Random Fields

Lecture Notes in Mathematics Edited by A. Dold and 13. Eckn'lann

534 Chris Preston

Random Fields

Springer-Verlag Berlin. Heidelberg New York 197 6

Author C h r i s Preston King's C o l l e g e C a m b r i d g e / G r e a t Britain

Library of Congress Cataloging in Publication Data

Preston, Christopher J Random fields. (Lecture notes in mathematics ; 534) Bibliography: p. Includes index. 1. Stochastic processes. 2. Measure theory. 3. Statistical mechanics. 4. Equilibrium. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 534. QA3.L28 vol. 534 [9A274] 510'.8s [519.2] 76-26664

AMS Subject Classifications (1970): 28A35, 6 0 G X X , 60K35, 8 2 A 0 5

ISBN 3-540-07852-5 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-38?-0?852-5 Springer-Verlag New Y o r k . Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

Preface

in the last decade there has been a lot of mathematical iuterest in models from classical e~ailibrium statistical mechanics; these notes describe some of this work. ~hey are concerned, in particular, with the properties of equilibrium states defined in terms of conditional probabilities. ~lis way of defining equi-. libri1~a! states is due to Dobrushin, Lanford ~ d

Ruelie; the formulation given

here is due mainly to F611mer. The approach taken will be fairly abstract, and will be done using the language and basic techniques of probability theory. It ~Ti]_l thus be assumed that the reader has some fs~iliarity ~ith things like standard measure theory, conditional expectations, the martingale convergence theorem, ~id probability kernels. Some of the deeper results will be obtained using standard Bore] spaces, but no previous ~ o w l e d g e of such objects will be required. These notes were written between 1974 and the present; the first six sections were written in the academic year 1974-75, while the attthor was a Fellow of Brasenose College, Oxford. The rest was written ~ i l e

the aubho~~ was

Fellow of King's College, Cambridge. The material has been much influenced b~" conversations with Harls ~611mer over the last three years, and many thanks are due to him.

Chris Preston King's College, Csmbridge, February, 1976.

RANDOM FIELDS

Section

i.

Introduction

. . . . . . . . . . . . . . . . . . . .

Section

2.

Random

and specifications

Section

3.

Existence

of Gibbs

Section

4.

Invariant

specifications

Section

5.

Lattice

Section

6.

Continuous

Section

7.

Specific

Section

8.

Some thermodynamics

Section

9.

Attractive

fields

models

models:

states

..........

33

..............

46

point processes

........

gain . . . . . . . . . . . . . . .

................

specifications

ii

..............

. . . . . . . . . . . . . . . . . .

information

1

..............

59

87

111

137

160

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

Index

198

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I.i

i. Introduction The intention of these notes is to set up a framework within which much of the equilibrium behaviour of a lot of models from classical statistical mechanics can be described. The main reason for doing this is to determine those features which are c o , o n

to almost all models from equilibrium stat-

istical mechanics. This will emphasise which kinds of behaviour are "general" as opposed to being consequences of a specific model.

(The type of behaviour

we consider will thus be qualitative rather than quantitative.) This approach will also help to unify subjects which are usually treated separately; in particular, the framework will cover both lattice and continuous systems. It will turn out that the framework looks a lot like the study of stochastic processes which are indexed by an arbitrary partially ordered set (rather than tile usual indexing by some subset of the real line). T~his suggests the possibility of adapting ideas and techniques from the theory of stochastic processes in order to give results about statistical mechanical models. In particular, we will use the language of probability theory, and will assume that the reader is familiar with the basic definitions and results of this theory (as found for example in Chapters I and II of Meyer (1966)). The approach that will be taken is based on the work of Dobrushin (1968a),(1968b),(1968c),(1969), These

Lanford and Ruelle (1969), and Ruelle (1970).

papers deal with the definition of equilibrium states for various

models and the properties of the equilibrium states; the reader is stongly encouraged to study them. The set-up which will be considered in these notes is a natural generalization of the Dobrushin-Lanford-Ruelle work, and is due to F~llmer (1975a),(1975b). A lot of the material which we will present is adapted from some earlier notes (Preston (1974a)) which dealt with a special case of Follmer's work.

1.2

2

Perhaps

the best way to start out is to describe a fairly simple

example and to use this to lead into the general lattice model, given as follows. a set of sites

(or locations);

d > I (where

Zd

are described

by a set

t ~ S

Y

, and that 0

and

the presence of a particle;

denoted by

+

and

in a positively cartesian

, with

+

(resp negatively)

product

X

=

~

S

S

is a

representing

will be

Zd

for some

space that have

there is an entity whose states

Y

suppose

is finite.

i , with

0

that all the For example,

representing

or alternatively (resp

set

in d-dimensional

Yt ; and for simplicity

have two points, denoted by i

and in most models

At each site

copies of the same space

and

There is a countable

denotes the set of points

integer coordinates).

set-up. The example

Y

are could

the absence

the two points could be

- ) representing

charged

Yt

when the entity is

state. The basic phase space is the

Yt ' which we will regard as a compact Hausd-

t g S orf space

(giving

ogy). Let

~

X

the product

topology and each

denote the Borel u-field on

Y

space, the evolution being governed by a Hamiltonian statistical

Hamiltonian which probability

mechanics

measures

on

for the model).

context

is due to Dobrushin,

above).

Equilibrium

right conditional

(X,~)

by formnlas our example,

as its phase

of some kind. O n e o f

represent without

the

from the

the equilibrium explicitly

const-

The way that this is done in the present

Lanford and Ruelle

(in the papers mentioned

states will be those probability

probabilities,

X

is to determine

states of the system (and to make this determination ructing the dynamics

topoi-

X .

Suppose that there is some system evolving with

first aims of equilibrium

the discrete

measures which have the

with these conditional

involving the Hamiltonian.

probabilities

given

We will explain what this means for

but to do thi~ we need some notation.

3

For each subset

A

of

S

let

1.3

X(A) =

Yt ' considered as a

~ tEA

compact Hausdorff space and let PA

be the projection from

thus a sub-o-field of observable)

F

from inside

X

_F (A) denote the Borel a-field on --o onto

X(A), and let

and it represents A . Note that if

denote the finite subsets of

S, and

is the set of events that are measurable

F(A) = pAI~Fo(A)). F(A) is

the events that are measurable A~

for

B C S

A ~ C

then

we put

from outside

A ~ C

probability measure

on

o~

and

y E X(S-A)

F A = F(S-A),

function of

Since for each

{o K}

F g F (A) ~o

y , a probability measure

probabilities

(i.I)

FA

A . Lanford and

the Hamiltonian will define a

as their (appropriate)

oY(F) i~ p

conditional

will be an F (S-A)-measurable =o .

will have the right conditional

if and only if

PA(P)

The equations

so

will

(X(A),F (A)), and the equilibrium states will be --o

those probability measures which have probabilities.

(or

F(A) C F(B). C

In terms of the above notation the hypothesis of Dobrushin, Ruelle is that for each

X(A). Let

=

I ~

dPs-A(P)(Y)

for all

A g ~ .

(i.i), although usually written in a different form, are some-

times referred to as the DLR-equations. The probability measure

a~

should be thought of as the equilibrium

state for a system evolving in the "finite vessel" A, governed by the same Hamiltonian,

but with the configuration

Since

is a finite set we can identify

X(A)

ect to counting measure), and usually where for

w g X(A)

gA(y,w)

outside

O~

O~

%

constrained

to be

y .

with its density (with resp-

will take the form

z-lexp gA(y,.)

is the energy of the configuration

w

in

X(A)

,

1.4

given that the configuration outside make

~

a probability measure,

A

thus

is Z =

y , (and where ~--

Z

is chosen to

exp gA(y,w) ). In the ease

w ~ X(A) when

Y

consists of the two points

ification of gA(F,G) =

X(A)

~

0

and

i , and there is an obvious ident-

with the set of subsets of

~(B)

(for

F~S-A

, GcA

A A , g

), where

will take the form

~ : ~.

> R

is the

BGFUG B~A # interaction potential corresponding to the Hamiltonian. At this point the reader is probably asking why the above definition is the right one. This is a very hard question to answer satisfactorily. The only real way of knowing that the definition of an equilibrium state is correct is to construct the appropriate dynamical system and check that the supposed equilibrium states are exactly the invariant states for this system. However, it is extremely hard to construct the dynamical systems corresponding t o m o s t models, and only recently has much been done on particular cases of this problem (see for example Lanford (1975) , Marchioro, Pellegrinotti and Presutti (1975)). The only justification we will make is that the definition is intuitively appealing, that most physicists would agree it is the correct one, and that the results which can be proved show that these equilibrium states have the properties which, based on physical experiments,

they should have.

For the model we have described above there are two ways of looking at the probability measures satisfying (i.i). Firstly, they can be regarded as those measures whose distributions

inside any finite set are in equilibrium

with respect to the distribution outside tile finite set. This is because PS_A(D)

is the distribution outside

A , and thus the right-hand side of (i.i)

represents the equilibrium state (inside side

A

is constrained to be

A ) of a system where the state out-

pS_A(D). Secondly, it is not too difficult to

5

check that the measures satisfying

1.5

(i.I) are e~actly those probability measures

which can be obtained as weak limits of sequences of the form

with

An

increasing to

kind of "thermodynamic

S

and

Yn g X(S-An)

Yn OA n

. This corresponds

, n ~ i ,

to taking some

limit": letting the vessel expand and at the same time

changing the boundary conditions.

This second way of looking at things will

perhaps help to convince any physicist

that the present approach does not really

differ from the "classical" way of doing things in equilibrium statistical mechanics. There is a better way (at least mathematically) model described above. For each

(1.2)

that is for each

X

and

F g ~

(1.4)

(where

~A(.,F)

if

probability measure

(1.5)

where

if

~A

~A

for all

is a probability kernel, and for each x g X

~A(X,.)

also satisfies:

F g ~ ;

and is

0

otherwise).

In terms of the

HA

a

will satisfy (i.i) if and only if

Ep( XF I ~A ) =

Ep( f I ~ )

by

~A(.,F) = XF ;

x c F p

)R

the

: Ps_A(X)XW s F }) ,

Then

It is clear that

then

HA : X x ~

is ~-measurable,

is ~A-measurable

F C ~A

XF(X) = i

X(S-A)xx(A)).

~A(',F)

is a probability measure.

(1.3)

define

PS_A(x) OA ({ w e X(A)

~A(x,F) =

(where we identify

A E ~

of formulating

HA("F)

p-a.e,

denotes the conditional

for all

A g ~ , F g ~ ,

expectation

(with respect to

p )

6 of

f

given the o-field

(1.6)

D(Ff]G)

1.6

; thus (1.5) can be rewritten as

=

.I zA(x,F) d~(x)

for all

A ~ C , G s FA , F g F .

G

The measures

{o~}

given by the Hamiltonian will satisfy a consistency

condition, which in terms of the

(1.7)

if

(where

~% zA

AC%

e ~

then

ZA

can be written as

~% ~A = ~%

'

is the probability kernel defined by

(z~ ~A)(x'F)

=

I ~A(Y'F) ~%(x,dy)

.

Note that, regardless of any physical interpretation, (1.7) is a natural assumption to make if the

7A

are to be interpreted as conditional probabilities,

since it just corresponds to the usual rule

E ( f I~

which holds

~-a.e. if

) = E ( E ( f J ~A ) I ~

AC%

because then

) ,

F=]C F=A 9

Now in the formulation of the lattice model just given the particular structure of the model plays very little part. All that we needed was a set equipped with a o-field C, ~C~A

a collection whenever

~ , an index set

{~A}AsC AC~

~

of sub-o-fields of

, and a collection

X

with a partial order denoted by ~

{~A}Ag~

which are decreasing in that of probability kernels sat-

7

1.7

isfying (1.3),(1.4) and (1.7). Given this, the problem is then to find out all about the probability measures on If

~ = {~A}AeC

satisfies

(X,~)

satisfying (1.5).

(1.3),(1.4) and (1.7) then we w i l l c a l l

a specification ; the corresponding equilibrium states (those probability measures satisfying (1.5)) will be denoted by

G(V)

and called the Gibbs states

with specification ~ . These notes are a study of specifications and their Gibbs states (although in fact we will relax (I~4) in most of what we do). This set-up includes ~ n y

models from continuous statistical mechanics, where usually

will be the bounded Borel subsets of R d (d-dimensional Euclidean space), and X

will be a subset of the integer-valued measures on

Rd

(representing config-

urations of particles in space). Although in the definition of a specification we only required

~

to

be any partially ordered index set, in practically all models from statistical mechanics

~

will be a collection of subsets of some underlying space. As in

the lattice model we will start with a space

S

(usually

on whether we have a continuous or lattice model), S o-field again

~

, and for each

(X,~)

so

~

~A

or

Zd

depending

will he equipped with a

we will have a sub-O-field

~(A) of

~

(where

is the basic phase space), representing the events that are meas-

urable from inside o-fields

A g ~

Rd

A . The o-fields {~(A)}A~ ~ will be increasing, and the

in the definition of a specification will be of the form

will be some suitable subset of

that are measurable from outside

~ , and

~A

~(S-A);

will represent the events

A .

As mentioned before, a useful way of thinking about specifications is in terms of generalized stochastic processes: the kernels

~ = {~A}Ae~

regarded as the transition kernels of a process indexed by case

~

would be some subset of

can be

~ . (In the usual

R , representing the time axis, and for

L ~

8

_~

1.8

would be the events determined by the process up to time

-t .) Some of the

basic problems which will arise with specifications will be concerned with the analogue of constructing the entrance boundary for a stochastic process. There are three parts to these notes. The first is Sections 2, 3 and 4 ; in which the basic properties of specifications are worked out. The second is Sections 5 and 6, which deals with exs~ples of specifications.

Section 5 deals

with lattice models, and studies the model we have already introduced (but with Yt

an arbitrary set). Section 6 is about continuous models, and in particular

about poin'~ processes. It is possible ! and maybe desirable) to read Sections 5 and 6 before the first part of the notes. The third part comprises Sections 7, 8 and 9, and treats some particufLar topics (which are mainly concerned with lattice models ). Let

G(~_~

be the set of Gibbs states for a specification

fundamental question is whether or not

~(~)

V . The most

is non-empty; and in Section 3 we

give sufficient conditions for Gibbs states to exist. Given that Gibbs states do exist for

~g , and noting that

we would i~ke to know whether

G(V)

is clearly convex set of probability measures,

G(Y_) has any extreme points, and whether

can be represented in terms of its extreme points.

G(Y0

(There are reasons for doing

this other than it being the reflex action of any mathematician who has had a course in functional analysis. The extreme points of

G(V)

should represent the

pure phases of the system, and these states should have particular properties which we would like to deduce from our framework. ) In Section 2 we characterize the extreme points of space then of them.

G(V)

G(V)

; and also show that if

(X,~

is a standard Borel

has extreme points, and a nice representation exists in terms

(The existence of extreme points and the representation are due to

9

1.9

FSllmer (1975a), and the proof is based on Dynkin's construction of a Martin boundary, which is given in Dynkin (1971a) and (1971b).) In Section 4 we consider the situation where there is a group of bijections acting on

X , and where the specification is in some sense invariant under the

group. It is then natural to study those Gibbs states which are also invari~ut.

(It

is not necessarily true that the Gibbs states are al~omatically invari~nt, and their failure to be so corresponds to the physical phenomenom of symmetry breakdown.) In most of the models the bijeetions on on the underlying space

S ; for example, if

X S

group usually taken is the group of translations.

are induced from a group acting is either

Zd

or

Rd

then the

The representation obtained

in

Section 2 will be used to give a representation of the invariant Gibbs states. The material in Sections 7, 8 and 9 deals mainly with lattice models, though Section 7 also includes results applicable to the models of Section 6. In Section 7 we give a variational characterization of invariant Gibbs states in terms of the specific information gain of one random field with respect to another. This follows an approach of F~llmer (1973) to proving more physical variational principles. We apply the results of Section 7 to give, in Section 8, a version of the Gibbs variational principle: the invariant Gibbs states are exactly those invariant states which minimise the specific free energy. This provides a generalization of a result of Lanford ~.d Ruelle (1969). Also in Section 8 we do some thermodynamics and, again following Lanford and Ruelle (1969), identify the invariant Gibbs states with the tangent functionals to the pressure (in a suitable B~uach space of interactions). In Section 9 we look at what could be considered generalizations of ferromagnets, or attraetivo interactions. We introduce this by putting a partial order

70

on

i. i0

X , and looking at specifications which ~ave certain properties with respect

to the order. We show that~ under suitable conditions, there exists more than one Gibbs state at exactly those places where there is a jump discontinuity in the density. Those readers who consider the present setting too abstract, and would like to consider alternate or preliminary material containing more specific examples, should look at the papers of Dobrushin, Lanford and Ruelle already referred to, and also at Spitzer (1971a), (1973), Lanford (1973), Georgii (1972), (1973), (1975), Sullivan (1973) and Preston (1974b).

These papers also many import-

ant topics which we hardly consider at all here, in particular the problem of determining when there is exactly one Gibbs state. Finally, it is worth noting that there are connections between the theory of specifications and topics in quantum field theory. The interested reader should look, for example, at Nelson (1973), Dobrushin and Minlos (1975), and Royer and Yom (1975).

Note: Professor Dynkin has just pointed out to me that the proof of Theorem 2.2 (which gives the representation in terms of extreme points) can be made much simpler using a general result from Dynkin (1971b), and the notion of a support system (see Dynkin and Juskevic: Controlled Markov processes and their applications, Moscow 1975, pages 306-307).

2.1

2. Random fields and specifications

This section works out the basic properties of specifications and their Gibbs states. We start with a set set

~

X , a o-field

F

of subsets of

which is partially ordered. The order on

(rather than o-field of

C

~

will be denoted by

used in the introduction). For each

% , denoted by

X , and an index

A E ~

~ ,

we have a sub-

~ A ' and we assume that the o-fields

{FA}AEC

are

decreasing, i.e.

(2.1)

X

F % C__ F ~

whenever

A _<

represents the basic phase space,

F

the set of all observables~ and the

~A

are subsets of observables, usually associated with regions of some underlying space. We will assume that the partial order is directed upwards (i.e. if AI, A 2 g ~

then there exists

A ~ ~

with

ably generated (i.e. there is a sequence then

A < A

for some

AI ~ A {An}n~ I

and from

A 2 ~ A ) and is count~

such that if

A g

n ).

n

Some notation is needed. If

~

is a sub-o-field of

denote the set of probability measures on r A : P~_F) i.e.

>

P(FA)

c P~_F) and

A g ~

for all

A ).

P~B)

then let

~ g P(~F) , F g ~ A " Elements of

P(~_F) will be

(sometimes the term stochastic fields is used instead). If then we can regard

ervable with respect to side

A E ~

then let

be the mapping got by restriction to the smaller o-field,

rA(~)(F) = ~(F)

called random fields;

(X,~) ; if

~

rA(~)

as the distribution of

~

obs-

A ,(which in most cases will mean observable from out-

2.2

12

As in the introduction we define a collection of probability kernels = {~A}AeC

to be a specification

(2.2)

~A(',F)

(2.3)

~A(.,F) = XF

(2.4)

~

If

~

~A

is

=

if

F__A-measurable for all

for all

7%

F g FA , A E ~ ;

whenever

A ~ % .

is a specification then we define

cation V

to be those random fields

(2.5)

A g ~ , F g ~ ;

E ( XF i ~A ) = ~A (',F)

~(~

, the Gibbs states with specif-

~ g P(~_F) for which

~-a.e.

for all

Let us consider for a moment the kernels this will give us some insight into conditions the kernels define

~ = {~A}AgC

tA : P~FA)

tile kerneI

~A

tA(V )

>P(~_F)

by letting

on the measure

~ s P(F_A)

= ~A(X,F)

~

satisfy:

tA(V) = ~ A

A g ~

we can

' where the action of

is given by

dr(x)

for all

F

which gives uS a probability measure

when we only know what is happening on the sub-

FA " With this interpretation

{tA}Ag ~

(2.3) and (2.4). Suppose that

J

as a predicting mechanism,

on the whole of

~-field

the

tA

as acting on measures;

just satisfy (2.2); then for each

(~A) (F) Think of

~A

J g ~ , F g % .

it would be natural to assume that

2.3

13

(2.6)

=

r A tA

(2.7)

if

A < %

(2.6) just says that

identity: P~F A)

--~P~FA)

then

tA rA t%

tA(~)

is an extension of

(2.7) says that the extension from

=

~A

for all

A g ~ ;

t% .

~

to the larger o-field;

is consistent with that from

~

9

The connection between this and the definition of a specification is given by the following proposition.

Proposition 2. !

Let

satisfying (2.2); let only if ~

be a collection of probability kernels

~ = {~A}Ag~ tA(~) = ~ A

satisfies (2.3), and

" Then {tA}Ag ~

{tA}AE ~

satisfies (2.6) if and

satisfies (2.7) if and only if

satisfies (2.4). Proof

(2.6) says that

tA(~)(F) = v(F)

for all

~ g P(F=A) , F e ~ A

; and

tA(~)(F) = I ~A(x,F) dv(x) . Thus for (2.6) we need

I for all F e~A

F g~A

~A(X,F) d~(x)

=

I XF(x) dv(x)

' ~ e P(_FA) ; while for (2.3) we need

~A(',F) = X F

; these conditions are clearly the same. Similarly, if

e P(F_F A) , F g ~

then

(tA rA t% (V))(F)

=

I~A(Y'F)

d(rA t%(~))(y)

A~

for all and if

2.4

14

=

I~A(Y,F) I~%(x,dy)d~(x)

=

II ~A(Y,F) ~%(x,dy) dr(x) ,

and thus for (2.7) we need

I I ~A(Y'F)~%(x,dy)d~(x)

for all

F g ~ , v s P~%)

Still thinking of

=

I~%(x'F) dr(x)

, which is the same as (2.4).

tA

82

as a predicting mechanism, the natural random

fields to look at are those which predict themselves, i.e. those which g P_(~

tA rA(P) = ~

for all

D s P~J

for

A s ~ . Writing this out in full we get that

is self-predicting if

(2.8)

D(F) =

I~A(x'F) d~(x)

for all

A g~ , F g~ .

