enew is independent. If this search fails, the visited subgraph (without enew) is minimally rigid, as we now show. Theorem (duster) When the (Gd + 1)th search for an augmenting path fails for a new edge enew, the failed BFS spans a minimally rigid subgraph of U, i.e. one with exactly ngd + mGd -- Gd well distributed edges. Proof:
By the same reasoning as in the Sugihara-Hendrickson theorem, we can see that the failed BFS visits a subgraph with exactly ngd + mGd edges, and that the Gd copies of enew are in this subgraph. Remove the Gd copies of enew, and the result is proven. [] This property of the algorithm allows us to identify rigid clusters, even when the system as a whole may not be rigid. This is very important in practical applications as already mentioned. In Hendrickson's original algorithm, newly discovered rigid clusters are given a common label. Using these labels one can save some CPU-time since, if a new bar enew has both ends connected to the same rigid cluster (the same label), then no test is necessary. In this case, enew is trivially redundant. The resulting algorithm can be shown to need O(n 2) time in the worst case (Hendrickson, 1992). Experimental results using this algorithm show (Moukarzel, 1996) that its average time-complexity is not much better than this. We now describe a modification of Hendrickson algorithm, which has much improved average behavior.
9.2.4
Condensation
As already discussed, each time a B FS fails to find an exposed node to which the (Gd + 1)th copy of a bar (a, b) can be matched, the set of nodes and bars visited forms a minimally rigid subgraph C(a, b). In Hendrickson's algorithm, these clusters are given a common rigid label and left in the system. It is possible to dramatically improve the time- and memory-requirements of the algorithm by condensing these rigid clusters (Moukarzel, 1996), i.e. replacing them on 79 by a single body-node with Gd degrees of freedom (An alternative to condensation consists in changing the internal connectivity of these rigid clusters in order to speed up subsequent searches, as described by Jacobs and Hendrickson (1997). This has the drawback of being more complicated than condensation (which consists in simply deleting all nodes forming the rigid cluster).). Subsequent searches are then done much faster since now one node must be checked instead of a whole subgraph.
2
f
e
b
c
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e
a ~,
,d ,
a)
':~-~' b) b c
c)
b
c
Fig. 9.4 An example of condensation in a two-dimensional B-framework (gd -- 2, Gd = 3). a) A redundant bar is found (dashed), and a new rigid graph is identified (shaded). The framework in this example consists of two separate rigid clusters which can rotate around node d. b) Direct reconnection: the rigid graph is condensed to a new body-node N', and bars (e, d), (g, d) and (c, d) are directly reconnected to N'. Direct reconnection of bars which are incident to a joint-node is not correct in this case. c) In order to preserve the rigid properties of the system after condensation, an auxiliary structure must instead be used to reconnect bars which are incident to a joint-node. This auxiliary structure is composed of the original joint-node (d in this figure), rigidly connected to N" by gd auxiliary bars (dashed). Condensation is implemented by first erasing all objects (nodes and bars) visited during the failed search, that is, all objects in C(a, b), and then reconnecting all external bars incident to them to a newly created body-node N'. In this reconnection step, a distinction is to be made between bars which are incident to a body-node in C(a, b) and those incident to a joint-node in C(a, b): the former are directly reconnected to N" while bars incident to joints in C(a, b) must be again connected to their joints (which are not deleted), and these in turn rigidly fixed to N" through an auxiliary structure made of gd bars. In order to see why auxiliary structures are necessary, consider the example shown in Fig. 9.4. Consider what would happen if all bars which are incident to node d were directly reconnected to N" (Fig. 9.4b): if next a bar (a, b) were added to this structure, a unique rigid cluster would be formed. But this does not reflect the rigid properties of the original structure (Fig. 9.4a), which would still be composed of two rigid clusters with a common joint d, after adding (a, b). The correct procedure for reconnecting bars which are incident to a joint (bars (c, d), (g, d) and (e, d)) is illustrated in Fig. 9.4c, and it is easy to see that the rigid properties of the system are unchanged by condensation in this case (Moukarzel, 1996).
9.2.5
Extracting the physics
As we have seen, the matching algorithm provides a way to identify redundant bonds or constraints. Here we will see how it is possible to obtain other physically important pieces of information from this algorithm.
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Each time a redundant bond is found, a new rigid (or connected) cluster is also identified. This rigid cluster is the set of bonds with respect to which the new bond is redundant, which is to say the bonds that would be stressed if the new bond were replaced by a spring. In the connectivity case, the matching algorithm identifies the loop which would carry a current if the redundant bond were a battery. Clearly, a naive application of this algorithm does not ensure that all rigid clusters will be identified. According to what we have discussed, only overconstrained clusters will be discovered for sure, i.e. those which have bonds in excess of rigidity. Thus condensation only happens for overconstrained clusters, and our labeling (each condensed set has a unique label) corresponds to a classification of the system into disjoint overconstrained clusters. Minimally rigid clusters (those rigid but not overconstrained) would thus in principle go unnoticed, since they have no redundant bonds, and the MA is only able to detect a rigid cluster when a redundancy is found. If one is interested in identifying all rigid clusters, i.e. given any pair of points one would like to know whether they are rigidly connected, there is a straightforward way to do so. The trick is simply to test a fictitious, auxiliary bond between these two nodes in the graph. If this auxiliary bond is found redundant (its last copy cannot be matched) then the nodes are rigidly connected. The fictitious-bond trick also serves to identify the backbone, i.e. the subset carrying the load between two given points, or, in connectivity, the currentcarrying subset. This is so because of the property already mentioned, that the subset visited during the failed search is the set of bonds that carry stress if the last, redundant, bond is replaced by a spring. A fictitious bond tested between two ends of a rigid sample will be redundant, and the set searched during the last (failed) BFS is the backbone. In the infinite percolating cluster, the red bonds are particularly important, as removal of any one of them disconnects the sample. The MA algorithm allows an easy identification of these critical bonds. When implementing the fictitiousbond trick across the sample, the rigid subgraph identified (the backbone) will, in general, be found to consist of overconstrained clusters (clusters that have been already condensed at least once) rigidly connected by bonds. These bonds necessarily provide a minimally rigid connection, otherwise some would be redundant and they would have been condensed. Therefore all bonds in this rigid subgraph are red bonds, also called cutting bonds. When studying the structure of glasses and amorphous solids one is often interested in their vibrational properties (Caprion et al., 1996; Dove et al., 1997; Fabian, 1997; Oligschleger and Sch6n, 1997). An important quantity is the number of soft modes orfloppy modes (Maxwell, 1864; Thorpe, 1983; Cai and Thorpe, 1989; Dove et al., 1997; Hammonds et al., 1997); the number of null eigenvalues of the dynamic matrix. Floppy modes correspond to infinitesimal deformations which cost no energy. Clearly, what we have discussed about infinitesimal rigidity
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in Section 9.2, is the static counterpart of these concepts, since floppy modes are simply non-trivial flexes. The number of floppy modes F equals the number of remaining degrees of freedom once all independent constraints have been taken into account and thus for joint-bar networks one has (9.2.4)
F=dn-Gd-b+R,
where b is the total number of bonds and R is the number of redundant bonds. Once we know the number of redundant bonds, the number of floppy modes is easily found from (9.2.4) (Franzblau and Tersoff, 1992; Jacobs and Thorpe, 1996; Duxbury et al., 1999). Early methods for counting the number of floppy modes were similar to those used to calculate the full density of states. In general, one relied in diagonalizing the dynamic matrix and then counting the number of zerofrequency eigenstates (He and Thorpe, 1985; Cai and Thorpe, 1989). Numerically this is an extremely hard task. A much better alternative consists in finding the number of redundant constraints with the help of the MA (Jacobs and Thorpe, 1995, 1996; Jacobs and Hendrickson, 1997; Jacobs, 1998; Duxbury et al., 1999) and using (9.2.4).