Since by the definition of conditional expectation (2.5) is the same as

(2.9)

~(G~F)

=

I ~A (x'F) dp(x)

for all

G g~A

" A ~

, F ~

;

G

it is clear that any Gibbs state is self-predicting (just by putting

G = X

in (2.9)). The converse of this is true, thus one of the first results we shall prove is that Let

~ g ~_~)

~ = {~A}Ag~

if and only if it satisfies (2.8). be a specification and let

E~)

denote the set of

probability measures satisfying (2.8). We will hence show that

E(~

= G(V) ,

but assuming this fact for the moment we can obtain some simple but useful

15

properties of follows that

G(V) . ~or

A ~ ~

tA rA(~) = ~

~A

G~)

~A = tA(P-~-A)) ; by (2.6) it easily

if and only if

Also from (2.7) we have that of showing that

let

T%~

~A

~ g~A

whenever

' and thus

E(~) =

~

~A "

A ~ % . Therefore the problem

is non-empty amounts in most cases to showing that the

are compact in some suitable topology.

introduction, where

2.5

X

(In the example considered in the

was compact, we need only show that the

~A

are closed

in the weak topology, and this becomes a continuity requirement on the Hamiltonian.) We will more or less use this approach in Section 3. This section will be mainly taken up with the study of the extreme points of

G~

. We assume that we have a specification for which Gibbs states exist

(leaving the existence problem for the next section). After showing that E~VJ = G ~ ) , fields in

G(V)

field, and 0

or

we will characterize the extreme points of that have trivial tail-field.

~ g P(~)

( F=~ =

G(~ ~

has trivial tail-field if for all

as those random ~A

F g F=~

is the tail-

~(F)

is either

I . Sometimes the tail-field is called the set of observables at infinity,

for example in Lanford and Ruelle (1969).) This characterization of the extreme points of

~_~)

can he found in various degrees of generality in Dobrushin

(1968a), Lanford and Ruelle (1969), Georgii (1973) and Preston (1974a),(1974b). Out of this result comes the fact that the elements of the tail-field, in the sense that if all

F ~ =ooF then

~i' D2 E G(V)

G(V)

with

are determined on

NI(F) = ~2(F)

for

~I = ~2 " It also follows that distinct extreme points are

mutually singular. The main result that will be proved about

G(V)

is that if

(X,F)

is

a standard Borel space (which means that there exists a complete separable metric space

Y

such that

~

is ~-isomorphic to the Borel subsets of

Y ; see

16

2.6

for example Parthasarathy (1967), page 133) then each element of

G~)

~_V)

has extreme points and

has a unique representation in terms of the extreme

points. (This is of course all under the implicit assumption that

G~)

is non-

empty.) This result comes from Follmer (1975a), and the method of proof is based on Dynkin's construction of an entrance boundary for a stochastic process (see Dynkin (1971)). Before we start proving results it will be convenient to slightly generalize the definition of a specification. This will be mainly because of the applications to continuous models. A well-known problem which arises in elementary probability is what to do about the definition of the conditional probability of A

given

B

when the probability of

B

is zero. An analogue of this occurs

when trying to construct specifications corresponding to some models: for certain

x g X

measure

and

~A(x,-)

A c ~

there is no natural way of defining the probability

because

x

ably be overcome by defining

is an "impossible" event. This problem could prob~A(x,.)

in a fairly arbitrary way (but making

sure that (2.2),(2.3) and (2.4) still hold: but this might be difficult); or in some models it can be overcome by replacing

X

with a suitable subset of

We will do something like the latter, but will keep the original set the "impossible"

x

we let

~A(x,.)

X .

X . For

be the zero measure, and then only look at

Gibbs states which give zero probability to the set of "impossible" events. The above will be formalized in the following way. Suppose that for each A g ~

we have

~A : X x ~

RA g ~ A >R .

R = {RA}Ag C

if:

(2.10)

~A(x,.)

(representing the "possible" event in

We will call

~ = {~A}AsC

~ A ) and

a specification with respect to

is a probability measure for each

x g RA , ~ C C ;

17

(2.11)

~A(x,F) =

(2.12)

~A(',F)

(2.13)

~A(',F) = X F N R A

(2.14)

~% ~A

Again we define

0

for all

2.7

x s RA , A g ~ , F g ~

is ~A-measurable for all

=

G(~

~

if

;

F g ~ , A e ~ ;

F g _FA , A g C ;

whenever

A ~ % .

, the Gibbs states with specification ~ , to be those

g P(F__F) for which

(2.15)

E ( XF [ F A ) = ~A(',F)

~-a.e.

for all

A g C , F g F .

(2.15) is the same as (2.5) and thus also the same as (2.9). Putting in (2.9) shows that if

~ a _G(V)

then

~(R A) = i

for all

G = F = X

A g C ; thus the

Gibbs states automatically give zero probability to the "impossible" events. Note that we get our original definition of a specification back if we take RA = X

for all

A ~ C .

Again define provided

tA

by

~ g P~_FA,RA) , where

tA(~ ) = v~ A ; if P~_FA,RA) =

~ ~ P~_FA) then

{ ~ g P(FA)

: ~(R A) = I } . (2.7)

still holds, and so does (2.6) provided P~_FA) is replaced by T A = tA(P~FA,RA)) _E~) = ~

; again we have

T A , where as before

~ = CA rA(~) E~)

tA(~ ) g P(~_F)

P~FA,R A) . Let

if and only if

D g T A , thus

denotes the random fields satisfying

AgC (2.8"). Also, once more we have

_T2~~

_TA

quence of this is that to show

~ g E_~)

whenever

A < 2~ ; an important conse-

it is sufficient to show that

~ g TA n

18

for all

n ~ I , where

{An}n> 1

Some notation: if

~

2.8

is any sequence that generates the order on

is a sub-o-field of

the set of non-negative finite measures on extend

tA

tA(~) = VZA if

f

to be an affine mapping from for all

9 ~ M+(~'A) . If

%

~

then let

C .

M+~_B) denote

(X,B) ; it will be convenient to 4

M IF_A)

to

M+(~__F) by letting

is a non-negative finite measure and

is non-negative and integrable with respect to

X

then

f%

will be the

measure defined by

? (fX)(F) =

~

f d%

F

Now let us fix a specification

~ = {~A}Ae~

with respect to

~ = {RA}Ag ~ ,

and where it is necessary to make sense we will always assume that G(V) is nonempty, tA

Lemma 2.1

will denote the mapping defined above in terms of

If

A g C , ~) g M+~F_A )

tA(XF~) =

Proof

From (2.6) we have

XFtA(v)

and

F g FA

zA "

then

9

rA tA(Xp~) = XF O RA9

and thus

tA(XF~) (X-F) = (XFV) (X-F) = 0 .

Therefore

XX_FtA(XF~) = 0 ; similarly

XFtA(Xx_F~) = 0 . Now

tA(~) = tA( XFV + XX_F V ) = tA(XF9) + tA(Xx_F~)

,

and thus

XFtA(~) = XFtA(XF~) + XFtA(Xx_Fv) = XFtA(XF~)

19

2.9

= ( i - XX_F )tA(XFV ) = tA(XFV ) - XX_FtA(XF V) = tA(XFV ) .

Proposition 2.2

G~)

= E~)

, i.e., ~ s G(V]

if and only if

satisfies

(2.8).

Proof

It is clear that

G E~A

and

Lemma 2.1

E(yJ ~

F g ~ . Since

G(V)

~ g EQVJ

, thus take

we have

XG~ = XGt A rA(~) ~ tA(XGrA(~))

~(Gf~F) = (XG~)(F) = tA(XGrA(~))(F)

and writing this in terms of

~(G~F)

= I~A(X'F)

I ~A(X,F) d~(x)

~A

and let

A e ~ ,

tA rA(~} = D , and thus by . In particular

,

gives

d(XGrA(~)l(x)

,

~ e E~)

= IXG(X)~A(X'F)

which is what we want.

drA(~)(x)

82

G

We will now look at the extreme points of s _G~)

which cannot be written as

DI # ~2 " Recall that the tail-field

D = ~(~i+~2 ) F

G~) with

was defined by

, i.e. those elements ~I' ~2 ~ G ~ ) F

=

~

and

F A , and

AeC we are going to show that if

~

has trivial tail-field.

Lemnm 2.2

If

with respect to

Proof

~ g G~)

is an extreme point of

then

if and only

The proof of this will proceed via three lemmas.

A g ~ , ~ Z M+~_FA)_ and v

G(V)

f

is non-negative and integrable

tA(f~ ) = ftA(~ ) .

For simple functions it follows from Lemma 2.1 by linearity;

for den-

20

eral

f

just use the usual approximation by simple functions and the monotone

convergence theorem.

Lemma 2.3

If

respect to

Proof

2. i0

~

82

A E ~ , p c P~_~) and then

We have

tA rA(gp) =

g

is non-negative and integrable with

Ep( g I ~

) tA rA(P) 9

rA(gp) = Ep( g I ~A ) rA(P) , as this is really just the def-

inition of the conditional expectation. Thus from Lemma 2.2 we have

tA rA(gP) =

Lemma 2.4

Let

respect to

only if

Let

~

and let

I g d~ = I

g = Ep( g I F

Lemma 2.3 Thus

p E G~)

~ , with

an F -measurable

Proof

tA(Ep( g I ~A )rA(P)) =

)

with

A g C ; then

g

Ep( g ] ~A ) tA rA(P) "

be non-negative and integrable with

(thus

gp g P_~J). Then

p-a.e, then

g = g

if and only if

g = E ( g I FA )

g = E ( g I --F A )

tA rA(gp) = g p . But by

~-a.e. because

gp c G(_V) then

cA rA(D) = ~

p-a.e. If

g = Ep( g I -FA )

g = E ( g I F=~ )

take an increasing sequence

i

n

)

gD g _G~) .

p-a.e, for all C

~ E G~.

A g C .

is countably generated,

p-a.e. (For example, one way to see this is to

{An}n>_l

martingale convergence theorem to

n >=

since

g = EB( g I F

F=oo~F A ; hence

This, together with the assumption that the order on implies that

if and

D-a.e.).

g~ g T A

if and only if

Conversely, if

gp E G ~

p-a.e. (or, equivalently, if and only if there exists

tA rA(gp) = Ep( g I F A ) CA rA(P) , and

gB g _TA

82

generating the order on

C , and apply the

{EB( g I FAn )}n>l ' using the fact that

21

2.11

The characterization of the extreme points of

G(V)

is one of the sev-

eral facts that are simple consequences of Lermna 2.4.

Theorem 2.1 only if

~

(I) If

~ g G~)

G(V)

~i' ~12 c G ~ )

~I(F) =

~i

~

is an extreme point of

G~)

with

and

~2

are determined on the tail-field, in that if ~2(F)

for all

F e ~=~

are distinct extreme points of

then

G~)

~i = ~2 "

then

~I

and

are mutually singular, even considered as measures on the tail-field; thus exists (4) If

F g F==~ with

H(V) ~

H~_) = { ~

if and

has trivia], tail-field.

(2) The elements of

(3) If

then

M+~_F_)

~2

there

~I(F) = i , ~2(F) = 0 .

is the cone generated by

: ~ ~ O , ~ ~ G(_V) }) then

H(V)

G(V)

in

M+(~F

(that is

is a lattice in its own order.

(For a definition of this see for example Choquet (1969), Volume II, page ]60. This result tells us that if G~

G(_V)

is compact in some suitable topology then

is a Choquet simplex; and we could thus use all the representation theory

for such simplexes. However, we will not take this approach.)

Proof with to

(I)

If

D

is not an extreme point then we can write

~i' ~2 g ~ ) ~

and thus we have

with respect to F= -measurable. > 0

and

~i # ~2 ' g

with

g

is non-negative and integrable

D

~

is

{ x ~ X : g(x) > ~ }

is not trivial on the tail-field. Conversely,

does not have trivial tail-field, so there exists

O < ~(F) < 1 . Put

g

is not ~-a.e. a constant; thus for some

O < ~({ x E X : g(x) > ~ }) < i ~ But

is in the tail-field, so suppose that

~I = g~ ' where

)

is absolutely continuous with respect

~ ; and therefore by Lemma 2.4 we can assume that Since

we have

~I # ~2 " ~I

~ = 89

~ = ~(F)

and let

~i =

F g F

~-I(xF~)' ~2 = (I-~)-I(Xx-F ~);

22

then by Lemma 2.4 we have =

~i

(2)

Let

then

~i

2.12

~I' ~2 g ~(~) and of course

+ (i-~)~2 ' and from this it is clear that ~I' ~2 g ~ ) and

P2

with

~I(F) = ~2(F)

~i # D2 " Now

~

is not extreme.

for all

F ~ F=~~ . Let

are absolutely continuous with respect to

~i = gl ~ ' ~2 = g2 ~ ' and by Lemma 2.4 we can assume that F=m-measurable. then

If

FI, F 2 g F=o~

Therefore

Thus

gl = g2 Let

(4)

Let

0

and

- g21 d~

=

~t-a.e., i.e.

i . Thus

Vl' ~2 c H ~ )

and

g2

are

I ( gl - g2 ) d ~

I

FI

F2

=

( gl - g2) d ~

0 .

~I = ~2 "

; but by (I)

~I

and

, let

D2

~ =

~I(F)

G(V)

. By (2) there exists

and

~2(F)

can only have the

are mutually singular on

~i + ~2 ' and let

~I = gl ~ ' ~2 = g2 ~ " By rescaling the measures ures, it follows from L ~ m ~

gl

~I(FI) = ~2(FI), ~I(F2) = ~2(F2).

be distinct extreme points of

~I(F) # ~2(F)

~ , so we can write

, F 2 = { x g X : gl(x) < g2(x)}

~I(FI) - ~2(FI) - ~I(F2) + ~2(F2)

~i' ~2

F E F=~o with values

and thus by hypothesis

llgl

=

(3)

F I = { x g X : gl(x) > g2(x)}

~ = 89

gl' g2

F=oo .

be such that

to make them probability meas-

2.4 that we can assume that

gl

and

g2

are

+

F -measurable.

Put

~

= max{gl,g2}~

~ ~

; then by Lemma 2.4

= min{gl,g2}~

+

we h a v e

+

~) , ~ - a HQV)

~- = ~I A ~2

and i t

is easily

checked that

~

= ~1 v ~2

and t h a t

82

Because of (I) in the above theorem it is worthwhile noting a fairly

,

23

2.13

standard result about random fields with trivial tail-field. We will say that a random field

~

has short range correlations if given any

then there exists

A s ~

such that for all

F g ~

and

6 > 0

G ~A

I~(FNG) - ~(F)~(G)I <

Proposition 2.3 (This can be found for example in Lanford and Ruelle (1969).) Suppose that of

~

F

is countably generated (i.e. there exists a countable subset

such that

~ = o(_A) , the smallest o-field containing

~ ). Then a

random field has trivial tail-field if and only if it has short range correlations.

Proof

If

~

has short range correlations then clearly

field, because if

F ~ F=~~

l~(F) - ~(F)~(F) I < 6 either

0

or

then

F e~A

for all

A e ~

and thus

i . (This part of the proof does not require

then there exist

has trivial tail-

6 > 0 , and this can only happen if

generated.) Conversely, suppose that

Fn g ~ A

for all

~

F ~ ~ , {An}n> I

~

~

~(F)

is

to be countably

does not have short range correlations;

increasing and generating the order on

~ ,

, such that n

lira { H(FflFn) " ~(F)H(Fn) lr+oo

Since ball in

F

is countably generated L (X,~,~)

LI(x,~,~)

0 .

is separable, and thus the unit

is sequentially compact in the weak-*-topology.

there exists a subsequence

lim J-~

} #

{nj}j> I

~ ( F n ~ G) = J

~ G

It is easily seen that we can take

and

g dD

g

g ~ L~(X,F,~)

for all

to be

Therefore

such that

G g ~ .

F -measurable and since we have

2.14

24

I

g dp-

p(F)\ g dD

F

g

=

cannot be p-a.e,

tail-field.

equal to a constant.

G~)

Therefore

p

does not have trivial

that there might be some kind of representation

in terms of its extreme points,

though in this generality we have no

way of showing that any extreme points exist. However, is a standard Borel space (i.e. there exists a complete Y

0 ,

82

Theorem 2.1 suggests of

#

lira { p ( F ~ F ) - I~(F)~(Fn ) } j_><~ n. 3 3

~

such that

Parthasarathy

F

is isomorphic

as a o-field

for

~(_VO

example that is likely to occur

exists.

(X,~)

(X,F)

separable metric

space

to the Borel subsets of

(1967), page 133) then we can prove that

and that a representation

if we assume that

G(V)

Y ; see

has extreme points

It is worth noting that in any

will certainly be standard Borel. The

next theorem gives us the analogue of an entrance boundary.

Theorem 2.2 (The hypothesis If

(X,~)

+ R

for each

F ~ ~

(2.17)

for each

x e X

of

;

for any

z(-,F)

p s ~(V)

(2.20)

and

if

~ e P_(~

}

then

F e ~

is needed in this theorem.) kernel

then

;

~(x,-)

is an extreme point

we have

) = z(-,F)

if we denote tile measure

A(x) = { y g X : ~Y = x

is F=~o-measurable

the probability measure

E ( XF I ~

(2.19)

is non-empty

with the properties:

(2.16)

(2.18)

~(~)

is a standard Borel space then there exists a probability

: X x F

~(Y9

that

~(x,.)

p-a.e.

by

x

;

and for

x g X

11x(A(x)) = i ;

D e G(V)

if and only if

Dn = p .

let

25

2.15

Before proving this theorem we will show that it enables us to construct a representation of alence relation on

G~

in terms of its extreme points. Let

X

defined by

onding quotient space and

p

B

mapping from

G(V)

Proposition 2.4

Suppose that

~

(X,~)

~

G~V)

{z} g L

Fo = p-l~) also if

A(x)~ on

F

and if

(X,L)

and let

o

p

X

then let

F

==oo

(X,_B~ then let

p(~)(F) = ~(p-l(F))

, we can consider

p

.

as a

(X,F_~o) .

and the probability measures on

F~

p

(X,L).

z g X .)

F

and for any

==co

F e F=~ ~

then

~(-,F) = Xp

then

A(x)~F

= ~ ). Let

; define a probability measure

By (2.8) we have

that sends an element

defined by

for all

; then

x ~ F

E P~_)

be the corresp-

is as constructed in Theorem 2.2. Then

==co

F=~~

X

X

be the equiv-

correspond under this mapping to the point masses

(X,_F_~) . (Note that

Let

onto

to probability measures on

The extreme points of

Proof

X

= ~y ; let

is a probability measure on

are determined on

is an affine bijection between

on

~

be the probability measure on G~)

x

is a o-field of subsets of

= { F C X : p-l(F) g ~ } ; if

Since elements of

if

the map from

to its equivalence class. If

P(U)

x ~ y

~

~

on

F e =F

(since if V

~(-,F) is x g F

then

be a probability measure

(X,F=~ ~

by

~(p-l(G)) = ~(G)

be defined by

~~

=

o

g G~)

I ~(x,F) d~(x) .

and it is easily checked that

is onto. Now by (2.20) we have

D(F) =

~(x,F) J

d~(x)

p(o)

= v ; hence

2.16

26

F s F , p g _G ~ )

for any

, which shows that

thus it immediately follows that

p

p

is determined on

F O ==co ; and

is one-one. The rest of the proof is clear. 82

We now start the proof of Theorem 2.2; the proof will be via a number of lenmms, the first of which gives us a crucial fact (and does not depend on

(X,~)

being standard Borel).

Lemma 2.5

Let

all

we have

A g C

g

be bounded and F-measurable, and let

rag -~ E ( g I F A )

where

Proof

N-a.e. ,

~rAg is the function defined by

We need to show that if

I ~Ag dP F (since it is clear that

~Ag

=

g dP

=

I F

=

=

,

P g ~(~)

I ~ A g d(XFP) =

g d(P~A)

then

is ~A-measurable ). Note that by Lemma 2.1 we have

(XFP)~A = XF(P~A), and that since

I ~ A g d~ F

(TAg)(x) = ~g(y) ~A(x,dy) 9

F g ~A

~ F

p g G(V) . Then for

I F

The next results require

P~A = p " Therefore

I g d(XFP)~A

= I g dXF(P~ A)

g dP

(X,~)

to be a standard Borel space, so let

us suppose for the rest of the section that this is true. We will use the foil-

27

2.17

owing fact about such spaces (see, for example, Parthasarathy if

~

(1967), page 145):

has uncountably many atoms then there exists a countable field

such that

~ = o~A)

and such that if

~ :A

~ R + is bounded then

~C ~

is a

measure (i.e. countably additive) if and only if it is finitely additive. Note of course that if

~ : A

> R * is a measure with

~(X) = 1

/

Caratheodory extension theorem and the monotone class theorem quely to an element of Now if particular

F

~

then by the ~

extends uni-

P~_F) .

has only countably many atoms then so does

F

and in

is conntably generated. In this case Theorem 2.2 is easy to

prove, and we leave it as an exercise for the reader to do so. (This case, in fact, never occurs in any of the models we will consider.) Thus in the following we suppose that

~

has uncountably many atoms.

Fix a countable field also f i x an i n c r e a s i n g = N

FA

9 For

A

having the properties described above, and

sequence F e A

{An}n.>l

I

define

lim ZA (x,F) n ->~176

otherwise.

O

I = { x g X : lim z^ (x,F) n -+~176

UN ^ n>l m>n

Lemma 2.6

Proof

I e F

if the limit exists,

n

=

~o(X,F)

R~ =

~ ; thus

n

A~C

Let

g e n e r a t i n g t h e o r d e r on

exists for all

F ~ A }~Roo , where

n

m

and

It is clear that

~(I) = i

for any

D g G(V)

.

I~ e ~=~ , and by (~.9) we have

~(Roo) = I for any

28

e G(V)

. For

F g A

I (F) e F=~, and thus that

V(I

(F)) = 1

let

I (F) = { x g X :

I g F for any

since

theorem we have

~(I (F)) = i .

82

2.7

Proof

If

x e I

then

By the definition

additive

on

~

; but if

~o(X,FIUF

=

2)

=

of

~

In fact

x

>R

I F=~o )

To(X,.)

Tx

denote

=

because

on R=o~

we will I

and thus

To(X,.)

on

also denote To(X,X)

2.8

is some arbitrarily

zI

.

is finitely

82

if

x g I ,

if

x ~ I ;

chosen element

has the properties:

ol

~

; since

x

this extension

= i . Now define

by

\,(F)

Lemma

; thus

then

~l(X'F) =

~

A

Zo(X'Fl) + ~o(X'F2 )

the measure

zX(F)

(where

~-a.e.

lira zA ( x ' F I U F 2 ) rr+oo n

to a measure

e P(~)

iT1 : X x F

2.5 and the martin-

on the field

FIeF 2 = ~

}; then

n

then let

extension

= E ( XF

we need only show that with

exists

We now need only show

. But by Lemmm

is a measure

lim { WA (X'Fl) + nA (x'F2) }

x g I

unique

, F g A

To(X,. )

lira ~A (x,F) n +oo n

is countable.

lim ~A (~ rr+OO n

FI, F 2 ~ ~

n

If

A

~ e G(V)

gale convergence

Lemma

2.18

P~_F) ).

has a by

x

If

29

2.19

(2.21)

for each

x g X

Zl(X,-)

is a probability measure;

(2.22)

for each

F g F

nl (-,F)

is _F -measurable;

(2.23)

for any

~ e _G(V)

and

F E F

~I(-,F) = E ( XF I ~

Proof

(2.21) holds by definition;

FgA

and hence hold for all

Let

I~

=

Lemma 2.9

Proof

)

~-a.e.

(2.22) and (2.23) are certainly true if

F e ~

by the monotone class theorem.

82

{ x g X : ~l(X,.) e ~(~) } .

I~ g F

and

~(I ~

= i

for all

~ e G~V) .

From (2.8) and the monotone class theorem we have that

x g I~

if and

only if

(2.24)

~l(X'F) = I ~An(Y'F) ~l(X,dy)

for all

n~

i , F g ~ .

Since both sides of (2.24) are F -measurable functions of only countably many equations, we get (2.23) shows that if

I ~l(X,F) d~(x)

G E F=~

=

G

I G

and thus (2.24) holds for

and

I

I ~ g F==~ . Now a simple calculation using

~ e GQVJ

then

~A (y,F) ~l(X,dy) n

~-a.e.

x , and there are

x e X ; hence

dD(x)

,

~(I ~

= i .

So far the proof has been fairly straightforward,

82

and has followed an

2.20

30

obvious path. Now comes a trick, due to Dynkin, which enables us to complete the proof. Let It = { x s I~ : ~l(y, -) = ~l(X,')

Lenmm 2.10

Proof

1% g F

==co

It =

~ FgA

, and

for

~(1%) = i

x e I~

then

x g I~

c

G~)

.

=

x g It(F)

we have

for

Zl(X,.)-a.e. y s X } .

if and only if

I { ~l(X,F) -~i(Y,F )

but since

for all

It(F) , where

It(F) = { x e I ~ : ~l(Y,F) = ~l(X,F)

Let

~l(X,')-a.e. y s X } .

}2 ~l(X,dy)

I ~I(Y'F) ~l(X'dy)

0 ;

=

nl(X'F)

and thus

I{ ~I(X,F ) _ ~[l(Y,F) }2 ~[l(X,dy)

=

I { ~I(X'F)2 - 2~I(X'F)~I(Y'F) + ~I(Y'F)2 } ~l(X'dy)

I

~I(Y,F) 2 ~l(X,dy)

- ~l(X,F) 2 .

Ther'efore

It(F) = { x g I ~ :I~I(Y'F)2 ~l(X'dy) = ~l(X'F) 2 } ; and in particular

It(F) g ~

. Now if

~ g_G(V)

then

31

2.21

I I~I(Y,F)2 ~l(X,dy) d;J(x) - I1[l(X,F)2 dM(x)

But

=

I~I(Y,F)21~I(X,dy)

=

I~I(Y'F)2 d~(y)

d~(x)

- I~[l(X,F)2 d~(x)

- I~l(X'F)2 d~(y)

I ~I(Y'F)2 ~l(X,dy) -

~I(X'F)2 ~ 0

=

0 .

for all

x E I~ (and thus for

~-a.e.

x c X ) since we have written this above as the integral of something squared. This shows that

~(It(F)) = I .

Lenmm 2.11

x e I*

Proof

If

Let

x g It

Therefore

~l(X,F)

define

~

E ( XF

is either

~

: X • F

I

is an extreme point of ~ ( ~

for

Hi(x,') . If

F=~ ) =

~I(.,F) = ~l(X,F)

field. Thus by Theorem 2.1

Now

~l(X,.)

and write

~I(-,F) =

but we also have

then

82

0

or

~l(X,.)

~R

XF

~-a.e.

.

F c F=~ then we have

~-a.e.;

and thus

I , and hence

~l(X,F) = XF

~l(X,-)

~-a.e.

has trivial tail-

is an extreme point of G ( ~

.

82

by

~l(X,F)

if

x E It ,

7rl(Y,F)

if

x s It ;

~(x,F) =

where

y

is some arbitrarily chosen element of

It . It is a simple matter to

32

check that

2.22

?r satisfies all the requirements of Theorem 2.2; and thus the proof

of this theorem is now complete.

3.1

3. Existence of Gibbs states

In this section we will find sufficient conditions on a specification ensure that

G~)

p g G(V)

with

~n g ~Q~) such that

~ = o~)

to

is non-empty. The techniques we will use also give suffic-

ient conditions to ensure that that given

~

G~V)

is sequentially compact (in the sense

' n ~ 1 , then there exists a subsequence ~(F) =

lim ~n (F) j~ J

for all

F g ~ , where

{nj}j> 1 ~

and

is a field

which we will define below).

It is worth pointing out that there exist specifications defined in a natural and reasonable way for which there are no Gibbs states. We will briefly describe an example of this. Consider the lattice model described in the introduction, but now suppose that

Y

is countably infinite. Let

and we will consider the basic phase space

X

S

be the integers,

to be the phase space (i.e. the +

~ample space) of a Markov chain with state space

Y . Let

Q : y x y

>R

be

a stochastic matrix; then there is an obvious way to try and define a specification using the conditional probabilities defined by

Q . Suppose that

all its entries strictly positive and that all its iterates

Qn

Q

n > 1

has are

finite. Then there are no problems in defining these conditional probabilities, and so we get a specification defined it, hence and

Q

RA = X

~

(and this is a specification as we originally

for all

A g ~ ). However, if

Y

is the transition matrix for a random walk (i.e.

function of

y-z ) then Spitzer (1974b) has shown that

is the integers,

Q(y,z)

is only a

~(~_V) is empty. This

example is considered in more detail in Section 5, along with a generalization of this result due to Kes~en (1975). As at the end of the last section we will work with standard Borel spaces, but now add some more structure to equipped with a partial order

~

(X,F) . Let

N

be an index set

which is directed upwards and eountably

34

3.2

generated. We will suppose that we have a family

{$}eCN

of sub-d-fields

of

satisfying:

(3.1)

if

(3.2)

~

~ ~ ~

then

~

C

~

;

is the smallest o-field containing

(3.3)

(X,_~)

(3.4)

if

U

~e

is a standard Borel space for each

e I =< Co.

...

and

An

is an atom of

e g ~ ;

_~

for

n > i , and if

n

AI ~

A2 ---~ ..., then

~ n

An

# ~ .

& 1

We are thus assuming that

(X,~)

is the projective limit of the spaces

{(X,B6)}6E ~ ; these assumptions will make

If we are in the situation where from outside

(X,~)

F = ~(S-A) =A

A , then we will usually have

measurable from inside

a standard Borel space.

N = ~

, the events measurable

and

~A = ~(A)

, the events

A 9 In such cases we will almost always have (3.1),(3.2),

(3.3) and (3.4) holding.

(For example, it is certainly true for the lattice

model considered in the introduction.) The important technical fact that follows from our hypotheses following

(3.5)

(see Parthasarathy

if

DO g P B~)

and

is the

(1967), Theorem 4.2, page 143):

{DO}OE N

are consistent

all

e < D , F r _~ ) then there exists

all

e c N , F E ~

D g P~')

(i.e.

be(Y) = D~(F)

such that

for

D(F) = D0(F)

. (Note that by (3.2) and the monotone class theorem

for D

unique. )

Let

A =

U

~

; thus

A

is a field generating

~ ; if we are in

is

35

the case when

~ = ~

and

~ A = %(A)

then

3.3

A

is called the field of local

observables.

Theorem 3.1 Suppose

(3.6)

Let

~ = {~A}AgC

be a specification

with respect

to

= {RA}A~s

that

for each

A E C

there exists A

g A n > i , with = ' =

n

AI ~

A2~

"'"

=

n>

1

snd that for some

x g

(3.7)

given

e ~ N

6 > 0

such that if

~

and

RA

we have

y > 0

then there exists

F g ~

with

, A E ~

, and

~(F) < ~

then

, and

e P_(Be) , A S ~

~(x,F)

< y

for all

~ ~ A .

Suppose also that

(3.8)

given

F ~ ~

~-measurable all

Then

funstion

f : X

y > 0 >R

then there exists

such that

O ~ N

I ~A(Y,F) - f(Y)

and a

I < Y

for

y g RA .

G(_V_)

is non-empty.

Note that for our original definition holds trivially,

and (3.8) just says that for

in the uniform closure of that are B~-measurable slightly stronger

of a specification

B(~)

for some

than (3.7) is

A g ~

and

(when F g ~

, the set of bounded functions e c N

RA = X ) (3.6)

from

~A(-,F) X

to

. Note also that a condition which is

is R

9

36

(3.9)

given

@ g N

then there exists

~ g P~B@)

{~%(x,.)}~>__A , considered as measures on continuous with respect to that if

F e ~8

with

~

then

and

(X,~),

(i.e. given

m(F) < 6

3.4

A g ~

such that

are uniformly absolutely

y > 0

there exists

~%(x,F) < y

for all

6 > 0

such

% ~ A ).

(3.9) should remind the reader of the hypotheses of the Dunford-Pettis theorem (see, for example, Meyer (1966), Page 39), and thus (3.7) and (3.9) can be regarded as compactness criteria.

The basic idea for the proof of Theorem 3.1 is simply to choose a suitable ~ountable subse~

~

of %

with

Z=

o~),

to ehoose

which (3.7) holds, and to find an increasing sequence

lim ~A (x,F) n+~ n ion to

~

exists for all

of an element

The choice of have either (i) ~

~

V

x ~

{An}n> 1

[~

RA

such that

F E D ; then to prove this limit is the restrict-

of

P(~) ; and finally to show that

depends on how big the

B@

many atoms. By (3.1), if (ii) holds for some

~ ~ G~)

are. For any

has a countable number of atoms, or (ii) _~ 8 s N

@ e N

e ,

Then, as in Section 2, for each

=

8 g ~

{An}n> I

there exists a countable field

e s N 9 ~0

and such that finite additivity implies countable addit-

ivity for bounded set functions on and

O g ~ ,

@ g N .

We first give the proof for the case when (ii) holds for all

~O = 0 % )

we

then it holds for all

we can, without loss of generality, ~ssume that either (i) holds for all

such that

9

has uncountably

> 8 ; and since we will only be interested in what happens for "large"

or that (ii) holds for all

for

~

. Choose increasing sequences

which generate the orders on

~

and

~

{@n}n>l

respectively. Let

37

D =

~

_~

n>

i

, and thus

~ = o(D)

3.5

. Now fix

x g

n

~

R%

satisfying

(3.7);

AEC

by replacing

{An}n> 1

if necessary with a subsequence,

lim ~A (x,F) n ->~~ n

exists for all

clearly finitely additive on

F e _D 9 Let

Ao

we can assume that

~(F) = lim ~A (x,F) rc->~176 n

; then

and hence countable additive.

~

Thus

is

D , res-

n tricted to

A@

, has a unique extension to a probability measure on

(X,Bo) m

m which we denote by the

Dm

n__> I

~m " (Note that we have no a priori reason to suppose that

are consistent, let

,

although

in fact it follows from the next lemma.) For

~n = ~A (x,-). n

Lemma 3.1

For any

m=> i

and

F g Be

we have

pro(F) = lim ~n(F)

.

m Proof

Let

y > 0

and choose

A E C

, ~0 g P ( ~

)

and

~ > 0

as given by (3.7)

m (with

e = O m ). Now by standard measure

theory we can find

such that

A g ~ m

both

Pm(AAF) < y

such that

A

and

00(AAF) < ~ (where

> A . Then for n ~

I ~ m (F) - Vn (F)

we have

I

I Pm (A) - ~n (A) + ~m (F-A) - Vn(F-A)

<

I Pm (A) - Vn(A)

.

- Pm (A-F) + ~n (A-F)

I + ~m (FAA) + Vn(FAA)

But by definition we have

Pm(F) = lim ~n(F) rr+oo

Choose

~ o

o

=

n > n

AAF = ( A - F ) U ( F - A ) ) .

82

<

I

I ~m (A) - ~n (A)

lim Vn(A ) = p(A) = Dm(A)

, and hence

I + 2y .

no

3.6

38

Lemma 3.2

for all

There exists a unique

F g A . (Thus

Proof

B

P ~ P(F_F) such that

is an extension of the

By Lemma 3.1 the measures

{Pm}m>l

= Bm

for all

m~

defined above.)

are consistent. Clearly we can ex-

tend this sequence to get a consistent family Be

P

B(F) = lim ~A (x,F) n

{p0}6g~

with

p o g P(B_~)

i . Therefore by (3.5) there exists a (unique)

and

B e P F(~_

m

such that

B(F) = Pm(F)

for all

, m~

F c ~

i . Hence from Le~mm 3.1 we get

m

B(F)

= lira ~ A

(x,F)

.

82

n-+~176 n

Lemma 3.3

B(RA) = i

for all

A ~ C 9

Proof

Using the notation of Section 2

~) g T A

then

Now

v(RA) = 1 . Thus if

~)n = ~A (x,.) e_T A II

that if

B(RA)

=

n => no

i

TT[ ~

~ g T%

then

and thus, given any

_TA

whenever

~)(RA) = 1 A e =C

A < % ; also if

for all

there exists

A < 7[ . no

such

n

then

~)n(RA) = 1 . Therefore by (3.6) and Lemma 3.2 we have

.

Lemma 3.4

B g G~V_) .

Proof

B(~)

Let

denote the set of bounded F--measurable functions from

X

R , considered as a Banach space with the uniform norm; let B~A) = { f g B ~ )

Then

: f

is ~ - m e a s u r a b l e

W

=

{ f e B~__F):

I f dB =

W

is a closed subspace of

for some

lim I f rr+oo

e ~ N } ; and let

d~)n } .

B~F) , and since Lemma 3.2 just says that

to

39 XF e W

for all

F ~ A , we certainly have

y > 0 ; then by (3.8) there exists for all

nl > no

such that

I I

and hence

A

f ~ B~A)

W . Let

such that

:A(Y'P) dD(y)

- I

: ~A(y,F) d~(y) =

n~

> A

~(R A) = i ; also

i ~ f dD - I f d~n i < y

"rrA(Y'F) d~n (y)

A E C , F E A

and

I ~A(Y, F) - f(Y) I < "f ~)n(RA) = i

for all

f E W , so there exists

for all

n>

nl 9 Thus for

n>

nI

I < 3y ;

lim I 7rA(Y'F) d~)n(Y) , rr+oo

(i.e. nA(',F) E W ).

then

I ~A(Y,F) dgn(Y) =

: nA (y'F) ~An(X,dy) =

ZAn(X,F)

~ ~A(y,F) d~(y) = D(F) . We have shown this equality

Therefore by Lemmm 3.1

for all

B(~ C

y g R A . Now from the proof of Lenmm 3.3 we have

n__>_ n o , and by Lemma 3.3 we have

But i f

3.7

F g ~ ; by the monotone class theorem it must hold for all

hence by Proposition 2.2

~ ~ G(V) .

F g ~ , and

82

This completes the proof in the uncountable atom case; so now suppose that for each

@ E ~

_~

has countably many atoms. Let now

set of all finite unions of atoms of before and such that

additive on

_~e

B@ ; let

D(F) = lim ~ A (x,F) n +~o n

denote the

{0n}n~ I , {An}n~ I , ~

exists for all

and thus extends to a measure on m

A0

~0

F E ~ . ~

be as is finitely

which we denote by m

~m "

40

~m(X) < I ; the thing that could go wrong is that we might

It is clear that have

~m(X) < I . However,

Lemma 3.5

Proof

~m(X) = i

y > 0

Let

3.8

the next lemma shows that this does'nt happen.

for all

m~

I .

A s ~ , ~ g P(B=9 )

and let

and

6 > 0

he as in (3.7)

m

(with

@ = 8m ). We can find

F g ~O

such that

~(F) => I - 6

and tbus

m

W%(x,F) ~ I - y y > O

for all

% ~ A . Hence

is arbitrary we get

~m(X) = i .

~m(X) ~ ~m(F) ~ i - y ; and since 82

The rest of the proof in this case is the same as before; we leave the reader to check the details. Note that the proof of Theorem 3.1 shows that if (3.6) and (3.8) hold then for any

x e

~

RA

satisfying

(3.7) we can find

~ ~ G~V)

and an

AsC increasing sequence

{An}n> 1

such that

D(F) = lim H A (x,F) ~

for all

F e ~

We should point out that in a lot of the examples we will consider Section 6 (continuous models) hold (for suitable we can construct

(3.6) and (3.8) will not hold; however,

x e X ) and thus Lemmas 3.1 and 3.2 still work. ~ ~ P~F~

with

~(F) = lim w A (x,F) n->oo n

before, but we have to use different methods

.

n

for all

to show that

(3.7) will

This means

F g ~

D e ~_~J

in

just as

. We will

look at this at the end of the section. If we assume a bit more than the hypotheses show that

G(~

is sequentially

compact

of Theorem 3.1 then we can

in the sense mentioned

before.

41

Theorem 3.2

Let

~ = {ITA}Ag~

3.9

be a specification with respect to

~ = {RA}Ag ~ 9

Suppose that (3.6) and (3.8) hold and that

(3.10)

6 > 0

given

such that if

Then given e G~)

Proof

O e N

and

y > O

F e ~

x)n s G ~ ) such that

then there exists

with

~ G~

then

D(F) = lim Vn.(F) j~o j

for all

F g A =

for all

{n.}j> I

, A e C

nA(y,F ) < y

, n > i , there exists a subsequence

Once we have got a subsequence

~(F) = lim 9 (F) j-~o nj

for all

{nj}j~ 1

and y g X .

and

F g A .

and

such that

~ e P(FO

then the same proofs as before will give us

; thus we need only look at the analogue of Lemma 3.1. Consider the

uncountable atom case; let sequence

{nj}j> 1

{Sn}n> I

such that

~A(y,F) < y

for all

and

~

~(F) = lim v

=

for all

00(F) < 6

0J e P ~ )

j-~o

y e X

then

nj

~(F) < y

be as before and choose a sub(F)

exists for all

for all

F e ~ . Now if

v e ~ A ; and since

~n e ~ %

A ~ ~ , n ~ i , the proof of Lemma 3.1 goes through just as before.

Similarly, in the countable atom case Len~na 3.5 has the same proof.

If (3.10) holds then (3.7) holds for any theses of Theorem 3.2 we have

G(V~

82

x g X . Thus under the hypo-

is non-empty, provided

~

RA # ~ 9

A condition which is a bit stronger than (3.10) is

(3.11)

given

{~A(Y")}y

O ~ N

then there exists

m s P%)

g X ' considered as measures on ( X , ~ )

continuous with respect to

m .

and

A g ~

such that

, are uniformly absolutely

42

3. i0

We now consider a more general situation which will be applicable

to

the continuous models to be dealt with in Section 6. As we have already mentioned, in most of these models

(3.6) and (3.8) will fail to hold,

and so we need some-

thing else to put in their place. Let

_~_

denote the set of countable intersections

(thus (3.6) is the requirement

that

RA E ~

be an increasing sequence of elements

Theorem 3.3

Let

~ = {WA}AEC

of

for each

~

of elements

A e ~ ). Let

of ~=

~

,

{Un}n>=l

.

be a specification.

Suppose that for some

x e X

(3.7) holds and also

(3.12) all

given

A e ~

~ > 0

then there exists

m ~ i

such that

WA(X,U)

~ i - 8

for

.

Suppose also that

(3.13)

given any

such that

A e ~ , F e ~ , ~ > 0

IwA(Y,F) - f(Y) l < 8

and

for all

m~

y e U

i

then there exists

, (where

B(A)

f e B(~)

is the set of

m

bounded f ~ c t i o n s

Then

~(~)

Proof ~(F)

that are 4 - m e a s u r a b l e

for some

8 e ~ ).

is non-empty.

As before we use (3.7) to find = lira ~A (x,F) n~

for alL]

A

increasing and

n

~ ~ P(F__) such that

F s A_ 9 (Note that (3.12) implies that

x c

n

~

RA.)

AeC

We will show that

~ ~ _G(~) . Put

~n = ~A (x,.)

. Let

A e C

, F ~ A , and

6 > 0 ;

n

choose for all

m > 1

such that

n => i , and since

~A(X,%) Um e ~

>= 1 - 6

for all

A e C . 'lhus ~n(Um) >__ 1 -

this implies that

~(U m) => 1 - ~ . There

43

exists

f a B(A )

assume that

n >n O

I

such that

[~A(Y, F) - f(Y) l < 6

0 <= f ~ 1 . Let

then

3. ii

n

o

Ip(F) - Pn(F) I < 6

wA(Y,F) dp(y) - ~(F) I

he such that

and

for all

A < A ~ n

I I f d~ - I f

y E U m , and we c~n

and also such that if o

d~n I < 6 . If

6 + ~(x-~m) + I If d~

n n>= o

then

.(F)I

-

<

26 + I If d~- p(F)I ~ 36 +' ~f dPn- ]J(F)I

<

4~ + Pn(X-%) + I lwA(Y,F ) dPn(y) -p(F)J

<

56 + I IwA(Y,F)~ A (x,dy) - p ( F ) i n

56 + I~A (x,F) -p(F) l ~ 66 . n

Thus

p(F) = IwA(Y'F)

dp(y)

for all

A e ~ , F e A , and hence

p s ~(~)

.

In most of the models in Section 6 we will have a condition stronger than (3.12) holding, and also something stonger than (3.7)- These conditions are:

(3.14)

given any

F ~

then

Um

6 > 0

then there exists

lira inf WA(X,F) ~> 1= Ae C

m >= i

for all

such that if

x e U

, where

F e A U

given

8 s N

that if

F e ~

with

and

y > 0

~(F) < 6

then there exists then

~

%

;

m~l

=

(3.15)

=

with

~ e P(~)

lira sup WA(X,F) < y A~C

and

for all

6 > 0 x e U=

such

44

(3.13),(3.14)

and (3.15) imply a compactness

~(U ) = i . %~is is because such measures,

Lemma 3.6 m~

i

propel~y for those

U s ~(~)

(3.14) and (3.15) give us uniform estimates

with

for all

as seen by the next two lemmas.