9.3 9.3.1
Rigidity percolation on triangular lattices Introduction
One of the first attempts to describe the mechanical properties of disordered systems using percolation ideas is due to de Gennes (1976), who suggested that the elastic modulus E of gels close to the gelation point might behave in the same way as the conductivity ]e ~ (p - pc) t of disordered systems near a percolation point (Stauffer and Aharony, 1994; Sahimi, 1995). It became clear later that these two problems are not equivalent, due to their differing tensorial characters. In the elastic problem vectors (forces or displacements) rather than scalars (electric charge) must be transmitted. In fact E behaves as (p - Prig) f , with f ~ t, where Prig is the rigidity threshold. It was numerical work by Feng and Sen (1984) which first suggested that the elastic-modulus exponent f is in general different from the conductivity exponent t. In this work the Born model (Born and Huang, 1954) of elasticity, - Z
+
(9.3.1)
(i,j)
was used, where ( U i - - /~j)[[,_k is the relative displacement of neighbors i j, respectively in the directions parallel and perpendicular to bond (i, j), and Kij is 1 with the stiffness of bond (i, j). Randomly bond-diluted lattices have Kij =
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probability p and K i j - - 0 with probability 1 - p. The present authors notice that conductivity and elasticity are only equivalent if a = b, in which case the elastic problem decouples into d independent scalar problems (A similar decoupling happens in central-force elastic systems which have zero-length springs (Note 2, p 317). This particular case of "scalar elasticity" is for example relevant for the mechanical properties of polymers and rubbers, in which central-forces are of entropic origin and can in some cases be similar to zero-length springs.). But elastic modulus and conductivity are not proportional to each other in general, as can be seen by taking b -- 0: this gives a central-force (CF) potential, in which case a higher rigidity threshold was found, and there was also evidence (Feng and Sen, 1984) that fCF > t. Later Kantor and Webman (1984) considered a potential including bondbending terms as well as central forces.
V = a Z
Kij(~li --
{tj)~ + b ~
KijKjk(6Oijk) 2
(9.3.2)
When b # 0, a change in the angle Oijk between a pair of adjacent bonds costs energy, as if the bonds had to be "bent". In two dimensions, the existence of BB forces makes the rigidity problem simple; any connected path is rigid, and therefore the geometry of the rigid clusters is dictated by simple connectivity. Therefore Prig -- Pc when BB forces are non-zero. For any non-zero b, the elastic properties near criticality are dominated by bond-bending forces (Kantor and Webman, 1984) because the energy involved in a typical bond-bending contribution scales down to zero faster with size, than that due to bond-stretching forces. Using the nodes-links-blobs model (de Gennes, 1976; Skal and Shklovskii, 1975; Coniglio, 1981 ; Pike and Stanley, 1981), Kantor and Webman showed that f8 8 >_ dv + 1 where v is the correlation-length exponent. This lower bound is definitely larger than t ~ 1.3 (in two dimensions), proving that elasticity and conductivity are not equivalent. Similar ideas were later considered by various authors (Feng et al., 1984; Roux, 1985, 1986; Sahimi, 1986), who refined the above bound to f88 < 2v + t, conjecturing that it might be an equality. This conjecture is consistent with the most accurate available simulations (Zabolitsky et al., 1986). Therefore, bond-bending (Kantor and Webman, 1984; Feng et al., 1984; Bergman, 1985; Sahimi, 1986; Feng, 1985; Zabolitsky et al., 1986; Roux, 1985, 1986; Wang, 1996b) elasticity is fairly well understood in two dimensions. In d > 2, bond-bending constraints are not enough to provide rigidity at the connectivity percolation point (Phillips and Thorpe, 1985), so that the situation might be different. The more subtle central-force rigidity problem has been studied by several authors (Feng and Sen, 1984; Day et al., 1986; Thorpe and Garboczi, 1987; Burton and Lambert, 1988; Roux and Hansen, 1988; Hansen and Roux, 1989; Guyon et al., 1990; Arbabi and Sahimi, 1993; Knackstedt and Sahimi, 1992; Jacobs and
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Thorpe, 1995; Moukarzel and Duxbury, 1995; Obukov, 1995; Jacobs and Thorpe, 1996; Moukarzel et al., 1997a; Jacobs and Thorpe, 1998), but the most basic issues such as the universality class have been controversial for some time. Early results indicate that the CF elastic problem belongs to a new universality class (Feng and Sen, 1984; Day et al., 1986), i.e. that both the geometrical (/3, v, y, or) and elastic exponents ( f ) are different from those of BB elasticity. On the other hand, there have been also claims on the contrary (Roux and Hansen, 1988; Hansen and Roux, 1989; Guyon et al., 1990; Hansen, 1990; Wang, 1996a), based on the argument that effective bond-bending forces would be prevalent at large scales. Other authors even suggested that the bond-diluted and site-diluted CF cases might behave differently (Arbabi and Sahimi, 1988, 1993; Knackstedt and Sahimi, 1992), i.e. that CF elasticity would not be universal in a broad sense. Some of the reasons why these early studies have been so inconclusive are as follows. 9 Lacking a combinatorial algorithm to identify rigid clusters, one had to rely on solving the force equations, which is extremely time-consuming and restricted the numerical analysis to relatively small systems (e.g. 80 • 80 (Hansen and Roux, 1989)). 9 The lack of good independent estimates for pC F seriously affects the measurement of critical indices (Hansen and Roux, 1989). 9 It has been suggested (Day et al., 1986) that CF rigidity is a particularly difficult task for conjugate-gradient iterative solvers, since there is an unusually large accumulation of roundoff errors. It should be noted that the central-force-rigidity problem is an idealization. Real materials have bond-bending terms and there are effective bond-bending and twist-energy terms introduced by temperature (Plischke and Jo6s, 1998; Jo6s et al., 1999). Nevertheless the central-force limit provides an important limiting case. In the regime between the central-force-rigidity threshold and the connectivity threshold "soft" materials are held together by bond-bending and twist-energy terms which can be quite weak.
9.3.2
Results from matching methods
The introduction of matching algorithms (MA) (Hendrickson, 1992; Moukarzel, 1996; Jacobs and Hendrickson, 1997) in the study of CF rigidity percolation has alleviated all of the problems described in the previous subsection. The geometrical properties of rigid clusters can now be obtained without knowing the stresses, which has enabled a large increase in the sizes accessible to numerical simulation. Matching algorithms were used in studies of two-dimensional systems containing 106 sites (Jacobs and Thorpe, 1995; Moukarzel and Duxbury, 1995; Jacobs and
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Thorpe, 1996; Moukarzel et al., 1997a) and more recently up to 1.6 x 107 sites (Moukarzel and Duxbury, 1999). It is possible to implement the MA in such a way that the exact percolation point is identified for each sample (Moukarzel and Duxbury, 1995; Moukarzel, 1996). Thus, measurements exactly at Pc can be done (Moukarzel and Duxbury, 1995; Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999) at all sizes, and one does not have to rely on data-collapse to estimate it together with the critical indices. This removes an important source of error (Hansen and Roux, 1989) from critical indices estimates. Regarding the convergence problems with iterative solvers, if the rigid backbone is identified with the MA, and subsequently the elastic equations are solved on the backbone alone, it has been found (Moukarzel and Duxbury, 1995) that the conjugate gradient performs much better, enabling the study of elastic properties on systems as large as 512 x 512. As a result of these recent studies, it has became clear that two-dimensional CF rigidity percolation does not belong to the universality class of connectivity percolation (Moukarzel and Duxbury, 1999) (and, consequently, of BB rigidity). Precise estimates for the correlation-length exponent v in CF rigidity percolation have been obtained from studies of bond- and site-diluted triangular lattices (Moukarzel and Duxbury, 1995, 1999): v cF -- 1.16(3) and (Jacobs and Thorpe, 1995, 1996) v cF = 1.21(6), which have demonstrated that CF rigidity is in a different universality class than connectivity percolation(where v -- 4/3). This difference is even more striking in tree models and in infinite range models where rigidity percolation is first order (Moukarzel et al., 1997b; Duxbury et al., 1999), whereas connectivity percolation is second order. In addition to CF rigidity, we also discuss a two-dimensional rigidity model in which each site is a rigid body with three degrees of freedom, and contiguous bodies are "pinned" at a common point by a universal joint, allowing the possibility of relative rotation. This site-diluted body-pin (BP) model (Moukarzel and Duxbury, 1999) is depicted in Fig. 9.5. Each joint or pin provides two constraints, and can therefore be represented by two bars, thus allowing the use of the MA. Although forces are not "central" in this model, we have found that CF and BP are in the same universality class. This is consistent with the idea (Adler et al., 1987; Guyon et al., 1990; Moukarzel et al., 1997b) that it is incomplete transmission of information what makes CF rigidity different from scalar connectivity. We also present results from a two-dimensional CF rigidity model, which we call the "braced square net" (BSN) and which has a first-order rigidity transition (Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). An undiluted square net with open boundary conditions is "at the verge of rigidity": having L 2 sites and 2L 2 _ 2L bars, there is a total of 2L - 3 "floppy modes" or infinitesimal flexes. This system can be rigidified by randomly adding diagonal bonds. Since the number of remaining floppy modes is not extensive, the density of diagonals Pd at
2 o
0
a)
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o
I i i r i i I i I I
o
o
o
0
0
0
0
0
0
9
o
0
b)
Fig. 9.5 Site-diluted square lattices, a) The joint-bar rigidity model. Each site is a point with two degrees of freedom, and each bond is a rotatable bar which fixes the distance between them, providing one constraint, b) The body-pin rigidity model. Each site contains a rigid body with three degrees of freedom, and neighboring bodies are pinned at a common point. Each such pin provides two constraints. the rigidity threshold goes to zero in the limit of large lattices. A similar behavior is obtained if the boundary conditions are periodic in one direction, or if rigid bus-bars are located on a pair of opposite sides (while keeping free boundaries in the other). The non-generic version of this problem has been studied by many authors (Bolker and Crapo, 1977; Dewdney, 1991; Obukov, 1995). Our numerical simulations correspond to the bus-bar boundary condition, and to generic rigidity. In numerical studies of percolation, when pc is not known exactly it is usual to take numerical measurements at closely spaced values of p, from which Pc, and also other information at Pc, are inferred later, through, for example, finite-size scaling. Since in the MA it is necessary to add bonds one at a time (in arbitrary order), we have found the following procedure more convenient: add bonds, one at a time, at random locations (this is equivalent to slowly increasing the bond occupation density p by the smallest possible steps). We check, after each bond addition, whether rigidity percolates. This check is not time consuming within the MA (Moukarzel, 1996), and serves to exactly detect the percolation point for each sample, thus allowing very precise measurements of critical properties. In order to detect when there is a system-spanning rigid cluster, we connect an additional fictitious bar or spring between the upper and lower bus-bars on opposite sides of the system. This auxiliary spring mimics the effect of an external load, and therefore the first time that a macroscopic rigid connection exists, a globally stressed region (the backbone) appears. We take an initially depleted lattice and add bonds (in the bond-diluted case) or sites (in the site-diluted case) to it one at a time as described above, and use the MA to identify the rigid clusters that are formed in the system. For the
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case of bond dilution, p is the density of present bonds, while in site dilution it indicates the density of present sites. We have used two distinct definitions of the critical point (Moukarzel and Duxbury, 1995). First, we determine whether an externally applied stress can be supported by the network, which we call applied stress (AS) percolation. Secondly we studied the percolation of internallystressed (IS) regions. At the AS percolation point, an applied stress is first able to be transmitted between the lower and upper sides of the sample. The stressed backbone is detected as a self-stressed cluster within the MA, because of the existence of the fictitious bond mentioned above. The IS critical point is defined as the bondor site density at which internal stresses percolate through the system. This means that the upper and lower sides of the system belong to the same selfstressed cluster (Moukarzel and Duxbury, 1995). This is easily detected within the matching algorithm (Moukarzel, 1996). The AS and IS definitions of percolation are in principle different, but we found (Moukarzel and Duxbury, 1995) that the average percolation threshold and the critical indices do not depend on the definition used for large lattices. Most of our simulations were done using the AS definition of the critical point. The rigidity thresholds can be estimated with good accuracy: pCsF = 0.6975(3) on site-diluted triangular lattices (Moukarzel and Duxbury, 1995" Jacobs and Thorpe, 1996), and pCF _ 0.6602(3) on bond-diluted triangular lattices (Jacobs and Thorpe, 1995, 1996). For the body-pin model on site-diluted square lattices, our numerical estimate is Pc -- 0.74877(5). We define the spanning cluster (Fig. 9.6) as the set of bonds that are rigidly connected to both sides of the sample. However, only a subset of these bonds carry the applied load. This subset is called the backbone. At the AS critical point, the backbone will always contain some cutting bonds (Stanley, 1977; Coniglio, 1981, 1982; Moukarzel and Duxbury, 1995), so named because the removal of any one of them leads to the loss of global load-carrying capability. Cutting bonds attain their maximum number exactly at pc (Coniglio, 1981, 1982). The backbone bonds which are not cutting bonds are parts of internally overconstrained blobs (self-stressed clusters). The spanning cluster also contains bonds which are rigidly connected to both ends of the sample but which do not carry any of the applied load. These are called dangling ends. This classification is standard in connectivity (scalar) percolation. The MA allows efficient identification of all these spanning-cluster subsets (dangling ends, backbone and cutting bonds). We analyze the size dependence of the following densities associated with the spanning-cluster: Ps, the backbone density, PD, the dangling-end density and Pec, the infinite-cluster density, exactly at the percolation threshold of each sample. In Fig. 9.7a-c, this data is presented for the three different cases in 3200, (b) L max 4096, Figs. 9.6a-c. Maximum sizes studied were" (a) L max 1024. and (c) L max -
-
-
-
-
-
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Fig. 9.6 Infinite percolation clusters, a) CF rigidity percolation on a site-diluted triangular lattice, b) Body-pin rigidity percolation on site-diluted square lattice, c) CF rigidity percolation on the braced square net. The system size is L = 64 and rigid bus-bars are set on the upper and lower ends of the sample. The backbone, is composed of "blobs" of internally stressed bonds (thick black lines), rigidly interconnected by cutting bonds (gray lines). Cutting bonds are also called red bonds. Removing one of them produces collapse of the system. Dangling ends (thin lines) are rigidly connected to the backbone, but do not add to the ability of these networks to carry an external load (current in the connectivity case). (From Moukarzel and Duxbury, 1999). 100 r
10 ~
10 .2
a)
/ 10
100
L
1000
10
b)
.............. iiii1 100
L
1000
10
r
r
100
L
9
9
1000
c)
Fig. 9.7 Density of backbone bonds (circles), dangling bonds (squares) and infinite-cluster bonds (diamonds) at the AS critical point, a) Rigidity percolation (g = 2, G = 3) on a site-diluted triangular lattice, b) Body rigidity (G -- g = 3) on a site-diluted square lattice. c) Rigidity on a randomly-braced square lattice. (From Moukarzel and Duxbury, 1999).