Suppose

such that

~

satisfies

D(U m) ~ I

Proo_____f Let

U e G(VJ

If

then we have

F DUm

3.12

- 6

with

(3.14). for all

Then, given any ~ s G(V)

w(U ) = i . Given

6 > 0

lim inf WA(.,F) => i - ~

6 > 0 , there exists

with

~(U ) = i .

let

m > i

~-a.e.

be as in (3.14).

, and since

Ass ~(F) =

IWA(X'F)

d~(x)

it follows

~(U ) > 1 - 6 , because m

U

~

Suppose

~ e P(~)

for all

V and

~ e G(V)

satisfies ~ > 0

with

Note that if

Theorem 3.4

~

(3.15).

~n(U ) = 1

Then, given

such that if

F s ~

% s N with

and

~(F) < ~

is the measure given by Theorem 3.3 then it has

for each ~(U ) = i

y > 0 , there then

~(F)

82

Suppose that (3.13),(3.14)

e G(_V) with

Proof

82

w(U ) = i .

Proo___f Same as Lemma 3.6.

with

e A..

~(F) ~ i - ~ . Thus

m

Lemma 3.7 exists

from Fatou's lemma that

and (3.15) hold for

V . Let

n . Then there exists a subsequenee such that

~(F) = lim ~n.(F) J+~ 3

~(U ) = i .

for all

~n e G(V)

{nj}j>_ 1

and

F s A .

Using Lemma 3.7 and the proof of Theorem 3.1 we can find a subsequence

< y

45

{n.} > j j=l

and

I~ E P(_F) -

such that

Lemma 3.6 and the proof of Theorem

3.13

~(F) = lim ~n (F) j~ j 3.3 show that

for all

~ E G(V)

F e A

. Then

; and it is clear that

4.1

4. Invariant

specifications

Let us look again for a moment st the lattice model described Suppose S

that

S , the set of sites,

is a group, and each

t e S

{Tt(x)} s = xt~ s , where at site

s E S

xs

acting on

t

: X

X . Let

Po~F)

integers. ~X

Then

by

x (i.e. the value

x ). We thus get a group

under these transformations.

will also be invariant;

T

the s th. coordinate of

in the configuration

that are invariant

Z d , the d-dimensional

induces a transformation

denotes

F_-measurable transformations

specification

is now

in the introduction.

denote

{Tt}t e S

of

the random fields

In most physical models the

for the lattice model this will mean that

T t (Y) ~

t

= ~

for all

A e ~

and where we consider

Tt

, y e X(S-A)

, t e S ; where

also as a mapping from

X(A)

A+t = { s+t : s e A }, to

X(A+t)

denote those Gibbs states with respect to this specification One might think that

~

being invariant would imply that

that any Gibbs state is automatically and the existence iant corresponds

of non-invariant

invariant.

. Let

that are in

G_o(_V) = G ~ )

However,

_Go~V ) Po~_F) .

, i.e.

this is not the case,

Gibbs states when the specification

to the physical phenomenom of symmetry breakdown.

is invar-

(Examples of

this are given in Section 5.) In the case considered happen that invariant

is countably

( Y

finite)

G(V)

is closed in the weak topology.)

infinite then this need not happen.

Consider

ainly gives a specification which is invariant. the number of invariant Q

But Kesten

However,

if

the example given

at the beginning of Section 3 in terms of the stochastic matrix

holds if and only if

it will always

Gibbs states will exist whenever any Gibbs states exist.

(This is true, anyway, when Y

in the introduction

Q . This cert-

(1975) has shown that

Gibbs states is either one or none, and that the former is positive recurrent.

On the other hand Spitzer

has given examples for which Gibbs states exist with

Q

null-recurrent.

(1974b)

47

4.2

Inmost models in classical statistical mechanics there will be a group of bijections of the phase space (usually induced by the group of translations acting on the underlying space). The specifications that arise are also, in a natural way, invariant under the bijections. Thus, as in the lattice models just considered, we have the problem of analysing the invariant Gibbs states. We will do this in this section. Let

H

be a group of ~-measurable bijeetions from

define what it means for a specification H-invariant then we will look at

~

X

to

to be H-invariant.

~o~V) = G(V)~ Po~_F) , where

the set of H-invariant random fields.

X . We will If

~

is

Po~_F) denotes

( II will be fixed for this section, so

we will not bring it into the notation.) We will characterize the extreme points of

G_~O

: under reasonable hypotheses they will be exactly the H-ergodic

elements of G~O

G o (~

. In the case when

(X,~)

is a standard Borel space and

# ~ , Po~F) # ~ , then we will show that either

every element of

G~)

~ ~ G~

and

is non-empty, or

is mutually singular to every element of

this is uniform in the sense that there exists all

Go~)

~(F) = 0

for all

be non-empty whenever it can be. If

F E ~

such that

Po~_F~ (and ~(F) = I

~ g Po_(~ ). This shows that ~o(V_) # ~

it in terms of its extreme points, provided

for

G (V_O will

then we get a representation of

Po_~J

has a representation in terms

of the H-ergodic random fields. At the end of the section we give sufficient conditions to ensure that ~o~(~

is non-empty. Some notation: let

4 -1

are ~-measurable;

if

~ : X---~X ~

be a bijection such that both

is a sub-G-field of

~_B) = { B g % : ~'I(B) g B } ; for

~ ~ PCB)

let

~

~

and

then let ~

g p(~(B))

be given by

48

(~v)(F) = ~(~J-I(F)). p f : X

>R

Let

B = ~(B)

~ : X • ~---gR

F e ~ , and ~

~(x,')

: X x F

>R

~ ~ {~A}Ae~

~OV = p . A function

f = f-~ ; similarly

B ~ ~

will be

.

be a kernel (i.e.

is a measure for each

I[(-,F)

is ~-measurable for all

x s X ); we can define another kernel

by

~(x,F)

If

will be called ~-invariant if

will be called ~-invariant if

called ~-invariant if

4.3

= ~(~-l(x),~-l(F)).

is a specification with respect to

~ = {RA}Ag ~

then

~A

also has the properties:

(4.1)

~A(',F)

(4.2)

~A(-,F)

We will say that e~

such that

is ~(~A)-measurable for all

= XF~(RA)

if

F e ~(FA)

A e

~(~.~) C

~

~A

and

(~)~A

measurable from outside

A , then

~(A) e C

implies that

V

F A = F(s-A)

"

, the events that are

S . In this case we will have

~(A) = { ~(t) : t e A } . The bijection

for all

~-invariant is that

=

then there exists

~J will usually be induced from a bijection

: S------~S of tile underlying space ~(F%) = --~(A) ' where

;

.

~ is ~-invarisnt if given any

If we are in the situation where

that

F e~

~

A e C , and thus the natural definition of

~I[A = I:~(A) for all

will be such V

being

A e C . This condition certainly

is ~-invariant as we have defined it above.

It is important to note that if

V

is ~-invariant then we must have

49

Lemma 4.1

If

such that

Proof

that

TA

% g ~

with ~

is ~-invariant then, given any

~(T%) C

Let

p E~

~

~ g ~ , there exists

be such that

~i = t%(~) , and

~(F%) ~

~A

~ E P(F~,R%)

and

(~%)wA = ~%

; suppose that

. Then a simple calculation shows

= tA(1) , where

Proposition 4.1

If

~(~)(~(x),F)

~

Now let

H

d~(x)

is ~-invariant and

Immediate from Lemma 4.1.

p ~ G(V)

then

r

be a group of F-measurable bijections from

denote the H-invariant elements of

Po~)

~(~)n

g ~_V)

.

X

X ; we

82

will call an object H-invariant if it is ~-invariant for all

G(%) =

to

~ E H . Let

P~F) , and for a specification

~

G(V)

and

~V)

~ e Po~_F) then we will say that

~-a.e. for all

D(F)

~

, for

is convex, we will first look at the extreme points of

. To do this we need some definitions: if

g P F(F) o --

let

is non-empty.

Since G o ~~

Po~)

.

Suppose from now on that we have an II-invariant specification which

% g

9

I(F) =

Proof

4,4

f

f : X

~R

is F--measurable

is p-almost H-invariant if

~ c H ; we also apply this term to

F g ~

if it holds for

is called H-ergodic if, given any D-almost H-invariant

is either

0

or

i .

f-~ = f

F g ~ , then

It is well-known (and easily checked) that the

H-ergodic random fields are exactly the extr~ne points of

Po~_F) .

XF .

50

If

H

is countable then we can leave out the

definition of is either

0

4.5

D

being H-ergodic;

or

i

thus in this case

for all H-invariant

"D-almost" D

in the

is H-ergodic if

~(F)

F g F . (We leave the reader to check

this.)

Proposition 4.2

~ g ~o(V)

is an extreme point of

given any D-almost H-invariant

F g F=F=F=F=~, then

particular any H-ergodic element of

Proof

g : X

Let

g~ g P (F) O

>R +

~V)

D(F)

is either

g

or

I . (in

is extreme.)

be bounded, F--measurable and with

if and only if

0

~ g dD = I . Then

is D-almost H-invariant, and thus by Lemma 2.4

--

gD g ~ _ V )

if and only if there exists

D-almost H-invariant and with

g = g

g : X

>R +

When way: let

~

H

with

g

~[

is countable the above result can be stated in a more elegant

denote the o- field of all H-invariant elements of

is an extreme point of is also true if

F==o-mea~urable,

D-a.e. The result thus follows in exactly

the same way as the proof of Theorem 2.1 (i).

H

G_o~_V) if and only if,

H

G ~)

if and only if

D

is trivial on

is separable in the sense that

H

~ ; then

D

F=~on I . (This

has a countable subgroup

such that H-invariance is the same as H -invariance.) O

O

In many cases we can improve on Proposition 4.2 because, if enough, then

F C ~

F C F== ( i n that

F

~

is nice

being D-almost H-invariant implies that we almost have is in tile completion of

F=~~ with respect to

speaking, this would say that, up to null sets, ~ C separates

H

if there exists a sequence of G-fields

D ). Loosely

F=~ . We will say that ~i ~

~2 C-

...

~

H

4.6

51

such that

A C C

~

there exists

Lemma 4.2

~ U n> i

is the smallest o-field containing

~ e H

Suppose that

with

H

~n

separates

) C_ F A .

~ , P e Po~_F) and

F--measurable and u-almost H-invariant. Then there exists and

~

n ~ i ,

, and for any

f :X ~

)R

with

is bounded,

~ = f

D-a.e.

F -measurable.

==co

Proof (The idea of this proof is due to F~llmer and can be found in Georgii (1973).) We want to show that LI(x,~,D) norm. Given

I] ED( f I FA ) -

But since

f

f = E ( f I F=~o )

y > 0 , there exists

ED( f i F=oo ) II < y

is D-almost H-invariant and

D-a.e. Let

A e ~

and

n ~ i

IIED( f I ~

p g P (F) O

g H

and

II

II

denote the

such that

) - f II < y 9

it follows that for any

--

we have

f - ED( f I B

and thus for any

II f -

E (f

IF

II ED( f I ~ % )

)

=

f - E ( f I ~(_B--n) )

~ ~ H

) II < II f -

ED( f I _Bn ) II +

) - E~/( f I F A ) II +

< 2y + II ED(f I ~(B))-ED(~

Now choose

~

D-a.e.,

such that

~-Bn) ~

IIEp(i [~CB~)-Ep(f i-FA)

ED( f I FA ) - EA( f I ~

) If

F A) II

~A ; then by Jensen's inequality we have

II ~ IlED(f ]~_B~n))-f ]I
4.7

52

and thus II ~ ( f

II Ep( f I ~ ) - f I[ [F)-f

Theorem 4.1

If

< 3y . Since

II = o , i e

H

separates

y > 0

f = E ( f i~

~

then

p C_Go~_V~

)

is arbitrary we get p-a.e.

82

is an extreme point if and

only if it is H-ergodic.

Proof

This is immediate

from Proposition

If we are in the case when

4.2 and Lemma 4.2

~ A = ~(S-A)

, and each

from a bijection

@ : S---~S

, then we will usually

An E ~

increasing

to

with

A , % a~

An

, there exists

~ g H

S .

Then

~

such that

is the finite subsets of

A~(%)

S , and

In the language of functional separates

=C

then

=oG (V)=

G ~V_) are compact lar models,

that

~ : X x F

)R

We want to show that

follows by assuming H

(X,~)

= ~(x,F)

that

is separable~

H

if, given any

= ~ . For example,

this will

Theorem 4.1 says that if

This result,

P o ~)_

H

and

for some more particu-

(1969), and Georgii

(1973).

is a standard Borel space, and in Theorem 2.2. Let

for all

F g ~

I(H) g F=~ , and to do this we need

by only countably many conditions.

~

Po(F)_ , at least when

is the kernel constructed

~(x,F)

~ B = =F(An) , where

is the group of translations.

can be found in Lanford and Ruelle

I(H) = { x g X :

that

H

in some suitable topology.

Let us now again suppose that

is induced

at the beginning of the section,

analysis,

is a face of

take

~ ~ H

I] will separate

clearly hold for the lattice mode] considered where

82

, ~ g H } .

I(H)

to be determined

For simplicity we will achieve this in what

is countable,

as we defined

though it would be enough to ass~Ime

it above.

(For example,

in most models

in

53

4.8

continuous statistical mechanics the group will be taking the countable subgroup to be

R d , and this is separable,

Qd , the points in

Rd

having all ratio~al

coordinates.)

Theorem 4.2

Suppose that

H

is countable. Then

(4.3)

I(H) ~ F=~~ , is H-invariant, and

(4.4)

G~V)

(4.5)

if

= ~

if and only if

~ g Po~F)

with

~ c G ~_(~ ; if

(Note that if

V

~

v(l(H)) =

~(I(H)) = I

D(F) = ~ ( x , F )

then

p(l(H)) = I

0

and

for all

for all

p

p g G~)

V g Po(~)

;

;

is defined by

d~(x) ,

is H-ergodic then

~

is H-ergodic then by (4.3)

is an extreme point of v(I(H))

is either

0

G~_V) or

.

i .)

We will break the proof of this theorem up into a number of lemmas. Note that we are not assuming

H

separates

~ , so the extreme points of

G l~)

need not be H-ergodic.

Lemma 4.3

For

~ g H

define

I(~) = { x g X :

Then

I(~) E_F_~ and

Proof by

~

Let

~

~(x,F) = ~ ( x , F )

p(l(~)) = i

for any

be a countable field with

in the definition of

for all

D g ~)

~ = o(A)

F g F } .

9

; then we can replace

1(9) , and thus we need only show that for any

54

F ~ ~

{ x g X : ~(x,F) = ~J~(x,F) } g _ F

for any

~ e G~V) . But

~(',F)

and

~(x,F)

dr(x)

, and that

~(',F) = ~ ( ' , ~ )

~-a.e.

~b~(.,F) are F -measurable functions, and

so the set on which they are equal is in

I

4.9

F

==co

= IXG(X)~(~-I(x)'~-I(F))

. Now let

~ e G(V) , G ~ F=oo ; then

dr(x)

G

I{X G

=

~(',~-I(F))}o~-I

~XG.~(.,~-I(F))

d(~-lV)

dv

I

=

~(., _I(F) ) d(~-Iv)

~-I(G)

= (~-Iv)(@-I(G)~ ~-I(F)) ,

by (2.18); since by Proposition 4.1

~-i

g G(V)

, and also

-I

(G) e F

But (~-Iv)(~-I(G)~-I(F))

=

= (~-Iv)(~-I(G~F)) = v ( G n F )

] ~(x,F) d~(x) ,

(again by (2.18)).

G

Thus

,~ ~ ( x , F ) d r ( x )

=

G

~ ~T(x,F)d]J(x)

for a l l

and hence

~(',F)

Lemma 4.4

I(H)

Proof

x g I(H) , ~ ~ H ; we want to show that

Let

G c F=oo,

G = ~(',F)

~-a.e.

82

is H-invariant.

~(x) g I(H) , i.e. we need

55

~(~(x),F)

= ~(~(x),F)

~w(~(x),F)

4.10

for all

~ e H , F g F . But

= ~(~-l~(x),~'l(F))

= (~-l~)li(x,~-l(F))

= w(x,~-l(F))

(since

= ~-l~(x,~-l(F)) =

~(~(x),F)

.

x g l(H) )

(again since

x g I(H) )

82

Together, Lemmas 4.3 and 4.4 give us (4.3), since

I(H) =

N

I (~')

9

~EH

Lemma 4.5 p = ~

Let

H

be countable,

let

9 r Po(~)

V(I(H)) = I , and let

, i.e.

P(F) = I ~(x,F) d r ( x )

Then

with

9

~ g G=o~V_) .

Proof

It is immediate from (2.8) and the fact that

x g X

that

p g G~)

. If

~ g H

then

(~p)(F) = p(~-l(F)) = IZ(x,~-I(F))

= I{~7~(',F)}'~ dV

=

~ I(H)

=

I~'(x,F)

~(x,F) d~)(x) =

z(x,-) e G(V)

I l(n)

~

dv(x) =

d(~v)(x)

=

1[(x,F) dv(x)

(~(x),F) d~(x)

I~(x'F)

=

p(F)

d~(x)

.

for all

56

Thus

D e P

CF) O

have

is an easy consequence

with

and thus

82

--

(4.4) g Po~_F)

9

4. Ii

~(l(H))

~o~V)

> 0

then

# ~ . Conversely,

~ g Po(~_~)

and

~(I(H))

of Lemma 4.5, because ~

if

E ~o(~_V) , where Go(~)

if there

exists

~ = {~(I(H))}-IxI(H)V

# ~ , then for any

~ g G~V)

we

= i . We have now only the second part of

(4.5)

left to prove.

Lemma 4.6

If

H

is countable,

if

V g P (F) O

and

~ = ~g

Proof

Let

F e F

,

then

~

G g F==~

with

F

is

an

extreme

be ~-almost

H-invariant

point

of

H-invariant;

and

is H-ergodic

V(Z(H))

with

= 1

--

G ~V)

since

~(G) = ~(F)

.

H

is countable

. (We can take

there

F =

~

exists

~(G)

, for

~gH example.)

If

~ g H

~(~(x),F)

Thus,

as

~(x,-) field,

x g I(H)

= ~(~(X),~(F))

V(I(H))

H-ergodicity

and

= I , we have

of

~

implies

is an extreme point and

F g F

; so

~(x,F)

since

= ~-I~(x,F)

7[(.,F)

that of

then,

: ~(x,F)

is ~-almost

~(" ,F) G~

~(F) = F , we have

is v-a.e,

for all

.

H-invariant;

hence

the

equal to a constant.

x g X , thus has trivial

can only take the values

0

or

But tail-

i . Therefore

==r

either or

~(-,F)

= I

~-a.e.

i . Thus by Proposition

The proof g P~_F_) then

~

or 4.2

~(',F) ~

= 0

v-a.e.,

and hence

is an extreme

point of

of Theorem 4.2 is now complete. = ~

that we can consider

if and only if ~

~ g ~C~)

as an affine mapping

Recall

~(F) G=~V)

from

.

(2.20)

. Thus T h e o r e m

from

is either

0

82

that if

4.2 tells us

{ v g po~_~) : ~(I(H))

= 1 }

57

to

G ~)

4.12

; this mapping fixes the points of

sends extreme points

G ~)

, and is thus onto, and it

to extreme points.

In all reasonable

situations

there will exist a representation

of

P (F) O

in terms of the H-ergodic measures. like this: if

E

denotes

the H-ergodic

an affine bijection

T : P(E,E)

subsets of

P(E,~)

E , and

denotes the point mass at exists;

let

E

The representation

~P (F)

denotes ~

elements of where

will usually be something

Po~_F) then there will be E

is a suitable o-field

the probability measures

for any

a g E , then

= { ~ g E : v(I(H)) = 1 }

on

E

(E,~)

and suppose that

E

of

. If

T(6 ) = ~ . Suppose

O

that

T

g E . Let O

= { E ~ D : D g E }

==O

--

and consider

the mapping

O

=

~.T

: P(~o,E)

It is not hard to show that extreme points.

E

is affine, onto, and sends extreme points

to

From this it is a simple matter to obtain a representation

in terms of (Eo,~o)

~C=o ~)- "

P(Eo,E=o)

, where

under the equivalence

(Eo,~)

relation

We will now give sufficient

is the quotient

, where

conditions

~ ~ 6

if

to ensure that

of

space got from E(~ ) = E(6~)

_Go~_V)

.

is non-

empty. Let us return to the situation of Section 3; thus we have an increasing family of o-fields H

{Be}ee ~

be a group of bijections;

(4.6)

if

8 s N

and

~ g H

satisfying

(3.1),(3.2),(3.3)

and (3.4). Again let

we will suppose that

then thei-e exists

~ g N

such that

~(Be) ~

B~

.

4.13

58

Theorem 4.3

Let

H

be a countable abelian group; let

= {WA}AeC

be an

=

H-invariant specification with respect to (3.10), and with

Proof

Let

n

~ =

~

RA # @ 9 Then

~ = {RA}Ag C

~_V)

satisfying (3.6),(3.8),

# ~ .

~O ; then by (4.6) we have

~ A ) C_ ~

for all

@ E H .

0 c

Since each and

H

and

~o

on all

--~O is countably generated, the order on

is countable, we can find a countable field )C

(X,~)

=oA

for all

~ e H . Let

topologised so that

F g A

is countably generated,

=oA C. =A

such that

=F = ~ )

M(X,P)= denote the finite signed measures

~n----~p

. This topology makes

~

M(X,F)

if and only if

~n(F)

>p(F)

for

a linear topological space (it is

==O

Hausdorff since the elements of M(X,F) are determined on 3.2 tell us that

G~)

is compact.

If

). Theorems 3.1 and

is a non-empty, sequentially compact subset of

but it is not hard to check that G~)

A

~ E H

then

M(X,~)

M(X,~) ,

has a countable base for its topology, so

~ : M(X,%)------~M(X,_F~

inuous linear map, and by Proposition 4.1 we have

~))

is clearly a cont~

G(V_) . The result

thus follows from the Markov-Kakutani fixed point theorem (see for example Dunford and Schwartz (1958), V. 10.6, page 456).

82

Finally, we leave the reader to check that the hypotheses of Theorem 4.3 can be weakened, and that satisfying (3.13),

_Go(~)

is non-empty if

~

(3.14) and (3.15) (provided

is an H-invariant specification U

~ r ).