It is clear from Fig. 9.7 that the BSN (Fig. 9.7c) has a qualitatively different behavior than the other cases. For the BSN, P s , P ~ and PD all have a finite density at large L, indicating that the rigidity transition is first order in this case (Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). In contrast, in both rigidity cases (Figs. 9.7a,b), P8 and P ~ are decreasing in a power law fashion over the available size ranges. However, the behavior of PD is more complex. Let us first discuss the behavior of the backbone density P s .
M . J . Alava e t al.
286
1.2
1.0 /i
0.8 o
0.8
o
~/v
~/v
/
0.6
/.-
0.4
jY
0.4 _------.
o
o
o
0.2
/ Z/
z/ z
0.0
a)
0.0
0.2
0.4 0.6 1/In(L)
0.8
0.0
1.0
0.0
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0.4 1/In(L)
0.6
0.8
b)
Fig. 9.8 The spanning cluster density exponent/~/v as numerically estimated, a) Rigidity percolation on a triangular lattice (L max - - 3200). b) Body-pin rigidity on a square lattice (L max - - 4096). These plots show - l n P ( L ) / l n L vs. 1/lnL, which intercept the ordinate axis at a value equal to the estimated leading exponent ~/v. Solid lines are fits using P(L) -- C1L-~/v(1 + C2L-C~ Different estimates result in each case from fitting the scaling of spanning cluster density P~ (triangles) and dangling end density PD (circles). This indicates that finite-size effects are still too large. (From Moukarzel and Duxbury, 1999).
Taking into account corrections to scaling, the fit PB -- C I L - e ( 1 + C2 L-~~ I to the numerical data for the CF and BP cases lead to ~gr/V -- 0.22(2). In consequence the CF and BP rigid backbones are fractal at Pc, with a fractal dimension DB -- 1.78(2) (Moukarzel and Duxbury, 1995; Jacobs and Thorpe, 1995, 1996; Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). Now we consider Pc~ and PD. In the rigidity cases there are strong finite-size effects and even at sizes of L -- 3200 (joint-bar rigidity, Fig 9.7a), and L -- 4096 (body-pin rigidity, Fig. 9.7b), it looks as though the dangling probability may be saturating, while the infinite cluster density continues to decrease. Since P ~ = PB + PD, and PB --+ 0 more rapidly than Pc~, then P ~ and PD must have the same asymptotic behavior. The analysis in Fig. 9.8 demonstrates that the current data is not sufficiently precise to illustrate that fact. Clearly the numerical results for the range of system sizes currently available are still influenced by strong finite-size effects, and the results depend on the analysis method chosen. The dangling-end density and the spanning-cluster density are not displaying their asymptotic behavior in Fig. 9.7. Either P o starts to go down, or P ~ levels up, for large lattices. A fit to the P ~ data of Fig. 9.7a,b yields f l / v - 0.147(5) (see Fig. 9.8a,b). But a similar fit of the dangling end density gives f l / v ~ 0.03 for the joint-bar rigidity case (Fig. 9.8a) and f l / v ~ 0.01 for the body-pin rigidity case (Fig. 9.8b).
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Similar investigations by Jacobs and Thorpe (1995, 1996) using equivalent numerical methods produced results consistent with ours, but they chose to interpret the infinite-cluster probability as being key, disregarding the dangling-end behavior. This led them to conclude that, at the CF rigidity transition the spanning cluster is fractal. If on the other hand one were guided by the Cayley-tree results (Moukarzel et al., 1997b) for rigidity percolation, which indicate a first-order jump in the infinite-cluster probability, it is natural to interpret Fig. 9.7a,b as indicating a saturation of the infinite-cluster probability at the dangling-end value of about 0.1 (Moukarzel et al., 1997a) and thus/~ -- 0. Clearly much larger simulation sizes are required to find ~ / v precisely. Now we turn to the calculation of the correlation-length exponent for rigidity percolation. At a second-order transition there is a diverging correlation length ~ [ P - Pcl -v. In rigidity percolation, the correlation length is of the order of the mean size of a rigid cluster. But instead of measuring cluster sizes, which is not computationally efficient, it is customary (Stauffer and Aharony, 1994) to find the exponent v of this divergence by measuring the width of the spanning probability distribution as a function of size L. The spanning probability pSpanning(p, L) is the probability that a spanning cluster exists on a system of size L and density p of present sites (or bonds), and its derivative with respect to p is the probability PL (pc) that a spanning cluster appears for the first time at density p. The width ~ r ( L ) - ~ < p2 >L - < Pc >2) of this distribution, according to finite-size scaling, behaves as o-(L) ~ L -1/v. In order to measure v, it is usual to do simulations at fixed values of p near the estimated critical point, and then extract the width of PL (pc) from the resulting estimate of pSpanning(p, L). We can do much better than this with help of the MA, since it allows us to determine Pc exactly for each sample. Therefore we measure ~r (L) as the sampleto-sample fluctuation in pc. An asymptotic analysis for o-(L) is shown in Fig 9.9a for joint-bar rigidity and in Fig. 9.9b for body-pin rigidity. From these figures we estimate 1/v = 0.85(2) for CF and BP rigidity percolation. This result is consistent with other recent studies of CF rigidity percolation (Jacobs and Thorpe, 1995, 1996). In the case of the first-order rigidity on the braced square net, the variations in Pc behave as L -3/2, in accordance with analytical results for this model (Moukarzel, unpublished data). The MA also identifies the red bonds at the percolation point, for the case of AS percolation. The number NR of red bonds scales at Pc as L x. Coniglio (1981, 1982) has shown that x = 1/v exactly, for scalar percolation. Numerical evidence suggesting that x -- 1/v also in rigidity percolation was first presented by Moukarzel and Duxbury (1995). It is possible to extend Coniglio's reasoning to the case of central-force rigidity percolation (Moukarzel, unpublished data). It turns out that x -- 1/v has to be rigorously satisfied also in this case, and therefore or(L) and 1/NR(L) must have the same slope in a log-log plot. Analysis of the
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1.20
1.4
1.10
1/v 1.2
1/v 1.00
1.0
0.90
0.8
a)
0.0
0.2
0.4
0.6
0.8
0
o o
~oo ~
A
j:
~
f f.....
0.80 0.0
1.0
A
o
iz
0.2
0.4
0.6
0.8
b)
Fig. 9.9 The thermal exponent 1/v as numerically estimated, a) Rigidity percolation (g = 2, G -- 3) on a triangular lattice, b) Body rigidity on a square lattice (G = g = 3). Two independent estimates result for each case from fitting the scaling of red bonds (triangles) and fluctuations in Pc (circles). The closeness of these estimates indicates that CF and BP rigidity are in the same universality class. (From Moukarzel and Duxbury, 1999).
number of cutting bonds is also presented in Fig. 9.9, and yields values of 1/v consistent with the analysis of variations in percolation thresholds described in the previous paragraphs. It has recently been demonstrated that the number of floppy modes behaves as a free energy for the rigidity-percolation problem, and of course also for the connectivity-percolation problem (Duxbury et al., 1999). The specific-heat exponent can then be extracted from the second derivative of the number of floppy modes. Data from the analysis of floppy modes in the bond-diluted triangular lattice are presented in Fig. 9.10. From this data, the specific heat exponent = - 0 . 4 8 ( 5 ) was extracted. This in combination with the exponent v = 1.16(3) violate the hyperscaling relation (dr = 2 - c~), though only by a small amount. Possible rational values which are close to the numerical estimates and which obey hyperscaling are: (v = 6/5, ot = - 2 / 5 ) ; (v = 5/4, ot = - 1 / 2 ) ; (v = 7/6, ot = - 1 / 3 ) . Assuming that rigidity percolation is conformal invariant it would be useful to find its central charge and hence to determine whether the exponents can be expected to be simple rationals. These results provide further strong evidence that rigidity percolation is not in the same universality class as scalar percolation. It is interesting to notice that CF and BP are in the same universality class, although the forces involved in BP rigidity are not at all central. It therefore seems appropriate to say that what characterizes the universality class of rigidity percolation (at least in two dimensions) is not the central character of the forces, but whether a bond provides complete information or not. CF and B P models both have incomplete transmission of
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60
o L=680 J::JL=960 9L=1150 Fit
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bond concentration p Fig. 9.10 The second derivative of the number of floppy modes on bond-diluted triangular lattices. (From Jacobs and Thorpe, 1996). information. One bond connecting a site A to a fixed "background" is not enough to fix all degrees of freedom of site A in both cases. If we keep having bodies on the lattice sites but now each bond is a rigid "beam", we recover the universality class of BB elasticity, because one such beam is enough to completely fix a body relative to another. Partial transmission of information along a bond is relevant to rigidity and also to diluted magnets (Adler et al., 1987; Guyon et al., 1990; Moukarzel et al., 1997b).
9.4
Connectivity percolation
As mentioned in Section 9.2.2, the simplest case in which the matching algorithm is useful is s c a l a r connectivity. This corresponds to gd = G d = 1 since now the only degree of freedom is the "electric potential" of the sites, and it is easy to see that a Laman-type theorem trivially holds. Therefore everything we
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derived for rigidity percolation can be applied, without conceptual modification, to connectivity percolation. Of course, for the much simpler problem of connectivity percolation there exist extremely efficient integer algorithms (Hoshen and Kopelman, 1976; Leath, 1976b; Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984; Hoshen et al., 1997) for the identification of connected clusters, so it would hardly make sense to apply the relatively more complex MA for this task. But the same cannot be said if we are interested in backbone properties or red bonds. In the physics community the burning algorithm (Herrmann et al., 1984; Herrmann and Stanley, 1984) is the most popular backbone algorithm, although this procedure is relatively slow and does not allow the study of large systems unless careful optimization is done (Rintoul and Nakanishi, 1992, 1994; Porto et al., 1997). For the particular case of planar graphs, a very efficient algorithm has been implemented (Grassberger, 1992a, 1999). There are also other efficient algorithms available for higher dimensions (Tarjan, 1972), though they have not been applied to percolation as yet. The MA is particularly well suited for backbone identification in general, and it has been used for connectivity percolation in two to five dimensions (Moukarzel, 1998a). These studies provided backbone exponents which are among the most precise presently available. Also very good precision is obtained for the thermal exponent 1/v and threshold values Pc in dimensions three, four and five. The MA for connectivity percolation follows from the procedures described in Sections 9.2.3 and 9.2.5 for the particular values Gd = gd = 1. But since the algorithm takes a simple form in this case, it is worth explaining it in the following simplified terms, which are specific to the connectivity case (Moukarzel, 1998a): consider a system of n sites i -- 1. . . . . n connected by an arbitrary set E of b bonds (i, j) connecting sites i and j. The matching algorithm can be thought of as a way to identify and remove loops, and is implemented using a directed graph D as an auxiliary representation of the system. In this directed graph, lattice sites are represented by nodes i and bonds by directed edges (i, j). We may think of each edge as an arrow. These can be pointing in either direction, subject to the constraint that no node has more than one arrow pointing to it. A node with an incoming arrow will be said to be covered by the corresponding edge. A node with no incoming arrows is uncovered. The directed graph D will be kept loop-less. In order to do this, each time a closed loop (a cycle, or circuit) is identified it will be removed from D, and replaced by a loop-node (a body-node in the terminology of Section 9.2.3). A loop-node is a node in 79 that represents a deleted, or c o n d e n s e d loop. Therefore, although initially there are as many nodes in 79 as there are sites in our system, as the algorithm proceeds we will delete some nodes and create loop-nodes. These loop-nodes will be given a loop-label, which is the minimum of all edge- and node-labels contained in the loop.