5.1

5. Lattice models

So far we have developed a genera], theory of random fields,

specifications,

Gibbs states. The time has come to examine some particular models. ion we will look at lattice models, introduction Let

S

be a countable

suppose that at the site

about the

set, representing

t e S

in the

Yt )'

a collection

of "sites",

we have an entity whose possible

described by the points of a set

In this sect-

and will study the model described

(but now making no assumptions

and

Yt " The cartesian product

and

states are

-~

repre-

Yt

t g S sents all possible configurations phase space F =

X . Suppose that

of the system,

Yt

and this will be our basic

comes equipped with a o-field

F t , the product o-field on

X . For

AC-S

let

=tF

Tr ~ ,

X(A) =

t ~ S

_F_o(A) =

; let

teA

~

Ft

, and let

PA

be the projection

from

X = X(S)

onto

X(A)

.

t gA Let

__F(A) denote the sub-o-field

depend

on

the coordinates

C let

in

F

observable)

, thus

FA

from outside

the order by

~

for

A s ~

let

finite subsets of

consists of the events

A . Since

(rather than by

we have a o-finite measure

consisting of those events which only

A , i.e. F(A) = (pA)-I(Fo(A)).

will denote the (non-empty)

F A = ~F(S-A)

measure);

of

mt ~A

C

~ on

S ; and for

A e C

that are measurable

(or

is ordered by inclusion we will denote

used in Section 2). Suppose for each (Yt,~t)

t e S

(which we will use as a reference

denote the product measure

~--[

~t

on

tea (X(A) '~o (A)) . We will start by assuming absolutely

continuous with respect

that we llave a specification to

~A

for each

A g ~

and

for which

o~

y g X(S-A)

is (where

60

OK

5.2

is the measure described in the introduction). This will impose conditions

on the Radon Nikodym derivatives of the

oK

(in order that (1.3),(1.4) and (1.7)

hold), and we will compute what these conditions are. From this comes a representation of the derivatives in terms of what, in a physical model, would correspond to a potential. We then reverse the procedure, and starting with a suitable potential we construct a specification. The properties of these specifications are then worked out in terms of the properties of the potentials. In this section we will only deal with specifications as we originally defined tbem (i.e. kernels satisfying (1.3),(1.4) and (1.7)). This is mminly for the sake of simplicity, and it is left as an exercise for the reader to see what modifications are necessary in the more general case. In fact, the more general case will be covered by what we do in Section 6. Some notation: it will be convenient to ~ i t e

(for

x E X , A~

X(A)

whenever

Let function on

S ), and in fact to write y E X(B)

{fA}Ag C X(A)

with respect to

with

YA

xA

instead of

for the projection of

be a collection of ~-measurable functions, fA(yx')

for fixed

y $ X(S-A)

, will represent the density o f

~A " (Here we have identified

~A(X,F) =

~

y

onto

B ~ A .

X

with

X(S-A)xX(A)

that we will continue to do.) We thus want to try and define

(5.1)

PA (x)

fA(Xs_A×w ) d~A(W)

, as a oK

, something

~A : X x ~

,

G

where

G = { w g X(A)

kernel we need:

:

Xs_AXw g F } . For this to make

~

a probability

)R

by

61 (5.2)

fA ~ 0

(5.3)

for all

5.3

A c~ ;

I fA(yxw) d~A(W) = i

for all

y ~ X(S-A) , A e ~ .

Suppose that (5.2) and (5.3) hold; it is then easily checked that (1.3) and (1.4) hold. Thus for ~ = {~A}Ae~ (1.7) holds; i.e. whether

to be a specification depends on whether

~%~A = ~% whenever

A C-% E ~ . A simple calculation

shows that (1.7) will hold if and only if

[

I fi(Xs-%Xw) I f%(xS-%xw%-Axz) d~A(z) d~i(w) F

(5.4)

I

f%(Xs_~Xw) d~7~(w)

for all A C_ % g C , x g X , F g _F_o(Ti ) .

F

In particular, (1.7) will hold provided

(5.5)

f%(x) =

fA(x)I f%(Xs_Axw) d~A(W)

for all

x g X ,A ~ % E~ .

(The converse of this is almost true, in that (5.4) implies (5.5) holds for ~-a.e. x% .) We can in fact replace (5~

by a simpler condition, given by the

following result.

Lerm~a 5.1 I

Suppose (5.2) and (5.3) hold. Then (5.5) is equivalent to if A ~

~~

and

x

~ g X with Xs-~ = ~S~A

(5.6) L

f%(~)fA(x) = f%(x)fA(~) .

then

62 Proof

5.4

Suppose that (5.5) holds and let A ~ % s C , x , ~ g X with

Xs_A = KS_A . Then

f~(~)fA(x) J~(Xs_A• %(w)

:

an~ thus f%(~)fA(x) = f%(x)fA(~)

But if

I f%(Xs-AXW) d~A(W) = 0

F(x)f~(~) ]f~%_~xw)%(w) provided

then

,

I f%(Xs-AXW) d~(w) > O .

f%(x) = f%(x) = 0 , and so we still have

f%(x)fA(x) = f%(x)fA(x) . Conversely, suppose (5.6) holds. Let

A~%~C,

xsX;

if w,z C X(A) then

fA(xs_Axw)f%(Xs_Axz)

Now by (5.3) there exists

by (5.3) we have that

w

s x(A)

f%(xs_AXw)fA(xs_AXZ) .

such that

I fTi(xs_Axz) dc0A(z)

fA(xs_A•

exists. Therefore

= j" fA (x)f% (xS_A• =

I fT[(x)fA(Xs-A•

d~A(W)

f~(x)

dmA (w)

.

Suppose from now on that we have a collection fundtions satisfying (5.2),(5.3) and (5.6). Let defined in terms of fA(x) > 0

for all

{fA}Ag C

> O , and thus again

{fA}Ag C

~ = {~A}Aa~

of =F-measurable

be the specification

using (5.1). If, instead of (5.2), we have

x E X ~ A E C , then ~e can obtain a representation of

63

{fA}As C

5.5

in a form which should be recognizable to anyone who is familiar with

the formulae of statistical mechanics. Fix

X

= { x e X :

x

and

b

b e X

as a reference point and let

differ at only finitely many sites } ;

O

thus

x g Xo

if and only if

Proposition 5.1 exists

a unique

Xs_ A = bs_ A

Suppose that function

for some

fA(x) > 0

V : X

-#R

for all with

A e =C .

x e X , A g ~

V(b)

= 0

and

. Then there

such

that

0

exp{V(x)} (5.7)

fA(x) =

for all

x e X

~exp{V(Xs_A

XW) } d~A(w)

Proof

Let

x g X

let o

A g C

~

with

Xs_ A = bS_~

o

and let

% e C

~

with

~

%~

A 9

Then by (5.6) we have

f%(x){f%(b)} -I

and thus we can define

(5.8)

Clearly

V(x) =

V

=

fA(x){fA(b)}-i

by

log{fA(x)} - log{fA(b)}

V(b) = 0 . Now let

x e X

, A s C O

Xs_ % = bs_ %

and

% ~

and choose

=

A . Then

exp{V(Xs_Axw)}

and thus by (5.5)

for any

=

,% -I f (Xs_AXw){f (b)} ,

A g ~

with

% s C

Xs_ A = bs_ A .

such that

64

fA(x)

exp{V(Xs_A•

=

Therefore

(5.7) holds.

Xs_ A = bs_ A

= {f~(b)}-IfA(x)

d~A(W)

{f~(b)}-lf~(x)

=

If

5.6

exp{V(x)}

f~(Xs_A•

d~A(w)

.

is such that (5.7) holds and if

V

x g X

with

o

then we must have

fA(x){fA(b)}-l =

exp{ V(x) - V(b) }

V(b) = O , V

Thus, together with the condition

.

is uniquely determined

Now although the above result tells us the form a specification

by (5.8). ~I

is likely

to take, in practice we really want the converse of this. We will usually be given a function

V : X

> R , and we want to define a specification

o

of it. It is easily checked that if fA : X

> R

o

then the

fA

V : X

is defined by (5.7) satisfy (5.2),(5.3)

~R

o

with

The function relative b

to

to

b , i.e.

V

and (5.6) , but of course with

V(x)

from

replacing

X~ X~

to

X ; and

to ensure that this can be done. energy of the system

is the work done to change from the configuration

x . In statistical

physics

as a sum of terms due to various

erent sites. We will mimic

(5.9)

V

fA

in (5.7) looks like the potential

the configuration

usually ~ i t t e n

on

V(b) = O , and if

(assuming that the integrals make sense),

X 9 The problem is thus to extend the definition of we will now look at conditions

in terms

the potential

interactions

this here, and suppose that

V(x) =

~

CA(XA)

,

V

energy is

between the diff-

can be written as

65

~A : X(A)

where

(5.10)

Let

~

) R

is ~o(A)-measurable

~A(w) = 0

if

w{t } = b{t }

5.7

and s u c h t h a t

for some

t ~ A 9

= 0 . Note (5.10) implies that the sum in (5.9) is only over a finite

number of non-zero terms for each convergence.

~A

x g X ~ , thus there are no problems about

can be regarded as the potential due to the IAl'body interac-

tion between the sites in

A . In fact, the assumption that

as in (5.9) is only a measurability checked that, if we define

(5.11)

~A(z) =

~A

~

assumption.

V

can be written

This is because it is easily

by

(-I)[A-AIV(bs_AXZA)

,

A CA

then (5.9) and (5.10) hold.

(Furthermore,

this

~A

is the unique solution to

(5.9) and (5.10).) Thus the only question that arises is whether (5.11), is

F(A)-measurable.

and only if for each

Now for

(5.12)

#(x)

It is not difficult to see that this will happen if

A g ~

A ~ C

V(bs_A •

define

=

~

: X(A)-

A :X o

g

)R

)R

by

~A(XA ) ;

A~(A)

where

~A ' defined by

~(A) = { A g ~ : A h A

V(Xs_Axw)

# ~ ) . Then we have

- V(Xs_AXbA)

and thus we can rewrite (5.7) as

= gA(xs_Axw )

is F o(A)-measurable.

68

5.8

exp{ gA (x) } (5.!3)

fA(x) = l expigA(Xs_AXw)] dmA(W)

But if the

~A

are nice enough then we can use (5.12) to define

x c X , and hence use (5.13) to define

fA(x)

for all

ileal I

=

sup

for all

x g X .

Suppose until further notice that the reference measures Let

gA(x)

t

are finite.

i~A(y) i

yeX(A)

Proposition 5.2

Suppose that for each

t g S

II~AII

~._

is finite.

Ac~({t}) (Note that

AE~({t}) just means the sum is over all finite

Then for each X , and if

A g ~

fA : X

A

containing

(5.12) defines a bounded, ~-measurable function )R

is defined by (5.13) then

{fA}Ag C

A

g

t .) on

satisfy (5.2),(5.3)

and (5.6).

Proof

We have

~-- iCA(XA)l = ~. AaZ(A)

t g A

~__

1 --I+A(XA) I lAnAI

AgZ({t})

i0A

=< t 8 A~ AgZ({t})

thus

~__

li:it by

~A(XA)

converges absolutely and uniformly. If we denote the

gA(x) , then clearly

gA : X

>R

is bounded and F=-measurable; we

67 can thus define

fh : X---->R using (5.13). It is inmlediate that (5.2) and (5.3)

hold; to show that (5.6) holds for exp{gA}; and if

=

Hence

fh

A C-% E ~ , x , x E X

exp{gT~(x)}/ exp{gT~(x)}

=

5.9

=

amounts to showing that it holds for with

then

exp{ gT[(~) _ g%(x) }

exp [ ~ { ~A(~A)- ~A(XA)}] ~AgZ (%)

exp{ gh(~) _ gA(x ) }

Xs-h = XS-A

=

exp{g~(~)}exp{gA(x)}

=

exp [ ~ { CA(~A) - CA(XA) }] As Z (h)

exp{gA(~)} / exp(gA(x)} .

=

exp{g%(x)}exp{gA(~)}

.

82

We now have an explicit method of constructing specifications in terms of the functions V

{~A}AgC . If

a potential (with base point

call

{~A}Ac~

V : X~ b ); if

)R

with

V(b) = 0

~A

is defined by (5.11) then we will

the interaction potentials corresponding to

satisfy the hypotheses of Proposition 5.2 and specification, then we will say that We will sometimes write states with potential

~(V)

~

instead of

~ = {~A}Ac~

then we will call

V ; if

{~A}Ac~

is the corresponding

is the specification with potential G~)

, and call

~(V)

V .

the Gibbs

V .

Conditions like (5.2),(5.3) and (5.5) probably first occur in Dobrushin (1968a). The proof of Proposition 5.1 is adapted from Preston (1974b); a similar kind of result can be found in Sullivan (1973). The condition on the

{~A}AcC

given in Proposition 5.2 corresponds to one which is often used in statistical physics; see, for example, Ruelle (1969), Lanford and Ruelle (1969).

68

Let

{~A}AeC

5. i0

satisfy the hypotheses of Proposition 5.2, and let

the corresponding specification.

We will show that

used in the previous sections; in particular

G(V)

~

be

satisfies the hypotheses will be non-empty (provided

we have the right standard Bore! structure). Suppose from now on that

(Yt,~t)

is a standard Borel space for each

then

{BA}AE~

~A = ~(A)

satisfy (3,1),(3.2),(3.3) and (3.4) (with ~ = ~ ).

Proposition 5.3

Proof

t e S ; thus if we let

G~V)

is non-empty.

We will show that (3.7) and (3.8) of Theorem 3.1 are satified. Let

and let

NA > 0

be such that

IgA(x) I ~ NA

for all

x a X . If

F g ~(A)

A g then

G

where

G = PA(F) ; and thus for any

~A(X, F)

Define

~ g P(B__ A)

holds with For

~

x ~ X , F E ~(A)

we have

exp{2NA}~A(PA(F))/ ~A(X(A)) 9

by letting

~(F) =

mA(PA(F))/ WA(X(A)) ; then clearly (3.11)

A = e . But (3.11) is stronger than (3.7), and so (3.7) also holds.

A , % g ~

define

g~ : X-

g~(x) =

~

~R

by

~A(XA)

A~Z(A)

A~% then

g~

is ~(%)-measurable, and

g~-----> A gA

uniformly as

~---->S

. Hence, if

69

F ~ =F(Ao)

for some

h%(x) =

Ao g =C and

h%

5.11

is defined by

[ I exp [gTl(xs_Axw) ( w ) l -} dmA i A

[~

exp{g%(xs_AA Xw) } d0~A(w)1 F O

where

Fo = { w E x(A) : xS_A•

g F } ; then

is a simple matter to verify that Therefore (3.8) holds.

h~

,,,

h%

is =F(AUAo)-measurable , and it

> ~A(.,F)

uniformly as

~

~S .

82

The calculation in the above proof in fact shows that (3.11) holds; and thus

G~)

has the compactness property given by Theorem 3.2. Recall that this

gives us the following: if {n.}.>. j

j

and

D g G(V)

~n ~ ~-~-) , n ~ i , then there exists a subsequence

such that

~(F) = lim ~n (F) j-~o j

l

for all

F g A , where

t._,J %(A). A~__c

A =

The most important examples of lattice models are when d ~ I , and when all the

(Yt,~t)

are copies of the same space

S = Zd (Yo,~)

this case there is an obvious way of defining an F-measurable bijection T

t

: X

>X

for each

(5.14)

Tt Ts

(5.15)

To(X) = x

(5.16)

Tt(~(B)) = =~(B+t)

where

= Tt+ s

t r Zd

sucb that

for all

for all

B+t = { s+t : s g B } .

s, t g Z d ;

x g X ;

for all

B e_ Z d

,

t g Zd

for some . In

70

We just let

(Tt(x)) s = Xs+ t , (i.e. the

(s+t) th. coordinate of d

5.12

s th. coordinate of

Tt(x )

is the

x ). There is also an obvious way of constructing

~

Z -znvariant specifications, i.e. specifications that are invariant under the group

{Tt}tEzd

same measure

~o

t ~ Z d ; then where

: start with reference measures

{~A}AcC

(Yo,_Fo)_ , choose

b g X

d . will be Z -znvariant if

~

(5.17)

where

on

~

~t

that are all copies of the

such that

bt

is independent of

is defined in terms of

{~A}AcC ,

d are Z -invariant in the sense that

~A(X) =

(x+t)s+t

Proposition 5.4

~A+t(x+t)

for all

A g ~ , x s X(A) , t g Z d ,

= xs

Suppose that

{~A}AE~

X

satisfy (5.17) and that

]I~AII

Ag~({t}) is finite for all

t ~ Zd 9 I f

=V i s t h e c o r r e s p o n d i n g

specification

then

G (~_V)

d is non-empty; i.e. Z -invariant Gibbs slates exist.

Proof

From the proof of Proposition 5.3 we know that (3.7) and (3.9) hold. Thus

the result follows from Theorem 4.3.

82

The proof of Theorem 4.3 also tells us that, under the hypotheses of Proposition 5.4, _Go(~V)

is sequentially compact in the sense described after

Proposition 5.3.

Note that

{Tt}tgzd

separates

Theorem 4.1 the extreme points of ~(V=)

~ , (as defined in Section 4), thus by

G=o(~) are exactly the zd-ergodic measures in

. Note also that all of the above remains true if

S

is any countable

7~

5.13

abelian group. Before looking at some specific examples of lattice models, we will consider one more general topic. This will concern finite range interactions, and more particularly, nearest neighbour interactions. We will say that a function

f : X-

>R

is ~-measurable if

f

is ~(A)-measurable for some

(Such functions are sometimes called tame.) Let

V

in terms of

has finite ran g_e if

{fA}Ae C

as before. We say that

~-measurable for each

A e ~ . If

V

~

A e ~

then there exists

(Recall that and

% e ~

~(B) = { A e ~

;I~AI I

such that if

: A~B

is finite for all

be a specification defined

is a potential, with

ponding interaction potentials, then we say that

V

{~A}AgC

A g ~(A)~(S-%)

then

fA

is

the corres-

has finite ranse if, given

# ~ } .) Note that if

A e ~

A e ~ .

~

V

IleAl !

then

~A = 0 .

has finite range is finite for all

Ae~({t}) t e S , since there are only a finite number of non-zero terms in this sum.

Proposition 5.5

Let

x e X , A e ~ ; let 5.1. Then

V

ll@Al I

tial

V , then

For

V

be the potential with base point

finite for all ~

A e ~ , and if

V

~

b

fA(x) > O

for all

given by Proposition

is any finite range potential is the specification with poten-

has finite range.

t e S

""

A e C

have finite range, and suppose that

has finite range. Conversely, if

with

Proof

~

let

7~ e C

be such that

f~t~~

is ~(At)-measurable.

If

t

and

% =

~

A t , and

A e C

is such that

A e ~(A)N~(S-%)

tea will show that

~A = O . From (5.8) and (5.11) we have

, then we

72

~A(y)

= ~

5.14

(-l) IA-Bi{ log fA(bs_BXY B) - log fA(b) }

B~-A

Let

s 8 AlIA , t s A~l(S-%) , and for

= BkJ{t} , B = BD{s,t}

(5.18)

B C- A - {s,t}

let us write

B = B~{s}

; then

fA(bs_BxYB)fA(bs_~Xy~)

I

(_l)iA_Bilog

~A(y ) =

fA(bs_~XY~)fA(bs_~XY ~)

B e_ A-{s,t}

fA(bs_BXY B)

f{S}(bs_BXY B)

But from (5.6) we get

and fA(bs_~Xy~)

fA(bs-BxYB) " fA(bs_gXyg) t ~ ~

-

f{S}(b S ~Xy~)

f{S}(bs-BxYB)

f{s} ;

and since

is F(%)-measurable,

and

f{S}(bs_gXy ~)

we have

f{S}(bs_BxYB) = f{S}(bs_~Xy~)

, f{S}(bs_~Xy~) = f{S}(bs_~Xy~)

.

Thus every term in the sum on the right-hand side of (5.18) is zero, and hence ~A(y) = O . Conversely, suppose that finite for all implies

A g C ; let

A e C

V

is a finite range potential with

and choose

~A = 0 . By (5.12) we have

% g C

gA(x) =

such that

IleAl i

A g__Z(A)fIZ(S-%)

~A(XA) , and therefore

g

A~_Z(A) A~ is ~(%)-measurable.

Let

~

Thus by (5.13)

fA

is ~(%)-measurable.

82

be a finite range specification corresponding to a finite range

,

73

potential

V

with

[[~A[ [

finite for all

5.15

A e ~

proof tells us that the "range of interaction" Hore precisely,

for

~(r(A))-measurable, A e ~(S-~(A))

Proposition

A e C

let

and let

implies that

5.6

For all

r(A)

r(A)

. An examination of the above

of

~

is the same as that of

be the smallest

be the smallest

set such that

set such that

fA

V .

is

A e ~(A)

and

~A = 0 . Then we have

A e ~

r(A) = ~(A)

; also

r(A) =

~

r({t})

.

t g A

Proof

The proof of Proposition

(5.19)

r(A) C_.

r(A)

~_.

5.5 tells us that

I ~

I r({t})

for all

A e C

.

tEA

In particular we have

r({t}) = r({t})

r(A I) ~

r(A2)

whenever

Thus if

t e A

then

t e S . Now it is clear that

AIC- A 2 ; and by (5.5) the same thing is true for

r({t}) ~

r({t})

for all

~

r(A)

r(A) C

tEA

, and hence by (5.19)

~(A) ~

~

r({t})

G . Assume that

G

has no multiple

and that there are only finitely many edges incident s

and

~A , the boundary of

t

~

S

are

edges, no loops,

to each vertex;

then we will call them neighbours.

For

if there is A ~

S

define

A , by

~A = { t e S - A : t

SAg

.

example of the above result is when the points of

the vertices of some graph

(Note that

.

tEA

An important

an edge between

r

if

A e ~

is the neighbour

of some

s e A } .

.) Call a finite range specification

~

nearest

74

neighbour

(with respect to

(i.e. if

r(A) C

if

r

= 0

whenever

A~A

whenever s , t s A

Proposition 5.7 potential

V

if and only if

V

Pro_____o~ Suppose

G ) if for each

A

is not a simplex of

with

s # t

then

II@AI I

t ~ r({s}) = r({s})

with

s # t

A

V

is a simplex with

AUrA .

D ( F 2 ~ F 3) > 0 ~(A)

nearest neighbour G

if

are neighbours).