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We start from a graph 79 initially containing n nodes and no edges, and add edges one at a time. Before adding an edge ab ~ E to 79, the following test is done in order to know if a loop is closed by ab. We attempt to reorient the existing arrows on 79, in order to uncover both a and b, without ever pointing two arrows to the same node. Since, by hypothesis, 79 without edge ab has no loops, it must always be possible to uncover one of them. Let us then assume that a is uncovered first. If after doing this b can also be uncovered (while keeping a uncovered), then the new edge ab does not close a loop, and therefore it is definitively added to 79. This means: we add an arrow between a and b, pointing to either of them (Fig. 9.1 1). If on the other hand it is not possible to uncover b while keeping a uncovered (Fig. 9.12), this necessarily means that a loop would be formed on 79 by the addition of ab. In this case, edge ab is not added to 79 but the following is done instead: starting from b (covered), we follow the arrows backwards. This will necessarily lead to a, thus identifying the new loop. All edges in this loop are
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given a common loop label Imin, which is the minimum amongst all node- and edge-labels in the loop (including nodes a and b and edge ab). Next all nodes and edges in the loop are removed from 79 and replaced by a node with label Imin. This is the condensation step. We now present results for scalar percolation on site-diluted hypercubic lattices in two to five dimensions (Moukarzel, 1998a). The largest sizes studied were L = 4096 in two dimensions, L = 256 in three dimensions, L = 60 in four dimensions and L = 26 in five dimensions. These upper limits correspond to approximately 12-17 million sites, and stem from the need to hold (in the present implementation of the algorithm) the whole system at once in memory, i.e. the algorithm is memory limited. The results described here were obtained on an Alpha500 workstation with 512Mb RAM. Since sites are added one at a time until the percolation point is reached, it is possible to measure, for each sample, the critical density pc of occupied sites. We
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assume that finite-size corrections in sample-averages Pc (L) behave as Pc (L) = pc(OO) [1 + a L -w~ (1 + bL-W2)] and fit this expression to our data with five free parameters. In doing so it is found that the leading correction exponent w] is consistent with the values of 1/v estimated by us (see later). So if now wl is fixed to be l/v, we obtain: pc = 0.59273(2) in two dimensions; pc = 0.31162(3) in three dimensions; pc = 0.19682(5) in four dimensions; pc = 0.1408(2) in five dimensions. These values are consistent with previous work (Grassberger, 1992a; Ziff, 1992; Lorenz and Ziff, 1998; Ballesteros et al., 1999). The fluctuation crt` = < p2 > t` _ < Pc >2, where < > t. indicates averages over samples of size L, is expected to scale as at. ~ L -1/v with system size. On the other hand Coniglio (1981, 1982) has shown that the number Rt` of red bonds grows at pc as L 1/v. Therefore by measuring the number of red bonds Rt` (as described in Section 9.2.5) and CrL we get two independent estimates for the thermal exponent 1/v. Using this method, we find values of v which are also consistent with previous work, though the memory requirements of the matching method mean that invasion methods (see Section 6.2) are more effective in finding pc, v and/~. However, the matching method is very effective in finding the backbone. Data for the fraction B ( L ) of bonds on the backbone at pc are shown in Fig. 9.13. Assuming power-law corrections to scaling of the form B ( L ) = )~L-IY/v(1 + aL-~176
(9.4.1)
we estimate (Moukarzel, 1998a) db = 1.650(5) (two dimensions), 1.86(1) (three dimensions), 1.95(5) (four dimensions) and 2.00(5) (five dimensions), which are consistent with results found using the burning algorithm and depth-first search (Grassberger, 1992a; Rintoul and Nakanishi, 1992, 1994; Grassberger, 1999).
9.5
Applications to soft materials
As stated in Section 9.2, there is no analog of Laman's theorem for bar-joint graphs in arbitrary dimensions. However, for body-bar systems an extension of Laman's theorem is proven and an extension of the matching algorithms has also been demonstrated (Tay, 1985; Tay and Whiteley, 1985; Moukarzel, 1996). In addition, and of practical significance, it has been proposed that glasses and proteins which have weak "twist forces" but strong "flex" forces in three dimensions can be analyzed using matching methods (Jacobs, 1998; Jacobs et al., 1999; Thorpe et al., 1999). In these materials, it is assumed that the twist force is negligible and is the cause of the flexibility of the molecule. To illustrate the concept, we present (Fig. 9.14) a segment of a typical biomolecule. Bonds around which there is a circulating arrow indicate that the twist energy around that bond is very low.
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In contrast, the central-force and bond-flex energies are comparatively large. In the rigidity algorithms, the twist energy is assumed to be zero, and we seek to determine which bonds are rigid with respect to each other. Often systems of this sort are called "bond-bending" networks. As a point of reference a ring of six two-fold atoms (with zero twist energy) bonded in a ring is still rigid. However, a five-fold ring is stressed (overconstrained) while a seven-fold ring is floppy. The bond-bending force can be represented by next-nearest-neighbor bars. The presence of these next-nearest-neighbor bars eliminates configurations such as Fig. 9.2a which cause a violation of Laman's theorem in three dimensions. For this reason, bond-bending networks in three dimensions can be studied using matching methods. Jacobs (1998) has developed a matching algorithm for this problem and it has been applied to glasses (Thorpe et al., 1999) and proteins in three dimensions (Jacobs et al., 1999). In chalcoginide glasses (e.g. SexGel-x) it has been suggested (Phillips, 1979, 1981; Thorpe, 1983; Phillips and Thorpe, 1985) that the rigidity transition point is correlated with their glass-forming ability and with a change in properties as a function of composition. The limiting cases, Se (which is two-fold) and Ge (which is four-fold) are "floppy" and "rigid", respectively. The rigidity transition predicted by constraint-counting mean-field
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Fig. 9.14 An amino acid sequence starting at the N-terminus of a polypeptide chain. The side groups (residues) are denoted by R. (D. J. Jacobs and M. F. Thorpe, unpublished).
theory is x = 2.4. The actual transition is very close to this point, but although the mean-field and Bethe-lattice calculations indicate a first-order transition, while the numerical analysis indicates that the transition is second order. The combinatorial algorithms provide unprecedented precision in testing the properties of chalcogenide glasses as a function of composition. Proteins and enzymes are often close to their rigidity threshold. Hydrogen bonds determine whether they are rigid or floppy and hydrogen bonding also determines much of the tertiary structure such as or-helices and /J-sheets. A systematic study of the effect of hydrogen bonding on the rigidity of biomolecules can be carried out using the rigidity algorithm. Typically, as hydrogen bonds are added the structure changes from floppy to rigid. An example of the rigid cluster structure of an important protein near the rigidity transition is demonstrated in Fig. 9.15. The practical consequences of this intriguing application of matching methods are currently being pursued (Jacobs et al., 1999).
10
Closing remarks
We have given an introduction to some very interesting algorithms which have wide applications in quenched disordered systems. There are several ways in which this area of research is attractive from an esthetic point of view. First, it is possible to find exact solutions to problems that seem pretty hard. That is always nice. Secondly, in this area there is very close contact between combinatorics, computer science and physics which makes the analysis of algorithms rigorous and complete, where possible. Finally, there is still a great deal to be done, both in finding and applying new techniques from the computer science community and also expanding the physics applications of the algorithms described here. Anyone studying a problem concerning the low-temperature equilibrium
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Fig. 9.15 Rigid regions of HIV protease. Only the main chain is shown for clarity. There is a single large rigid region shown in black and various flexible pieces shown in alternating light and dark grey colors. Hydrogen bonds are shown as dashed lines. (D. J. Jacobs, A. J. Rader, L. A. Kuhn and M. E Thorpe (unpublished)).
phases of disordered systems should know about and should seriously consider using optimization methods. Many of the algorithms that are needed are standard packages which can be downloaded from the intemet (see Note 1 at the end of the References) (e.g the LEDA library). It used to be assumed that models of "non-trivial" glassy behavior required NP-complete models such as spin-glass models in three dimensions. But that is clearly not correct as the random-manifold, random-surface and multipledirected-polymer problems are "glassy" (e.g. chaotic) but they are solved by polynomial algorithms. It is thus important to search for polynomial models of other glassy problems. Even more importantly it would be useful to compare polynomially solvable models with those that are NP-complete in order to identify the factors which make the problem computationally hard. Once those factors were identified it would be possible to decide if they are "relevant" from a physics perspective. It is likely that in some cases the features which make a problem computationally hard are not important physically, while in other, perhaps more interesting problems, the opposite would be true. An unexpected, to us, area of opportunity which results from the mappings described herein, and the much more extensive body of results in the mathematics
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literature (Lov~isz and Plummer, 1986; Ahuja et al., 1993a, b; van Lint and Wilson, 1996), is to develop new rigorous results for disordered systems. The flow and matching representations of disordered systems introduce a vast array of theorems and results, which are largely unused in the disordered-systems community. These results form the basis of many theorems in combinatorics, number theory and in combinatorial optimization. It is likely they can also have a broad impact in rigorous and analytic approximations to the physics of disordered systems. Finally, it is likely that ideas generated by solving these physics problems will, in turn, affect the way computer scientists think about bounds on algorithmic convergence. For example in physics we know that proximity to a critical point strongly affects the convergence of many algorithms. It is then natural to ask whether computational time is sometimes a critical variable, with a singular behavior near critical points. Similarly the structure of the growth front during greedy algorithms should be related to computational efficiency. Several fruitful ideas along these lines are already in the literature (see Monasson et al., 1999).
Acknowledgments
MA thanks Eira Sepp~il~i for providing several figures and Attilio Stella and the Dipartimento di Fisica, Universita degli studi di Padova for their kind hospitality. V. Pet~ij~i and M. Stenlund are thanked for collaborations, the Academy of Finland for several grants, and the Nordic Research Institute for Theoretical Physics (NORDITA) for hospitability and support during 1996-1998. PMD thanks Bruce Hendrickson for stimulating conversations about graph algorithms, and Paul Leath, Sorin Bastea, Don Jacobs and Mike Thorpe for profitable discussions and collaborations. He thanks Jan Meinke for providing several figures and the DOE for support under contract DE-FG02-90ER45418. CM acknowledges financial support from FAPERJ. HR wishes to express his special thanks to U. B lasum, J. Kisker, F. Pfeiffer, G. Schrrder, T. Knetter, N. Kawashima, V. Pet~ij~i, M. Diehl, M. Jiinger, G. Reinelt, and G. Rinaldi for a very fruitful collaboration on various topics treated in this review. He is also grateful to the J. von Neumann Institute for Computing (NIC) c/o Forschungszentrum Jiilich and the Institut fur Theoretische Physik of the Universit~it zu Krln for generous hospitality and support. Moreover, he acknowledges important financial support from the Deutsche Forschungsgemeinschaft (DFG).
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Note 2 A similar decoupling happens in central-force elastic systems which have zero-length springs. This particular case of "scalar elasticity" is for example relevant for the mechanical properties of polymers and rubbers, in which centralforces are of entropic origin and can in some cases be similar to zero-length springs.
Index ,-circuit, 34-5 ,-clusters, 44 ,-path, 34 Abrikosov lattices, 235-7, 238, 240, 261 Admissible arcs, 164, 165 Agreement percolation, 91-4, 100-11 Aizenman-Higuchi theorem, 98 Aizenman-Wehr theorem, 214 Algorithms, complexity of, 149-50, 163 Almost-Markov property s e e quasilocality Amenability, 38 Applied stress (AS) percolation, 284 Ashkin-Teller model, 104 Assignment problem, 179 Augmenting path algorithms, 163, 179 Augmenting path theorem, 162, 176-7 Automorphisms of lattices, 8 Axial next-nearest-neighbour Ising model (ANNNI), 197 s e e a l s o frustrated magnets Backbones, 152 Bak-Sneppen evolution model, 195 Bernoulli percolation, 31-6, 42-4 bond percolation, 36, 47-8 site percolation, 32-6 s e e a l s o dependent percolation Bethe lattice s e e regular tree Biconnected paths, 152 Bipartite graphs, 177-9 Bipartite matching algorithm, 265 Blossom, 180 Boltzmann-Gibbs distributions, 10 s e e a l s o Gibbs distributions, finite volume Bond diluted Ising model (BDIM), 113-116, 204 Bond percolation, 36, 47-8 Boolean models, 124 Born model, 279-80 Braced square net model (BSN), 282-7 Bragg glasses, 235, 241,258-61 Breadth-first search (BFS) algorithm, 151 Burning algorithm, 152, 188-9, 290 Burton-Keane uniqueness theorem, 43-4 Capacitated network, 160 Cayley tree s e e regular tree Chaos, 216-17, 230-1,249-52, 259 Chemical distance, 152
Chinese postman problem, 225 Cluster algorithms, 67-9 Cluster size, 35-6 Cluster theorem, 276 Complete analyticity, 84 Configuration, 7-8 Connectivity percolation, 152, 188-90, 289-93 Connectivity percolation problem, 188 Connectivity theorem, 271-2 Continuum percolation, 123-5 Contour loops (CLs), 260 Convex cost problem, 168-75 Convex cost programming, 182-6 Counting condition, 272 Coupling, 21-3 optimal, 22-3 Coupling inequality, 22 C-random cluster measures, 60, 62-5 Critical behaviour, 32-5, 37, 195-6 Cutting plane algorithm, 226-7 Cycle cancelling algorithm, 171 Dependent percolation, 37-42 Ising model, 39-41, 65-6 s e e a l s o Bernoulli percolation; random-cluster model Depth first search algorithm, 152 Dijkstra's algorithm, 154-6, 172 Diluted anti-ferromagnets in a field (DAFF), 197, 207-12, 216-21 s e e a l s o random-field Ising magnets (RFIM) Directed cycles, 154 Directed polymer in random media (DPRM), 187, 190-5,204-7, 234, 237, 241-6 Disagreement percolation, 73-8, 121-3 Disordered elastic media, 241,252-64 Disordered flux arrays, 235-4 1 Disordered systems, 112-23 Distance labels, 153 Dobrushin-Lanford-Ruelle (DLR) states, 10-11 s e e a l s o Gibbs measures Dobrushin's uniqueness condition, 77-8 Domain wall renormalization group (DWRG), 232, 248, 249-50 Dual linear programming problem, 183 Edmonds' algorithm, 180-1,233 Edmonds' theorem, 181 Edwards-Anderson model, 221-3, 228
Index
Edwards-Sokal coupling, 48-9, 53, 58-9 Elastic glasses, 234-5, 239-41,252-5, 258-64 Euclidean matching problem, 197, 233-4 Exact simulation, 69 Exponentially weak-mixing, 72, 77, 84-91, 116-23 Extremal dynamics, 195-6 Flow algorithms, 158-74 Flow augmentation lemma, 173-4 Flow network, 160-2 Flux lattice, 235-6, 239-41 Flux-line lattice s e e Abrikosov lattice Forest fire algorithm, 188-9 Fortuin-Kasteleyn-Ginibre (FKG) inequality, 25-6 Fortuin-Kasteleyn model s e e random-cluster model Free boundary condition, 52-3 Frustrated magnets, 197 Fully packed loops (FPLs), 260-1 Gauge glasses s e e vortex glasses Gibbs distributions, finite volume, 12 Gibbs measures, 9-13 correlation decay, 72 extremal, 13-14 infinite volume, 11, 12-13 multiple, 13-14 as phases, 14 structure of set of, 13-14 uniqueness, 72-8 Gibbs systems, computer simulation of, 67-9 Ginzburg-Landau theory (GL-theory), 236-7 Gluck's theorem, 268 Graph, locally finite, 6 Graphical distribution, 79-82 Graph theory, terminology, 6-8, 148-9 Greedy algorithms, 150-1,155, 195-6 Grey measure s e e graphical distribution Griffiths regime, 116-17 Growth algorithms, for percolation, 189 Hamiltonian, 9 relative, 10 XY-, 238 Hard core (lattice gas) model, 18-20, 77, 108 agreement percolation, 103 stochastic domination, 30-1 Hendrickson's matching algorithm, 270 modified, 276-9 Hendrickson's theorem, 273 Holley's inequality, 24-5 Hoshen-Kopelman algorithm, 189 Hottest-bond algorithm, 195 Imbalance, 171 Independent percolation s e e Bernoulli percolation Indicator function, 9 Infinite cluster, 32-5 Infinite clusters, number of, 42-7 Infinitesimal rigidity, 266-8, 267 Integer linear programming (ILP), 182 Internal stress (IS) percolation, 284 Invasion algorithms, 189-90 Invasion percolation, 69