~

is nearest neighbour

A g ~

s , t

then

with

not a simplex;

not neighbours. Hence

A g ~({s}) V

A

, A ~ ~(S-r({s}))

is nearest neighbour. If A C AUrA

; thus

82

g G(V)

then

A g _C and

;j is often called a

F 1 g F(A)

, F 2 ~ F(~A)

,

then

~.'(FI~ F 2 ~ F 3 ) D ( F 2)

says that

t

V

is a simplex of

A c ~ . Then

and

A E ~(A)

is nearest neighbour and

F3 g ~(S-(AD~A))

Thus if

and

; and since

Markov random field. This is because if and

( A

CA = 0 . Conversely, suppose that

r(A) = ~(A) C-

If

C

is nearest neighbour, and let

t ~ ~{s}

and

s

finite for all

s , t E A

A ~ ~

is ~(At.JSA)-measurable,

be a finite range specification corresponding to a

there thus exist

we must have

fA

is nearest neighbour.

~

and so

A ~ ~

) ; call a finite range potential

Let

with

5.16

=

D(F2f~ F 3 ) ~ ( F I N F 2 ) 9

we can write this as . ~(FIIF 2 ~ F 3) = ~(FIIF2)

is independent of

~(S-(A~$A)),

given

~(~A)

, which

9 This is an

analogue of the usual Markov condition (though here it is spatial, rather than temporal). Proposition 5.7 is a form of a much over-proved result, versions of which can be found in Averintsev

(1970), Spitzer (1971), Harmmersley and Clifford

(1971), Suomela (1972), Sherman (1973), Preston (1973) and Grimmett

(1973).

,

75

5.17

Now for some specific examples of lattice models.

I. The Ising model

This is supposed to be a model for a d-dimensional magnet.

Let

S = Z d ; at each site

and

- ; thus each

Yt

t E Zd

there is a "magnet" Which has two states,

is a set with two points in it. We could let

+

Yt = {+'-}

but it will be more convenient (for the author) to let

Yt = {O,I} . This allows

us to interpret

as the site being empty

1

as the site being occupied, and

O

(and so we could consider this as a lattice gas rather than a magnet). For A C. Z d

we will make the obvious identification of

of subsets of

A

(i.e. we identify

y e X(A)

with

define potentials we will take the base point to be X~

can be identified with

potentials

{~A}Ag~ ; if

~ . Let %r'A

e ~

V : ~---~R

X(A)

with

~(A)

{ t e A : Yt = i } ). To (i.e. all

O's ), thus

be a potential with interaction

then by (5.10) we have

~A(%) # O

= A ; thus we can regard the interaction potentials as a potential by letting

, the set

~(A) = ~A(A). This gives us the formulae connecting

V

only if ~ : ~-

and

~

JR in

the form

V(A) =

~

~(A)

A~A

In terms of t g Zd

~I

,

~(A) =

~

(-I) IA-AIv(A) .

A~-.A

~

~(A) I

the hypotheses of Proposition 5.2 become that for all must be finite. Let uS assume that this is true, and thus

teAgC = ~(V)

is non-empty. The main problem for this model is to determine for which

potentials we have If

V

I~(V) I = 1 , i.e. exactly one Gibbs state with potential

is zd-invariant then we also want to know if

,=,IGo(V) I = I ; and whether

V

~

76

G (V) = G(V)

. These

problems

5.18

are extremely

diificult;

the results

that are kno~r

~O

often have

involved

and very clever proofs,

some of the results

(5.20)

(Dobrushin

and much is not known.

in order to give some idea of the complexity

(1968c))

Suppose

< I

there exists

'~', (IAI-1) I~(A)I 5 ~/

state

of the situation.

such that

for all

2e

We will

t E Zd .

tcAaC Then

i~(V) i = i . (This result

says that the interaction

Suppose

A

all

(5.21) with

between

the sites

from now on that the potential g ~

, t E Zd

(Lebowitz

IAI ~

means

and Martin-L~f

diam(A)

= max{

d

.

is Z -invarlant,

(1968)) Im-nl

(1972))

l(v)l :

that the interaction

(Ruelle

is weak.

i.e. ~(A+t)

for

= ~(A)

9

2 9 Then

(5.22)

S = Z d .) The condition

does not depend on having

I

Suppose

if and only if

is attractive,

Suppose

~(A) _>_ 0

lo(V)l :

for all 7% E

i . The condition

C on

or that we have a ferromagnet.

d = i , and for

: m , n g A } . Then

~

that

A C.Z

I~(V) I = 1

let provided

l~(A) Idiam(A)

O~A~s is finite.

This says that in one dimension

able interactions.

(5.23)

(Dyson

we have

i~(v)1

= 1

for most reason-

However:

(1969))

Suppose

d = I ,

r

= 0

for

]A I ~ 3 , #({O,n})

= n -~

77

for

n > O , and

~({0}) =

- ~_

5.19

n -~ , (where

n>

~ > i ). Then

a < 2 . (Note that from (5.22) we have

[~(V) I = I

if

#(A) = O

if

, . Z d -invarlance

~({O,t})

for all

is determined by

~({O})

and

(5.24) (Ruelle (1971) , Lebowitz and Martin-Lof for all

t # 0 . Then

]~(V) I = 1

2~({O}) +

~

> I

if

a > 2 .)

Now we will only consider pair potentials, i.e. ~

[G(V)]

1

IAI ~ 3 . By the 0 ~ t g Zd

(1972)) Suppose that

~({O~t}) ~ 0

provided

#({O,t})

#

0 .

t#O

In terms of magnets (and we are again in the ferromagnetic case) the condition 2~({0}) + ~

~({O,t})

=

0

means the absence of an external field; thus

t#O I~(V) I = i

in the presence of a non-zero external field.

(5.25) (Ruelle (1971) , Griffiths (1964), Dobrushin (1965)) for all

t # 0 , that

2~({0}) + ~ ' - - "

~({O,t})

=

Suppose

~({O,t}) ~ O

O , and that for

% > O

V%

t#O is the potential corresponding to I~(V~)I > 1 . Further, if if

~({O,t}) ~ y > O

d ~ 2 , then there exists

we also have

%~ . If

p = p(~)

l~(V%)l > i for all

such that

l_G_~(Vp)I > 1 ). The parameter

%

and

t E Zd

~ > ~ with

then llti] = i , and

l~(Vp) l > 1 , (and by (5.21)

is inversely proportional to the

temperature, thus the result says that a ferromagnet, in the absence of an external field, exhibits spontaneous magnetization at low enough temperatures.

78

If

V

is as in (5.25) with

5.20

for

~({O,t}) ~ y > O

Iltll = i , and if

d > 2 ,

then letting

c

we have

=

0 < %c(~)

A particular

(5.26)

d = 2

inf{ ~ > O : I~(V%)I

< m

and

I~(V%)]

i

if

O

otherwise,

~({o})

the actual value

to be a major unsolved

For

(Griffiths

repulsive

VI

(5.28)

d > 2

(1964),

~

interaction

e

is known.

the value

=

1

-

=

~ > %c

(Dobrushin

is unknown,

Dobrushin

(1965))

Let

corresponding

(or an anti-ferrom~o.gnet). G (VI) # ~(VI)

(1968c))

Suppose

and the case

result

d = 3

~

to If

be as in (5.26), 1} . If d > 2

. (This is about

~ < O and

of

is con-

=

then we have a

the simplest

that

~({0})

otherwise~

-dx

+

and for any

~ < -I

if lltll=l, =

O

-d .

(It comes from a famous

breakdown.)

~({o,t})

if

problem.

be the potential

I~(Vx) [ > 1 , and also symmetry

, ..[G(V%)I= > 1

~c

IInil = 1

of

sidered

let

% <

llt~ = I ,

=

(1944).)

s R

if

that

Onsager

(5.27)

= I

> 1 }

case of this is:

Suppose

~({O,t})

For

(r

p

.

(~)

then

example

of

c

79

If

% > O

d > 2

then from (5.24) we have

then there exists

M > 0

5.21

[G(V) I = 1

such that if

whenever % < -M

O ~ 0 . However,

then

IG(V) I > 1

for some

For other results on the Ising model the reader should look, for example, Ruelle

(1969) , Gallavoti

2. A model with

S

(1972) and Georgii

an infinite tree

is much simpler

We will again have some

N ~ 1 , where

vertex.

TN

(1974a) and Preston Yt = {O,I}

deleted.)

involved,

for each

u

and

I ~({s,t})

N+l

graph which becomes disconnected

are homogeneous;

the

S = TN

for

edges incident to each when any edge is

on this graph, and will

thus there are only two parameters

v , where

v

if

s

and

t

are neighbours,

=

~({s})

O

The corresponding

The results about

t g S , but now

We will look at nearest neighbour potentials

assume that the potentials

it is

(1974b).

is the infinite tree with

(A tree is a connected

than the Ising

can be made with it. However,

still complex enough to exhibit some interesting behaviour. model are taken from Spitzer

in

(1973).

This model

model, and fairly explicit constructions

if

= u

otherwise,

potential will be denoted by

we want to know for which values of

u

and

v

V(u,v)

. As with the Ising mode]

do we have

l~(V(u,v))[

= 1 .

For this model a comp]ete answer can be given. A crucial fact about this model seems to be that there is an explicit method of constructing we can construct

a lot of random fields;

at least one e]ement of

enough so that for each

~(V(u,v))

.

u ,

v

80

Let

5.22

O < p < i , O < q < i , and consider the matrix

as the transition matrix of a Markov chain with the two states be the unique stationary distribution (i.e. subgraph isomorphic to X(A) =

--~-

Yt

p l-p ) l-q q

M =

O

~M = ~ ). Now if

and

A ~. T N

Z , then we can define a probability measure

corresponding to

M

and

~ :

~A

1 ; let is a

~A

on

is the stationary distr-

t ~ A ibution of the doubly infinite chain, as a measure on its sample space.

It is

not hard to check that these measures are consistent, and they uniquely define a random field an element of

~(p,q)

on

~(V(u,v))

X . A simple calculation then shows that

~(p,q)

is

, where

=

logIIl-q~N+l

I l-p~]

(5.29)

= logI (l-p)(l-q) Pq 1 Let sends

F : (O,I)• (p,q)

to

~R 2 (u,v)).

r

be defined by (5.29) (thus

of more than one point for some l~(V(u,v)[ > I 9 In the case that

and only if

s30

(u,v) v ~ O

G(V(u,v))

. If

xl x

F-l(u,v)

consists

then we will have of course (an attractive interaction), the results

(1972) can be used to show that

lY-l(u,v)[ > 1 . But

I~

is the map that

is easily seen to be onto, thus as stated above,

we can construct at least one element of each

of Lebowitz and ~ r t i n - L ~ f

F

II'-l(u,v)[ > 1

l~(V(u,v))I > i

if

if and only if the equation

81

has more than one solution w i t h

x > 0 . If

(5.30) has only one n o n - n e g a t i v e

solution;

then there exist values solutions.

of

u

for w h i c h

Note that the case

N = I

5.23

N = 1

or

but if

fN+l~ 2 ~ I , then

exp(v) ~

N > i

and

(5.30) has either

2

/N+i~ 2 ~N-~I

exp(v)

>

or

non-negative

3

has already been covered by (5.22),

since

T1 = Z . If F(p,q)

v < O

= (u,v)

{ (u,v)

(a repulsive

with

: v < O }

by

v < O

interaction)

if and only if

{ (p,q)

q x

then

IF-l(u,v) l = 1 ; also

p + q < 1 . Thus w e can r e p a r a m e t r i z e

: p + q < I } . Define

N

+

~ : R+ -

~R +

by

(l-q)

~(x) (l-p) x N

Using a construction showed that if

similar

p + q < I

only if the equation

p

to the one used to get

and

~o~(x)

+

(u,v) = F(p,q) = x

and

if and only if

(u,v) = F(p,q), ~(x) = x

then

~o~(x)

In both cases we therefore l~(V(u,v))I

positive solved,

solution.

= i

though the answer

3. Countable

state Markov

of some countable

set

= x

solution,

Y

for each

if and

(namely

x = i ).

if solution

and this happens

surprisingly

S = Z

result:

~o~(x)

the p r o b l e m of when

Let

solution

= I

if and

solution.

have the following

is perhaps

chains

IG(V(u,v))[

(197~a)

has only one positive

if and only if the equation

For this model

Spitzer

p + q ~ 1 . This is b e c a u s e

has only one positive

only if (5.30) has exactly one positive

then

, then

has only one positive

In fact the above still holds when p + q ~ I

(5.27),

= x

if

(u,v) = F(p,q) has exactly one

I~(V)[ = i

has thus been

complicated.

and suppose

that

Yt

is a copy

t E Z , We will look at nearest n e i g h b o u r

82

5.24

specifications

in this case (with

Z

neighbours

n

n+! ). We will see how the Gibbs states

of

being

n-I

and

considered

with Markov chains having state space

Y

as a graph in the usual way, the

(and regarding

axis). The results stated are taken from Spitzer ional material can be found in F~llmer Put the o-field measure. X

Let

{fA}Ag C

g X , A g ~

also that

. ~

fA

iance, thus

~(Y) satisfy

Choose a base point V

A

gives that

V

is of the form ~

for all

b g X

b

other than a potential.

f{O}

{n}

or

by

, where

probability x

t g Z .

~{0}

x , z c u

t e Z

5.1, and let

and

define

f

be

5.7 cont-

the proof of this

. The Z-invariance ~{0,i}

(for some {~A}As~

Then, although Proposition

{n,n+l}

Thus of

~A = 0 V

therefore

" to represent

X,Z

x = W_l , y = w 0 , z = w I

is ~({-l,O,l})-measurable).

the conditional ing state was

For

{fA}Ag c . Suppose

for all

given by Proposition

In the present case it will be convenient

since

and

= ~

t

for all

, and that we have Z-invar-

A g C , x g X

such that

fA(x) > 0

is a nearest neighbour potential.

is determined by

fx,z(y) = f{O}(w)

given by

that the reference measure was finite,

result still shows that unless

and suppose that

for all A g ~

interaction potentials.

ained the hypothesis

(1975). Addit-

(1975b).

(5.3),(5.6),

be the potential

the corresponding

time

Y , and use counting measure as the reference

is ~ ( A ~ A ) - m e a s u r a b l e = fA(x)

as the discrete

(1974b) and Kesten

will denote the specification

fA+t(x+t)

S Y ). Let

on

Z

Lie in

: Y----~R

state is

and the next state will be

z . Let

by something

by

(and this is well-defined

Thus, for any

that the present

V

~ g G~)

, fx,z(y)

is

y , given that the precedQ : Y • Y

-~R

be given

5.25

83

(5.31)

{f~,y(~) }-If~,y(X)

Q(x,y) =

Then a simple calculation shows that

(5.32)

fx,z (y)

where

=

{Q2 (x,z) }-lQ(x,y)Q (y,z)

Q2(x,z) =

Q(x,w)Q(w,z) . Thus

Q

determines the

f

X~Z

, and

wgY vice versa. But it is not difficult to see (using (5.8) in the proof of Proposition 5.1) that we have

*{o}(X)

I

=

log [

{Q(~,~) }2 ]

(5.33)

:IQ(x'Y)Q(~'~)] ~{O,l}(X,y)

Thus

Q

determines

V

=

log [Q(~,y)Q(x,~)J

and hence

V

as

fA

of

X .) Note that one consequence of

{fA}AE~

is ~(AUSA)-measurable this means that

is determined by We will use

Q

(since

Q

V

V

determines fA

fA

on

X

O

, and

is determined on the whole

being determined by the

f

X~Z

is that

f{O} .

to describe the specification. It is easily checked that

has the properties:

(5.34)

Q(x,y) > O

(5.35)

Qn(x,y) < ~

for all

for all

matrix multiplication, i.e.

x , y g Y ;

x , y g Y , n ~ i , (where

Qn(x,y) = ~ Qn-l(x,z)Q(z,y) zgY

Qn

).

is defined by

84

Thus our specification defines a matrix sely, if

Q

Q

5.26

satisfying (5.34) and (5.35). Conver-

is a matrix satisfying (5.34) and (5.35), then defining

V

through

(5.33) gives us a specification for which (5.32) and (5.33) hold (but not necessarily (5.31)). Let

F

denote the set of all functions

satisfy (5.34) and (5.35). Then to each ~(Q)

Q g F

Q : y x y.

~R

which

there corresponds a specification

~ and the specifications we get this way are exactly the ones satisfying the

conditions we started with. This correspondance is not, however, one to one; but it is not hard to show that if exists

k > 0

and

v : Y

Q , Q g F

~R

, with

Q(x,y) =

If

then we will say that

~(Q) = ~(Q)

~(Q) = ~(Q)

v(y) > 0

Q(x,y)v(y) %v(x)

(5.36)

then

for all

for all

Q

and

Q

Q

are equivalent.

Q

~

by the matrix

Q

is

behaves like the transition matrix for a M a r k o v

chain (recalling the interpretation of However,

y g Y , such that

x , y 8 Y .

The reason for representing the specification because (5.32) suggests that

if and only if there

fx,z(y)

as a conditional probability).

is not usually going to be stochastic;

(though in many cases it will

be equivalent to a stochastic matrix). Suppose

P g F

and aperiodic. P

is stochastic; then by (5.34)

P

has to be irreducible

will thus have a stationary probability distribution if and

only if it is positive recurrent. If

P

is positive recurrent, and

~ : Y~

.>R

is the unique stationary distribution, then we can define a Z-invariant random field

~ = D(P)

by

n

~({ x : x k : tO , Xk+ I = E1 ,..., Xk+ n : ~n })

:

~(~0 ) ~

P(~j-I'~j ) "

j=l

We call

~(P)

the stationary Markov chain defined by

P . It is easily checked

5.27

85

that

~(P) g G(V(P))

. The following facts are known:

Y

(5.37) (Dobrushin (1968a); see also Spitzer (1971)) Suppose Q ~ F

then there exists a unique stochastic

equivalent; and

is finite. If

such that

P g F

~(P)

~(~_V(Q))I = I_Go(~(Q))I = I , with

P

and

Q

are

being the single Gibbs

state. Further, we have

P(x,y)

where

%

Q(x,y)r (y) ~r(x)

=

is the largest eigenvalue of

for all

Q

and

x,

r

ygY

,

is the corresponding right

eigenvector.

Suppose from now on that

Y

(5.38) (Kesten (1975)) If

is infinite.

Q E r

latter occurs if and only if Moreover, in this case

P

Q

then

I~_V(Q))I

is either

0

or

I . The

is equivalent to a positive recurrent

is unique, and

D(P)

P E r .

is the single Z-invariant Gibbs

state.

(5.39) (Kesten (1975))

Suppose

P 8 r , and suppose there exists

Q s F 6 > O

is not equivalent to a positive recurrent and

m > 1

such that for all

x g Y

m

~ .

Qn(x,x) > 6 .

n=l

Then

G~V(Q))

is empty. In particular, this will hold if

Y = Z

and

Q

is

equivalent to the transition matrix of a random walk; a case proved in Spitzer (197~b).

86

(5.40)

(Spitzer

(1974b))

then both

l~(g(Q))]

I~o~(q))l

= 1 .)

(5.41)

(Spitzer

I~V(P))I

= i

(1974b))

If and

Q e F

is equivalent

I~C~(Q))I

There exists

= ~

then it cannot be equivalent

Note that in this case w e could not have then any translate of Z-invariant).

p

is still in

to a positive recurrent

are possible.

a null-recurrent

= ~ . In this case w e also have

null recurrent

5.28

GoQV(P))

(By (5.38) we [lave

P g F

is empty,

such that since if

to a positive recurrent IG(~_V(P))I = 1 , b e c a u s e

~C~(P))

P g F

, (but of course

if D

P

is

P g F . P g ~(~(P)) cannot be

6.1

6. Continuous models: point processes

We noted that the Ising model, considered in the previous section, regarded as a model for a lattice gas; where at each site presenec,

and

0

the absence of a particle

1

could be

represented

the

(or molecule of gas). A more realis-

tic model for a gas would have the molecules lying in

Rd

rather than

Z d : we

will now generalize the Ising model to include this case. We will thus be concerned with the distribution of configurations of particles, with the particles lying in some underlying space point processes. applications Let arbitrary,

S . Such probability measures are often called

It should be clear that this kind of framework will have wide

in describing phenomena from physics, biology, and other fields. S

denote the space in which the particles lie; we let

though in practice this space will usually be

comes equipped with a o-field

(6.1)

if

A I , A 2 e_C

(6.2)

if

A g ~

(6.3)

there exist

~

, and let

then

and

~

S

be fairly

R d . Suppose that

be a subset of

~

S

such that

AIUA 2 g C ;

A g ~

with

A C.% A

then

A n g =C , n => i , such that

A g ~

S =

;

A n , and such

~_~ n > I

that if

A g C

then

A ~

A

for some

n .

n

Usually we will have

S = Rd

Borel sets. Our basic space on

(S,~)

(6.4)

S X

the Borel subsets of

and

C

the bounded

will be the set of all integer-valued measures

satisfying:

x(A) < oo

Rd

for all

A c C .

x

88

(By integer-valued

we mean taking values in the set

thus made the assumption particles

lying in

that if

A g ~

Already an implicit

(6.5) all

is separable,

~.~

nm 6x

1

Xl,X2,...

in any

i.e.

This is justified because we now add

~

is countably

generated,

and

{t} g ~

for

, where

A g ~

6x

is

the

elements of

S

; also

6

~ 6

x

which has

if

y

~

: B ~

n

m

A } , considered

defined by

replaced by X(A)

"A g C

FA = ~(S-A)

for

x

m

and

measures

with

Ar

on

, and

B g S

~A

= x(B) A ~ ~

for

" let

'

for

x.'s 3

F(A)

, where

{~A}AgC

. For

(A,SA)

A E ~

let

A , and let

which satisfy

(6.4)

F (A)

be the o-field

{ x e X(A)

: x(B) = m } ,

be the o-field of subsets of

PA : X

to

m = 1,2, . . . .

A"). Let

generated by sets of the form

~(A) = (pA)-l~o(A)) PA(X)(B)

of

, nm e {0,1,2,...}

as a o-field of subsets of

denote the se~ of integer-valued

m = 0,i,...

x

x ~ y . This measure clearly corresponds

We now need to define the o-fields

of subsets of

mass at

with only a finite number of

at

"A g C"

consists exactly of elements

point

particles

~A = { B g ~

given by

X

m

are distinct

the configuration

with

has been made between configurations

(6.4) and (6.5) it is easily seen that

m>

(with

measures.