319
Ising model antiferromagnetic, 17-18, 30-1, 47-55, 107-8 agreement percolation, 102-3 application of random cluster measures to percolation, 65 computer simulation, 67-9 dependent percolation, 40-1 ferromagnetic, 4, 15-17, 55-9, 75-7, 87-8, 106-7 agreement percolation, 94-100 diluted, 115, 116 stochastic domination, 26-30 random, 117-23 s e e a l s o random-bond magnets; random-field Ising magnets (RFIM); random Ising magnets; triangular Ising solid-on-solid model (TISOS) Ising spin glasses, 197, 221-34 Kardar-Parisi-Zhang (KPZ)equation, 187, 192, 194 Kruskal's algorithm, 158 Label correcting algorithms, 153 minimal path, 156-7 Label setting algorithms, 153 Laman's theorem, 270, 289-90, 293-5 body Laman theorem, 271 Lattices, 6-8 Law of large numbers, 14 Leath algorithm, 189 Lily pond model, 124 Linear programming, 181-3 Loops, 260-1 Loops s e e contour loops (CLs); fully packed loops (FPLs) Markov chain Monte Carlo method, 68 Markov random field, 11 s e e a l s o quasilocality Matching algorithms, 175-81,272-6 Matching problems, history of, 175 Matchings, terms, 175 Maximum-cardinality bipartite-matching algorithm, 179 Maximum-cut problem, 225-7 Maximum-flow problem, 160-7, 199-200, 208-11, 255-6 Maximum matching on general graphs, 179-81 Maximum-weight algorithm, 175 Maximum-weight forest (MWF), 155 Maximum-weight-matching problem, 179, 184 Minimal-energy-interface problem, 196 Minimal path, 153-7 Minimal-path algorithm, 192-5 Minimal-path problem, 194--5 Minimal-path tree, 153--4 Minimal-spanning trees, 155, 157-8 Minimum-cost-flow problem, 167-9, 175, 183, 248-52, 254-5 Minimum-cost path, 153 Minimum-cut/maximum-flow theorem, 161-2 Minimum-cut problem, 226 Minimum-energy interface problem, 196, 199-200 Mixed integer linear programming (MILP), 185-6
320 Monte Carlo method, 68 Negative cycle cancelling algorithm, 169-70 Negative cycle cancelling theorem, 171 Non-deterministic polynomial (NP) algorithms, 150 Non-deterministic polynomial-complete (NP-complete) algorithms, 150 Observables, 8-9 quasilocal, 8 Open cluster, 32, 36 infinite, 32, 42-7 Open path, 32, 36 Order parameter, detection of phase transition, 13 Overlap, 217 Papangelou intensities, 127-8 Percolation theory, 4-5 see also Bernoullis percolation Perfect simulation, 69 Periodic elastic media, 234-5 Phase transitions, 13, 13-14, 28-9, 55, 92-4, 105-11, 113-16, 125-9 Pirogov-Sinai theory, 5, 88, 92-3 Plaquettes, 109-11 frustrated, 222-5 Poisson blob model, 124 Poisson processes, 123-5 Poisson random edge model, 124-5 Polynomial (P) algorithms, 149-50 Positive correlation, 25, 28, 47 Potential lemma, 173 Potts model, 18, 55-9, 108, 109 computer simulation, 67-9 constant spin clusters in, 100-2 random, 113-15 and random cluster model, 47-59 stochastic domination, 30 Power law singularities, 116 Predecessor labels, 153 Preflow, 164 Preflow-pustdrelabel algorithm, 165-6 Preprocess procedure, 165 Primal linear programming problem, 182-3 Prim's algorithm, I57-8, 190 Probability measures finite energy, 43 monotone, 25 with positive correlations, 25, 28, 47 weak topology of, 9 Propp--Wilson algorithm, 69 Pseudo-flow, 171 Pseudo-polynomial algorithms, 163 Push-relabel algorithms, 163-6 Quasilocality, 11 Quenched magnetization, 114-15 Random-bond magnets interfaces in, 197-207 see also random-surface model Random-cluster model, 47-55, 59-65, 70-2, 78-84, 100-2, 104, 113-14, 119, 126
Index
Random-connection model, 124-5 Random-field Ising magnets (RFIM), 196-7, 207-21,207-27 see also Ising model Random fields, 9 Random Ising magnets, 113-16, 196-234 see also diluted anti-ferromagnets in a field (DAFF); Ising model Random-phase sine-Gordon model, 239-4 1,252-3 Random-surface model, 241,252-5, 261-4 Random-surface problem, 255-6 Reduced cost optimality theorem, 173 Reduced costs, 153, 172 Regular trees, 6, 66-7 Residual costs, 168 Rigidity percolation, 279-89 Rigidity theory, 264-95 Roughness exponent, 200 Sandwiching inequality, 27-8, 61 Scaling algorithm, 166 Search algorithms, 151-2 for percolation, 188-9 Self-organized criticality, 189-90 Sensitivity analysis, 166 Shortest-path algorithms, 153-7 Single-site heat-bath algorithm, 68-9 Site percolation, 32-6, 42-7 Site random-cluster measure, 70-2 Solid-on-solid model (SOS), 41,252-5 continuum, 198 see also triangular Ising solid-on-solid model (TISOS) Spanning clusters, 284 Spanning trees, 153 Spin glasses, 197, 221-34 Stochastic domination, 21, 23-31, 78-84 Strassen's theorem, 23-4 Strong duality lemma, 183 Successive shortest-path algorithm, 163, 171-4, 246 Sugihara-Hendrickson theorem, 274 Superconductors, 235-4 1 Super-rough phase, 243-6, 258-61 Swendsen-Wang algorithm, 69 Tail tr-algebra, 8, 13-14 Tay's theorem, 271 Transfer matrix algorithm, 193-4, 200 Travelling salesman problem, 185-6 Triangular Ising solid-on-solid model (TISOS), 256-8 see also solid-on-solid model (SOS) Two-level systems (TLS), 218-20 Uniqueness theorem, 43-4, 76-9 Vortex glasses, 234, 238-9, 246-8 Widom-Rowlinson model continuum, 125-9 lattice, 20, 30, 70-2, 77, 103-4, 108 Wired-boundary condition, 52-3 XY spin glass, 239