S .

t g S .

the form

X(A)

at the same point of

i.e.

that

(S,~)

Under

and

.) We have

then there can only be finitely many

identification

and integer-valued

the hypothesis

{0,i,2,...,~}

A . Note that we have not excluded multiple occupancy,

there can be more than one psrticle

of particles

6.2

~X(A)

x g X , B c S A . Now let

X

is the projection F = ~(S)

and

. This gives us a collection of o-fields with the right

,

89

properties,

with

from outside

~A

representing

the events

6.3

that are observable

A .

As in Section 5 we would like methods of constructing the models we now want to consider it will be necessary definition of a specification to (2.14)).

(i.e. as a collection

are actually hard spheres of diameter configuration

in which two particles

that in other models a catastrophic

satisfy

specifications.

For

to use the more general

of kernels

satisfying

(2.10)

There are two reasons for this. The first is that in some models we

want a priori to exclude certain configurations;

conditioned

(or measurable)

(in fact most realistic models),

than

any

r . The second is

certain configurations

have

and so we do not want to define probabilities

on them. These configurations

(6.4), the number of particles

are usually

in

A

those that, although

increases

effect of one of these configurations

force at all points of

if the particles

r , then we do not want to consider are closer together

effect on the system,

and tile resultant

for example,

S . (Of course,

too quickly as

they

A~S

,

is to produce an infinite

the set of such configurations

would have

zero measure with respect to any probability measure having some physical meaning.) The best way to handle these situations ized definition

appears

to be to use the general-

of a specification.

We take the same approach as used in Section 5. First we work backwards and get a representation

for certain specifications

starting with a suitable potential, Some notation: projection of

PB,A(X)

Lemma 6.]

from

X(B)

if onto

. An important

Let

A , B E S

we try and constuct

A , B g ~ X(A)

in terms of potentials;

. If

with

A~

x g X(B)

B

a specification.

then let

A~B

= @ . Then

PB,A

denote the

then we will write

technical fact about our present

with

X(AUB)

then

xA

instead

set-up is:

is isomorphic

to

90

X(A) x X(B) we have

Proof

If

under the mapping

F (AUB)

= _FFo(A) • ~o(B)

.

then we write

A g ~

for the corresponding

suppose we have a finite measure

Suppose also we are given

R A ~ ~f~ , and that

denotes the zero measure, RA = PS-A(RA ) Let

then

{fA}Ag C

0 g RA

RA c ~o(S-A)

and

element ef

for all

I fA(xs-AXY)

(X(A),~o(A)).

A g ~ Since

(where RA~

~A

0 ' if

9

functions. We can, as in

~A : X x ~----->R

dwA(Y)

on

R A = (Ps_A)-I(RA)

be a collection of Z-measurable

~A (x'F) =

wA

i.e. the absence of any particles).

Section 5, attempt to define a kernel

(6.6)

yXz

. For each

we put

(XA,X B) ; also, under this isomorphism

Easy exercise.

y g X(A) , z g X(B)

X(A~B)

x ~->

6.4

by

'

G

where

G = { y e X(A)

X(S-A) x X(A) . For

(6.7)

fA > 0

(6.8)

~fA(zxy)

(6.9)

fA(x) = 0

If (6.7),(6.8)

: xs_AxY e F } , and where we have identified ~A

X

and

to satisfy (2.10) and (2.11) we need

for all

A ~ C ;

d~A(y) = 1

if

for all

z g RA ' A g C ;

x ~ RA , A ~ C .

and (6.9) hold then

~A

also satisfies

(2.12) and (2.13)

; thus

6.5

91

for

~ = {~A}Ae~

(2.14), i.e.

to be a specification with respect to ~ = {RA}AE~

~%wA = w%

whenever

A~%

we need

g ~ . So far this looks much the same

as in Section 5; however, in showing that (1.7) is equivalent to (5.4) we needed tile lact that

~

= ~AX~_A

, and it would ~mke things a lot easier if this was

still true in the present case. We will call the measures

{~A}AsC

independent if whenever

A1 , A2 a

~AlXWA2

(under the identification of

with

AI~A 2 = ~

with

X(A I) • X(A 2) ). Then the same computation which connected (1.7) and (5.4)

then

WAIUA 2

now gives us that, if the

=

{~A}AEc are

X(AIUA2)

independent, then (2.14) will hold

provided

(6. IO)

f~(x) = fA(x) I f%(Xs-AXw) d~~

for all

x g X , A~%

g C .

Again, the converse is almost true in that (2.14) implies that (6.10) must hold for

~-a.e. x~ . Therefore, given independent

{0~}~g~

and

{fA}A~~

satisfying

(6.7),(6.8),(6.9) and (6.10), we can construct a specification ~ = {~A}Ae~

by

means of (6.6). This would be fine provided we have a way of constructing suitable independent defining

{~A}As~

{ ~ A } A ~ . Luckily it turns out that the most natural way of (in terms of a given measure on

(S,~)) mmkes them independent.

After we have obtained a representation of our specifications we will go through the construction of these measures. Suppose then we have

{~A}Ag~

independent and

{~}Ag~

satisfying (6.7),

(6.8),(6.9) and (6.10). Note that if (6.7),(6.8) and (6.9) hold, then a slight modification of the proof of Len~na 5.1 sho~s that (6.10) is equivalent to

92

if

A~%

g C

and

x

~ ~ X

6.6

with

Xs_ A = Xs_ A

then

(6.11) f%(~)fA(x) = f%(x) fA(~)

and

(6.12)

if

A C%

g ~

and

fA(x) = O

((6.9) and (6.12) show that

R

then

cannot be arbitrary if any

exist.) We will obtain a representation for = { y g X : y(S) < ~ } , thus only finitely many particles,

XF

~ 0

{fA}Ag C

are to

{fA}Ag C . Let

consists of those configurations containing

(equivalently, X F

which are supported in some element of exp(-~)

f%(x) = O .

consists of those elements of

~ ). We will make the convention that

.

Proposition 6.1

Suppose that

unique function

V : XF

fA(o) > 0

>[-~,~)

with

for all

A g ~ . Then there exists a

V(O) = O , and such that

exp{V(x) }

(6.13)

fA(x) =

for all

x ~ Z~R

A .

l exp{V(Xs_AXw)} dmA(w)

Proof

This is the same as the proof of Proposition 5.1.

82

The above all looks very much like the lattice models of Section 5. This is not surprising, since we could actually formulate our present situation as a lattice model. The assumptions about sequence

A n , n => I , of elements of

C

are such that there exists a disjoint

=C , with

S =

U n ~ i

An . Lemma 6.1 then

X

93

shows that

X

6.7

is the cartesian product of the

esponding product of the

~o(An)

X(A n) , and that

~

is the corr-

. The model we are looking at in this section is

the same as this lattice model. However,

the present model is, in most cases, much

worse behaved than the models considered in the previous section. As in Section 5 we would like to reverse the procedure that gave us Proposition 6.1, and construct a specification from a given potential. will first construct some independent

In order to define

~A

However, we

{~A}AgC .

it will be convenient,

for

A g ~ , to write

in a different way. By (6.4) there can only be finitely many particles we can write

X(A)

as the disjoint union

~

in

X(A)

A , so

Xn(A ) , where

n ~ O Xn(A) = { x s X(A)

: x(A) = n } . Xn(A)

involving exactly

n

( n

particles,

so

consists of all configurations

Xn(A)

looks something like

in

A•215215

times), except that in the product each configuration occurs up to

(since, for example, in

A•

($,n)

and

(N,~)

~n(A)

denote the product of

n

copies of

ting of a single point); put an equivalence relation (~l,...,~n) ~ (nl,...,nn) ~i = No(i)

for

alence relation;

then

In(A)

is equipped with the o-field which we denote by which we call

A ~

if there exists a permutation

i = l,...,n . Let

In(A)

n!

(with on a

~ = N

"~O(A)

~Tn(A) of

~A " so we can give ~ n ( A )

consis-

by defining

{l,2,...,n}

Xn(A)

. Now

I(A)

with

A

the product o-field,

S_n(A) , This induces a o-field on the quotient space let

or

be the quotient space under this equiv-

is isomorphic in a natural way to

B (A) . Finally,

times

both represent the same config-

uration, and this occurs either once or twice, depending on whether not). Let

A

be the disjoint union

D

In(A)

~n

n>

In(A)

O

, '

94

and give

l(A)

morphic to

the o-field

B(A)

6.8

generated by the

B (A) . I(A)

X(A) ; moreover, it is not hard to see that, under this isomorphism,

(x(A),Fo(A))

is isomorphic to

(I(A),B(A)).

Now there is a natural way of constructing measures on we start with a measure on %(A) < ~ For

for all

n ~ I

(S,S)

:

A g ~ , and let

the product measure

induces a measure

%~n)

let %A

%~n)

=

%A

Lelmna 6.2

Proof

{%A}AgC

(S,S)

on

if

with %

to

(A,~A) .

(~n(A),~(A)) ; let

%~0)

denote

. Then any combination of

. The particular combination

defined by

n! n>

exp{-%(A)}

times)

(In(A),~(A))

(I(A),~(A))

exp{-%(A)}

(Note that the factor

(I(A),~(A))

denote the restriction of

(I0(A),_Bo(A))

which we will use gives us a measure

%~

be a measure on

on the quotient space

will give us a measure on

(6.14)

%

%AX%AX...x% A ( n

the point mass on the single point in the

is thus iso-

0

makes

%A

a probability measure.)

are independent.

We leave this as an exercise.

82

In probabilistic terms we have done no more than (almost) constuct the Polsson point process corresponding to measure

(pA)

-i

* (XA)

on

% : if we identify the measure

(X,F(A)) , then

Section 3); thus if (3.4) holds (with unique

%

g P(F)

such that

PA ( % )

* {~J~}Ac~

C =N = ~A

%A

with the

are consistent (as defined in

, F(A) =_B A ), then there exists a for all

A g C . %

is called the

95

Poisson point process

corresponding

to

6.9

% .

We will now start w i t h a potential

V : XF

and from it try and define a specification. on

(S,~)

x ~ y

as above.

means

there exists

that

x

z E X

(6.15)

There

is a natural

with

V(x) =

~(y)

place any restrictions (5.11).

This would

of

and

V

to

V , because

~

that

V

~

, where

x ~ y

can be w r i t t e n

if

as

that in S e c t i o n 5 this did not

could be defined V

V(y) = -co . ~

in terms of

did not take the value However,

V

using

-~o , but if

it is not that difficult V(x) = - ~

will be called the in__teracLion potential

conditions

are going to be put on

w e deal w i t h will be measurable.

embedding

iA

: X(A)

~oi A

~

corr-

in order that w e can p r o c e e d

The first assumption we have to make is to ensure

everything

~X F

: X(A)

The next question

like

(thus, as measures,

can be w r i t t e n as in (6.15) if and only if, whenever

w i t h the construction.

we w o u l d

is a measure

V .

Various

(6.16)

y

~(0) = 0 . Recall

still be true here if

y ~ x , then

esponding

k

= 0 )

,

it does then things are a bit more complicated. to check that

' where

V(O)

x

with on

(with

X , w h i c h w e denote by

x + z = y ). We suppose

~

~ : XF-----~[-c~

~A = %A

order on

is a s u b - c o n f i g u r a t i o n

y<

where

Let

~[-~o,~)

For each

. We will always assume

>~-oo,=)

that arises

~exp{V(xs_AXw)}

A g ~

there is an obvious

that

is F=o(A)-measurable

is the existence

d~A(w)

that

for each

A g ~

of the integrals

to exist for all

x c Xy

and

.

in (6.13):

A g ~

.

96

6. i0

The standard condition which is used to make this happen is to assume the potential is stable (see for example the account in Ruelle (1969), Section 3.2). The potential for all

V

is called stable, if there exists

n ~ i , x a XF(n)

Proposition 6.2

If

V

, where

N c R

V(x) ~ nN

such that

XF(n) = { x ~ XF : x(S) = n } .

is stable then

lexp{V(Xs-A•

d~A(W)

exists for

all X ~ X F , A ~ C . Proof

Let

we have

x a X F , A E ~ , and suppose that

xs_AxY a XF(n+m ) , and thus

the definition of

~A(Xn(A)) =

V

V(xS_A•

being stable). Let

x(S-A) = m . Then for ~ (n~)N

~ = exp{-X(A)}

(where

N

y c Xn(A) is given by

; then noting that

~ { X ( A ) } n , we get

I exp{V(Xs_Axw)} d~A(W)

s

(expmN ) {X (A )expN }n

~ " {exp(n+m)N}WA(Xn(A)) n~=O

=

m exp{ m N +

X(A)expN }

<

~

82

n&O

We have defined

xA

as a measUre on

(A,~A)

for

x e X , A s S ; but

it will be convenient (to avoid more notation) to also regard it as a measure on (S,S) (i.e. as an element of Similarly, we will also regard A ~

X ); thus we let xA

XA(B) = x(A[~B)

as an element of

X(B)

for

for all A , B E S

B e~ with

B ; it will be clear what use of the symbol is being made in any particular

9

97

6. ii

circumstance. Let

V

be a stable potenti~.; in terms of it we want to construct a

specification, which entails defining

{RA)Ae ~

and

{fA}A~~ . Now corresponding

to (5.12) and (5.13) we can, at least formally, write (6.13) as

exp{gA(x)} (6.17)

fA(x) l exp{gA(xs_Axw) } d~A(W)

where

gA(x) =

~

@(y) ; since this just amounts to multiplying the

r

y<x

yCA)>o numerator and denominator in (6.13) by

exp{-V(Xs_A)} . Of course, this need

make no sense, since the integral could be zero, but it suggests the approach we will take in defining

g~:A x-

fA

For

A

~ e J~ with

A ~

~

define

~[-~,~) b~

glCx)=

0(yl; y ~x~ z(A)~0

(where a sum over the empty set is taken to be zero).

A g~

is clearly ~-measur-

able (and is in fact ~(~)-measurable). Let

R Ao

=

{ x s X : lim g~ (x) A+S

exists

}

~{

x g X : g~(x) A = -~

(where the limit in the first term is required to be finite);

for some

A ) ,

6.12

98

and define

g

A lim g~(x)

: RA~

+

o

R A = { x c X : xS_A•

a RA

such that

+ and

put

for all

gA(xs_A•

w s X(A)

=< N m

. N o w let

, and there exists

for all

w ~ Xm(A)

N e R

, m --> i } ;

-

R A = R A n R A , where

RA = { x a X : r

> -~

for all

y c ~,

with

y < Xs_ A } .

+

Clearly

R A ~ ~i

L e m m a 6.3

; note that

+ x g RA

If

The existence

b e i n g stable implies that

X F ~-. R A .

then

exp{-~(A)} s

Proof

V

lex~{gA(xs- A•

of the integral

d~A(w) < ~

follows

of Proposition

6.2. The lower b o u n d comes

exp gA(Xs_A•

= i

and

mA({O})

from the estimates

in the p r o o f

from the fact that

= exp{-~(A)}

(with

0

the zero m e a s u r e

in

x(A) .

-

L e m m a 6.~

If

Proof

A~-~

Let

A ~-- A e ~

e ~

and

then

RA~R~

+

+

e-_ R A .

x ~ X , z e X(A)

; if

it is easily seen that

(6.19)

g~(Xs_A z) = g~(~s_A•

+ g~(Xs_A)

E ~

with

A~A

then

99

6.13 ~

Suppose that

x s RAaR~

w ~ X m (A) , m => i ; let also we bays

+ S

g~

~ let

N

be such that

m ~ = x(A-A)

. Since

(xS_A) < ~ . Thus if

g A ( x s _ ~ ' ) a Nm

x s RA

z s Xm(A)

for a l l

we have

g~(x S A ) > -~ ;

xS_A•

o e R A , and letting

then

in (6.19) we get

~

A g

(XS_A>;z) = g~(XS_A•

- gA(xs A) ~

<= N(mo*~) + IgA(~)1

<= { Nm o + IA(~s_A)l }m

;

+

and hence

x s RA .

Proposition 6.3

82

S~aDpose that

V

is stable and define

exp

fA : X------~R

gA(x) if

~exp{gA(xs_AXw)] (6.20)

xeR

A ,

d~A(W)

fA(x) =

0

[Then

by

{/t}Ae~

satisfies (6.7),(6.8),(6.9) and (6.10); and thus defines a

specification via

Proof

otherwise.

(6.6).

It is clear that

{~}Aa~

satisfies (6.7),(6.8) and (6.9). Instead of

checking (6.10) we will show that (6.11) and (6.12) hold. We first look at (6.12). Suppose bilities:

A~A (i)

s C

and

~(x)

x @ R , (ii) A

= O

for some

_ x c I{A , but

x e X ; then there are three possi+ ~ ~ R A , and (iii)

x s R A , but

100

6.14 ~

gA(x)

....

If (i) then either

x ~ R~ , or

x ~ RUA and

gA(x)

= -~

; if' (ii)

+

then by Le~na 6.4

x ~ R~ ; and if (iii) then

all cases we have

f~(x) = 0 , ~ d

A f~A

e ~ , x , ~ c X

with

x

of the above equation gA(x)

X

are zero;

if

= -~ , and so we can assume

greater

than

zero.

; we want

= fA(x)fA(R)

and

gA(x)

gA(x)

= -~ . ~lus

We now look at (6.11);

R A n R ~ , since otherwise = -~

then,

fA(~),

fA(x),

both sides

as above, we w o u l d have ~(x)

and fA(~)

are all

~

gA(~) + gA(x) = g^(x) + gA(~) .

But this holds

e C

since for shy

with

~ A

we have

~

r w<=~

wS_y

w(~-)>o

w(.~)>o

_>---" w

s

w(A)>o

ws w(A)>O

r

+ 7_~ w
w(A)>O

wS_~ w(A)>O

r

let

to show that

Thus what we need to show becomes

-

in

.

are in

that

and

(6.12) holds.

xS-A = xS-A

fA(R)f~(x)

We can assume that both

hence

x ~ R~

+

~

r

w 5_Ys_A

w(]O>O

w s ~S_A ~(~)>0

(writing

x~=y

,

101

6.15

~

=

4(x)

Let us now look at some examples; what follows is mostly Ruelle Borel

(1970) 9 Let

S = R d , =S

sets. We will take

(S,~)

and

a

X

r

is translation

with

@

t o be

is a p o s i t i v e

r , by which we mean that

be the B o r e l subsets

real number.

r

invariant.

= 0

m

R d , and

denotes

~

from

the bounded

Lebesgue m e a s u r e

on

Suppose that we have a pair p o t e n t i a l

whenever

x(S) > 2 ; and suppose

Then there e x i s t s

even (i.e. ~(u) = r

[

am , where

of

adapted

6 e R

and

@ : R+

also that .~(-~,~]

), such that

r

= 6

if

r

= -@(Ul-U2)

x(S) = i ;

(6.21)

[

to

from

by

x(S) = 2 , where

the two points

Note that changing X

if

~

{exp(6-B)}X am

to

to

6

Let

and

V

assume that

denote the p o t e n t i a l

u2

are

x .

has exactly the same effect in (6.6)

. Thus we can, and will,

(exp6)am).

in

uI

6 = 0

as c h a n g i n g

(by c h a n g i n g

defined in terms

of

X r

(6.15). The first thing we must now do is to find conditions

that

V

Section

is regular. 3.2,

The conditions

on

~

to ensure

that w e state are taken f r o m Ruelle

(and proofs of the assertions

em~ be found there).

V

w i l l b e reg-

ular if any of the following hold:

n

(6.22)

~

n

~

~(~i-~j)~ o

for all

=

n > 1 , Ul,...,u n ~

i = l ~ = l

(6.23)

~ >= ~ ,

where ~

is positive

definite;

(1969),

Rd

6.16

102

(6.24)

there exist

0 < rI < r 2 < ~

42 : [r2,~)-------~(0,~)

il

td_l~l(t ~ 9

, with

dt

=

~

@i ' @2

i

,

0

and if

r

i~f

'

decreasing, with

td-l@2(t) dt < ~

;

>-~

, *(~)hr

for

lul ~ r 2 9

0&lul

for all

& r I , and

Rd

I@(u) l ~ l~2(lu I)

~

~i : ~ 0 ' r l ~ - - - ' ~ ( O ' ~

r2

u~:

If

and

is upper se~Licontinuous then (6.22) is equivalent to

V

being stable.

(6.24) is the most useful in practice; it is due to Dobrushin, Fisher and Ruelle. Physically it says that the interaction has to be both sufficiently repulsive at short distances, and to die off quickly enough at large distances. If

V

is stable then we get a specification from Proposition 6.3. We

would like to know when

G(V)

of

let

R d , and for

e ~ N

satisfy (3.1),(3.2),(3.3)

is non-empty. Let B

==~

N

denote the compact subsets

= =F(e) . It is not hard to show that

{B0}es N

and (3.4); tbus we are in the set-up of Section 3. For

most potentials the hypotheses of Theorem 3.1 fail; but there is one class of potentials for which everything is fine. This is the extreme case of an interaction being repulsive at short distances, when there exists ~(u) = ~

for

r > 0

such that

0 < lul < r . If this happens then we have a hard--core potential,

and this will describe situations in which the particles are hard spheres of diameter

r

(with the points in our configurations representing the centres of

the spheres). Suppose that

V

is a hard-core potential, that

inf us R d

and that

@(u) > -~ ,

103

(6.25)

there exists

and a positive

s > r

6.17

I~(u)l <__%(lul)

~ td-l~o(t ) dt < ~ such that

~o : [ s , ~ ) - - - - - ~

decreasing

for a l l

lul

>s

.

S

Then clearly

(6.24) holds, thus

a specification (3.6),

is regular and so Proposition

V . It is fairly easy in this case to show that

(3.8) and (3.11). %]n~s by Theorem 3.1 we have

Theorem 3.2 Rd

V

~(Y)

has the sequential

acts in a natural way on

X , and

d . . and the fact that R -invarlance set of points

in

Rd

~

property.

Further,

is Rd-invariant.

coordinates)

V

satisfies

is non-empty~

and by

the group

Thus by Theorem 4.3

d 9 is the same as Q -znvariance

having rational

For proofs of these results,

compactness

G(=VJ

6.3 gives us

we get

(where

~(V)

and much more about the hard-core

Qd

is the

is non-empty. case, see Ngu~en

and Zessin (1975).

If the potential

is not hard-core

then things are much harder,

most cases (3.6) and (3.8) will fail to hold. However, then (3.7) holds with

x = 0 , and for suitable

if

{Um)m> I

r

satisfies

(3.12)

(with

and in (6.24) x = 0 ),

(3.13),

(3.14) and (3.15) will also hold. We will look at how these facts are

proved.

All the hard estimates needed are taken from Huelle

(1970).

Note that since we are dealing with a pair potential we have

y s xR

y ~ xA

y(A)>0 Let

k

> 0 , B n > 0 , with n

~

~

v g x~

y(A)~(~-A)=z k

+ ~ n

as

n + ~ ; and for

n > i

let

104

W n = { x s X : x( An-An_l)

=< ~n } ' where

at the origin

A 0 = @ ). Let

(and where

6.18

is the cube with

An W =

~ ~

side

W n , (thus

W

2k n

centered

restricts

the

n~l number

of parSicles

@2 : [ r 2 ' ~ ) to the whole

Lemma 6.5

in various

regions

of space).

)(0,~)

; it will be convenient

R

r

of

by defining

Suppose

@

satisfies

In (6.24) we are given a function

to extend the domain

= @2(r2)

(6.247

and

for all

~

of' this

function

t < r2

r

< ~

for all

n>l ~ 0 . Then for any

Proof

Let

C

e A , ~ ~ C for some on

be a cube, then

)[ ~ C

for which

such that

An ~ o

W

R Ao

centered

and

g~

at the origin,

I~-nl >__ r 2 ; suppose

then it will happen

{ x ~ X : gA(x)

x s W

A c C

that

for all

C A ~

)gA

uniformly

with

C OA

has side C . Thus

= -~ ) , and so we need only concern

gA( x) ~ -~ C ; let

for all

n => no

and

~ e =C . Let ~ ~ A~

no

--

An ; suppose

r

and thus

- g~(x)

< m~n

2c . If A g~[ ~ g

A

A g~(x)

ys y (A )=y (Am+l-A m )= i

that

if

= -~

uniforlmly

with those

be the smallest

y%x~ y(A)=y(~-~):l

IA

W .

and such that

ourselves

Then from (6.26)

g~(x)

on

A g~(x)

integer > -~

105

Therefore,

if also

x c W

then (since

AIx) I

<

x(A) ~--~

@2

6.19

is decreasing)

@2(km-C+r2)~m+l

.

m > n

But

x(A) <_ ~

~n ; and thus

gA

~gA

imiformly on

W .

82

n>n o

Lemma 6.6

Proof

If the hypotheses of Lemma 6,5 hold then for any

Let

x s W , then by the proof of Le~rma 6.5 we have

w ~ X(A) . Also, for

A ~C

(where

g~(Xs_AXw ) = g~(Xs_A• A

w(A)

C

A c ~

W C_ R~ .

xs_Axw E R oA

for any

is the same cube as in Lemma 6.5), we have

A - gC(Xs_A•

@2(kn-c+r2)~n+l

A + gC(Xs_A•

+

A gC(xS_AXW)

.

n ~ I

A gC(Xs_AXW) = gA(x c _Axw) , and since (6.24) implies stability, there exists

But N >_ 0

such that

gAc(xS,AXW) ~ N{x(C-A)+w(A)}

difficult to check that x(C) <__ ~ n<

gn

and

N

for all

can he chosen independently

inf ~(u) > -~ ). Thus, since u

n o

exists

~l

N

such that

of

, (and it is not

x e W , because

x(C-A) <__ ~

~n ' there

n>n =

A g~(Xs_A•

can b e c h o s e n i n d e p e n d e n t l y

w e X (A) , and hence m

w ~ X(A)

_<_ Nm

of

x s R+ . A

for all

~

x s W ). T h e r e f o r e 82

o

C , w ~ Xm(A) (and note that

gA(xs_Axw) ~ Nm

for all

106

As before _let A =

[J

> -~

D

~

aud

W

are in

B(A~_) denote the set of bolmded

F(@) , an0 let

functions that are F( O )-measurable for some D = { x r X : r

6.20

for all

y c ~.

0 r N . Let

with

y < x ) . It is clear that both

.

Lelm~a 6.7

Suppose treat the hypotheses of Ler~la 6.5 hold. Then for any

F r A

5 > 0

all

Proof

m~d x

E W ND

For

-there exists

f E B(~)

such that

A s ~ ,

I~A(X,F) - f(x) I < 6

for

,

A r ~

with

A ~ A

define

h~ : X

>R

by

A

exp gS(xS_A• d,~A(w) A exp g~(Xs_A•

where

F = { z r X(A) : Xs_A•

to show that x r Wf~D

h~---~WA(.,F)

s F } . Then uniformly on

d,~A(W)

h~ s B(A) , and so it is sufficient WglD . But

WnD~-

R A , thus if

then

A ~ exp g (Xs_A•

~A(W)

A[ ~ exp g ~xS_A•

d~A(W)

~A(x,F) =

and since

Q ~ F (A)

S exp g~(Xs_A• A

d~A(W) ~ exp{-~(X)}

we need only show that for any

107

A I exp g~.(xs_A• %(,.)

> ]' exp gA(xs_,x,.,) %(w)

G

uniformly that

G

in

x s W~D

lexp(~)

lexp

. Using the uniform

- exp(q) I ~ l~-qlexp

A

(max{~,q})

from Lenma 6.6 and the fact

, we get

- exp g (~sA•

=< Ig~(Xs_A•

- gA(xS_Axw) I exp{~w(A)}

6(X)~(A)exp{Nw(A))

I

estimates

A

g~(Xs_A•

The proof

6.21

is thus complete,

w(A)e:>mp{iYfw(A)}

, where

6(X)'--'~O

as

~--'~S

9

because

dtOA(W )

~

=

m exp{Nm}

{I(A) )m ml"

<

~

82

m>=l

Let now

kn = n

An-An_ 1 ). S i n c e

~2

6 n = (2n) d - (2n-2) d , (so

is decreasing

< ~"

@2(n-~)nd-i

and

for all

and

It)2(t)td-1

> 0 , and thus

is the volume

of

we h a v e

~J2(n-~)~n+l

< ~

for all

n>l

e > 0 . For

n > 1 , m > 1

let

Un, m = { x E X : X(An-An_l)

I

9 Then by Lemma 6.7 the

Le~ma 6.8 exists

dt < ~

~.~

n > 1

6n

The

In :~ I

U

m

satisfy

} , and let

(3.13).

m

satisfy

such that

U

< m6n

(3.12) with

~A(0,Um)

~ i - ~

x = 0 , i.e. given any for all

A s ~ .

8>0

there

108

6.2~ ~n

Prcof

Fix

n ~ I

for the moment; we can write

Cj , where tile

An-An_ I = j=l

C. J

are disjoint, and where the closure of each

C. J

is a cube with unit side. Now

by Ruelle (]-970), Corollary 2.8~ we have there exists only on

~'2 ) such that for any

wA(O,E(M))

<

A e ~

and

p > 0

and

q ~ 0 (depending

M > 0

exp{-(pM2-q)~n }

,

n where

E(M) = { x ~ X :

{x(Cj)) 2 > M2Sn } . B u t

j=l ~n U

=

n~m

{xcX:

n < mS n } 3 {

x(Cj)

x E X :

j=l

{x(Cj)}2 < m28n j=l

(by Schwarz's inequality)

=

and thus

x

-

S(m)

;

IrA(O,Un,m)~ i -- exp{-(pm2-q)~n }

9 Therefore, since

WA(O,D)

= 1

we have

~A(O,~) & 1 - ~

e~{-(pm2-q)Sn }

n>l

and clearly

~

exp{-(pm2-q)~n )

-7 0

as

n--~

.

82

n ~ i

Lemma 6.9

(3-7) ho]ds for

x = 0 ; i.e. given

8 s I~ and

y > 0

then there

,

109

exists

m ~ P(~0) , A ~ ~

and

6 > 0

w~(O,F) < y

for all

Proof

e ~ N ; we can write

Let

6.23

such that if

F ~ _~

with

~(F) < 6

then

A O A .

X

as the disjoint union

~

Xn(e) , where

n>O Xn(8) = { x s X : x(8) = n } ; and we have measure to put on

(X,~)

(%1

is

(pc)

-1

Xn(e) s B=~ . The natural reference

*

(le) ; and

(x o) = exp{-x(e)}

~

~

,

n>O

(n)

where

~e

WA(O,')

= (ps)-l(l~n)){ " Let

as a measure on

(X,~)

A e =C

with

A~

8 ; let

m => i . Consider

, and look at the restriction of this measure to

X (e) . Theorem 0.2 of Ruelle (1970) shows that this has a bounded density with m respect to

~e(m) (and the bound is independent of

Lemma 6.8 shows that there exists

on than

~

> 0 , with n ~

A ). Further, the proof of lim ~ = 0 n n-~

e ), such that the probability (with respect to n

particles in

e

is less than

a

n

(and depending only

WA(O,.) ) of finding more

. From this it easily follows that (3.7)

n

holds with

im! ~8(m)

~ = ~

for suitably chosen

6

m~d

n

m = 1

Pr_~osition 6.4

If

@

satisfies (6.24) then

!(X)

is non-empty.

82

110

Proof

This follows i ~ e d i a t e l y

6.24

from Theorem 3.3, using Lemmas 6.7, 6.8 and 6.9.

82

The proof that (3.14) (resp. (3.15)) holds is much like the proof of (3.12) (resp. (3.7)). The proofs are, of course~ much harder; and a more careful analysis of Rueile's estimates is needed. We leave the interested reader to make this analysis. Given that (3.14) and (3.15) hold we have that the conclusion of Theorem 3.4 is valid, i.e. we have the sequential compactness property for those for which

~(U ) = i , where

U

=

~ m>

D e ~(~)

Um . (Those random fields with

p(U ) = 1

i

correspond to what Ruelle calls the te___m~ered measures.) Finally, the remarks after Theorem 4.3 show that if

@

satisfies (6.24) then translation invariant Gibbs

states exist, ~ud the translation invariant tempered Gibbs states also have the sequential compactness property.

It is possible to show that Gibbs states and translation invariant Gibbs states exist, and that the tempered Gibbs states are sequentially eompact~ under less restrictive conditions than having a pair potential given by

~

satisfying

(6.24). Ruelle's estimations are done for a class of interactions which contain what we have considered as a special case, and the proofs we have given also work for this class of interactions (the superstable, lower regular interactions). details we again refer the reader to Ruelle (1970).

For

.~. S~ecific information ~ain

We have described a framework which is supposed to have something to do with the equilibrium behaviour of models from statistical mechanics, and as we stated before this c~u only really be justified by constructing the appropriate dynamical systems and looking at their equilibrium states. However, some justification for these definitions would be obtained if we could show that the Gibbs states we have defined satisfied suitable variational principles. For example, for the Ising model (as we have defined it in Section 5)~ given by a zd-invariant potential satisfying the hypotheses of Proposition 5.2, Lanford and Ruelle (1969) have sho~m that the d

.

Z --~nvarlant Gibbs states are exactly those zd-invariant random fields which minimize the specific free energy. ~lis is a variational principle which was first formulated by Gibbs, and is generally accepted as being the right one for this kind of model. Lanford and Ruelle's result has been given other proofs, and extended to more general lattice models (with

Yt = {0,i}

replaced by an a r b i t r a ~

finite set)

in Thompson (1973), Holley (1971)~ Spitzer (1971b) (for 1-dimensional models), and F611mer (1973). Fr

showed that the variational principle can be restated in

information theoretic terms (an approach which was also used by Sullivan (1973)). For zd-invariant random fields h(~,v) , of h(~,v) = 0

~

with respect to

if and only if

~

B , v

he defined the specific information gain,

v , and proved that if

v

was a Gibbs state then

was also a Gibbs state.

In this section we will show that F611mer's result goes through in our present set-up provided the specification is nice enough. We will define the specific information gain, h(~,v) , for H-invariant randcm fields, with

H

~ , v s Fo(~) (where

a suitable group), ~ d

Po (F) =

are the

show that if

v s G(V)

112

and

~

satisfies certain conditions, then

7.2

h(v,v) = 0

if and only if

V c G(V)

.

To give an idea of what we are going to do let us state the results for the lattice models considered in Section 5. Suppose that

S = Z d , that the

are all copies of the same space

~

(Yo,F=o) , and that

(Yt,~t)

is a specification given

by a potential satisfying the hypotheses of Proposition 5.2; suppose that zd-invariant in the sense of (5.17). If subsets of

Z d ) and

(X,~(A)) ; if

p e P(~)

~ , ~ ~ P(~)

then let

A e ~ (and in this case ~A

~

is

is the finite

denote the restriction of

~

to

then 3et

d~

if

ZA

is absolutely continuous

t d~ A 1

~A(V,9) =

with respect to

~A ' and the integral

exists, otherwise.

Let

1 --

hA(~t,9) =

)

IAI show that if

if and only if

of cubes in

.

.

D , 9 s Po(~)

with respect to

the existence of energy.

.

lim hA(B,~) = 0 A

lim hA(B,~ ) A

cubes), then we denote the limit by ~

A

We will

v E G~V) , then

.

(where the limit is taken over the set

~ ).

If for

gain of

IA.I denotes the cardinality of

~ , ~ s P (F) (the zd-invariant random fields), and O

e G(~)

where

'

h(~,v)

h(w,~)

exists (where the limit is over all

and call it the specific information

~ . in the next section we consider the question of

and also its interpretation in terms of the specific free

113

7.3

For simplicity we will start by working with what looks like a general lattice model, (though it will include most of the models considered in Section 6). At the end of the section we will indicate the modifictions required for other models. Until further notice (where

d ~ i

meaning as

~

will denote the non-empty finite subsets of

is fixed), ordered by inclusion, i.e.

A ~" A . Again we have a basic set

we are given a ~-field

~(A)

measurable from inside

A ), such that

A e ~

we let

~

will assume that

= ~(zd-A)

of subsets of

, 8rid take

Z = o (~Z(A)

I

Tt : X - - - - ) X

=

now has the same

X , and suppose that for each

ACZ d

X (representing the observables

~(AI) c ~ ( A 2 ) ~(Z d)

whenever

AI~A

to be the basic o-field

2 . For ~ . We

.

We will further suppose that for each bijection

A ~ A

Zd

t ~ Zd

we have an F=-measurable

such that

(7.1)

Ts Tt

Ts+ t

for all

(7.2)

T0(x) = x

(7.3)

Tt(Z(B)) = Z(B+t)

where

B+t = { s+t : s ~ B } .

for all

s , t s Zd ;

x e X ;

for all

B~Z

d

t e Zd

Most of the objects we will look at will be translation invariant, i.e. zd-invariant in the sense of being invariant under the group

Some notation: if P(~(A))

; if

~ e P(Z d)

A~Z then

d ~A

{Tt}taZd .

then we will often write

P(A)

will denote the restriction of

instead of ~

to

~(A)

,

114

i.e.

PA e P(A)

we have

is such that

Pzd_A = rA(~)

then we will write ect to

~A(F) = B(F)

. If

kl, k 2

k I << 12

X 2 , and if

7.4

for all

are finite measures

to mean that

kl << 12

F s ~(A)

kI

; thus for

A r

defined on the same o-field

is absolutely continuous with resp-

dk I ---- will denote the Radon-Nikodym derivative

then

dk 2 of

kI

with respect to

Let

12 .

p , v a P (Z d)

and

A s C ; define

0

log

~ dp dv A /

if

PA << VA

and the integral exists,

otherwise.

I Let

hA(~,v) =

lira

hA(~,v)

~

)

where

IAI

denotes the cardinality of

exists then we denote the limit by

h(p,v)

A ; if

and call it the s~eeific

AtZ d information gain of limit to exist if

p A

with respect to

v . In fact it is too much to expect the

ranges over all elements in

C , so we will restrict the

above limit to be taken only over the set of all "cubes" in ~e mem~ any element of

{ (hi,..

C

9

C , where by a cube

that is a translate of a set of the form

, n a) E Z d : I n i l

for some non-negative integer

<__ N , i = 1 , 2 . . . . .

d }

N . It is not hard to check that

~A(~,~) _t 0 (and

we will verify this below), and thus we have

h(H,v) k_ 0 (if the limit exists); we

are interested in the consequences

h(~,~)

of having

: 0

115

7.5

For the first result of this section we will require our specifics~ions to satisfy a condition which looks something like (3.8); and for the lattice models of Section 5 it will be equivaAent to (3.8). As usual let

~ =

~(A)

, and then

Ac~ the condition we will need is:

(7.4)

given any

A~zd-A

A s ~ , F e A

and

and an ~(~)-measurable function

Another way of putting this is: let from

X

to

R

that for each If

V

> 0 , then there exists

BA(~)

g : X---~R

and

F e A

ZA (',F)

such that

with

IZA(-,F)-g I < 6 .

denote the set of bo'muded functions

which are F(~[)-measurable for some A e C

A s ~

A ~ zd-A ; then (7.4) says

is in the uniform closure of

BA(A)

.

is a specification defined in terms of a potential satisfying the hypotheses

of Proposition 5.2 then the proof of Proposition 5.3 shows that

V

satisfies (7.4).

Our results will thus include the specifications considered by Lanford and Ruelle (1969), FSllmer (1973), and Sullivan (1973). Of course, (7.4) does not include the models from Section 6, but we will be able to deal with these separately after having dealt with specifications satisfying (7.4).

Theorem 7.1

Let

p , v ~ Po(Z d)

V = {~A}AsC

with

be a specification satisfying (7.4); let

v s _G(V) , and suppose that

the lim inf is taken over the set of all cubes in

Let ~A << VA

~ , v

lira inf hA(~,~) = 0 , where A C . Then

~ e _G(_V) .

be as in the statement of the theorem. By hypothesis we have

for arbitrarily large cubes

A , and thus this must hold for all dP A

(since if

~A << VA

and

~ C_A

then

p~ << v~ ); let

gA :

. Now if d~ A

As

C

7.6

116

HA(~,v) < ~

then we can write

I l~

where

r

= (log (

is strictly convex, HA(P,V)

< ~

for ~ ~ 0

gA dPA = ~(log gA)gA dvA = ~ r

> 0 , r and

= 0 . Let

~(~) = 0

~(~) = r

if and only if

,

- (~-i) ; then

~ = 1 . Thus if

then

~A(~,~) = ~(gA) d~=

Ii dp +

Ii dv

dv

~{~(gA) - (%-l)}

Therefore in partiaular

HA(~,v) ~ 0 ; also by reversing the above steps we get

that if

then

I W(gA) d9 <

Now if

A~A

HA(W,v) < ~

and again

then it is easily checked that

HA(~,v) = ~ ( g A

g~ = EV(gAI~(A))

thus by Jensen's inequality

Therefore for all

HA(U,v )

is an increasing function of

A ; thus we must have

A e ~ (since it is true for arbitrarily large cubes). Let

A , ~ c ~

with

A~A

= r

and define

qA,A : X

~R

by

) dv .

,

and

7.7

117

if gz(x) r o

gAu~(x) / g~(x)

,

%,~(x) 0

Thus

qA,A

9-a.e. on

otherwise.

is ~(AUA)-measurable, { x r X : g~(x) = 0 }

(7,5)

Let

A , A s C

gAUA

= 0

we have

gA qA,A = g A l A

Lemma 7.1

and since it is easily checked that

~-a.e.

with

Aria = @ ,

and

let

F

r

~(AUA)

.

Then

~(F) = F

Proof

~(F)

Lemma 7.2

Let

=

I i dWAU~ F

=

I q~,~ g~ d~A~J~ F

A r C

lira ~+zd-A Proof

Let

F

A C_ Zd-A

and

=

~ gAk)~ d V A U ~ F

=

I qA,~ g~ dv " F

F r A . Then

I g~ dv = IWA(',F) du 9 F

with

A e C

and large enough such that

F s F(Ak] A) . Then

118

Let

~ > 0 , then using (7.4) there exists

~(A)-measurable

function

as above~ but also with

=A(.,F)

-

g : X

)R

A ~

=A(.,F)

=

-

=

{~A(.,F)

Ev(XFI~(A))

- gt

and an a

~

9 ~et

- ~({~A(.,F)

- g}i%(~))

.

~-a.e., and hence we can find a version o f

E (XFIZ(A)) I ~ 26

such t h a t t h i s h o l d s everywhere ( j u s t by c h o o s i n g a v e r s i o n such t h a t

IE~(XFIZ(A)) - gl s 6

everrwhere).

Therefore

F

Putting together Lemmas 7.1 and 7.2 gives us that if

A e ~

and

F E

then

p(F)-

~A(',F)dB

=

lim A+zd-A

~ {qA,~ - l}g~ dv . F

But by Proposition 2.2 and the monotone class theorem we have only if

be

Ev(~A(.,F)]~(~))

~A('~F ) - g + g - Ev(~A(.~F) IZ(A))

I~A(.,F) -

A~zd-A

I~A(', F)

such that

=

Thus

with

A ~ A ; then

~(xFIZ(~))

- g}

7.8

p ~ G(V)

if and

119

~(F) = I~A(''F) a~

~erefore

for all

7.9

A ~ g,

F g A 9

the proof of Theorem 7.! reduces to showing that

lim

~ (qA,A - l}g~ d~

Z+z~-A

=

O

for all

A e ~ , F e ~ ;

F

and since we know that the limit exists it is sufficient to show that

(7.6)

lim inf ~ ~lqA, ~ ~ llg ~ d~

:

0

for all

A s __C .

~.+zd-A We will in fact show that

(7.7)

P

lim inf

~Y(qA,~)g~ dv ~

=

0

for all

A e ~ ;

~§

since the following lemma tells us that (7.6) is a consequence of (7.7).

Lemma 7.3

Given

y > 0

ability space, and if

~(f)

Proof

dl < 6 , then

there exists f : Y

~If-l,

.>R

~ > 0

such that if

is any prob-

is non-negative, B--measurable, and such that

dl < y .

This is a simple estimation, comparing the functions

we leave the reader to do it.

(Y~B,X)

82

We now set about establishing chat (7.7) holds.

W(~)

and

I$-iI

;

7.10

120

Lamina 7.4

Let

A , A e C

with

A(~. = @ . Then

d~

Proof

=

i~l~

Lemma 7.5

qA,~ dw

Let

=

~l~

AI, .... An g C

qA,~ d ~ A ~

I {l~

be disjoint, and for

k rk = ~

=

qA,~}gA~)~ dVAk)~

k = 1,...,n

put

n Aj ; l e t

A e ~

with

j=l

D

j-=l

A. C_ A J

Then

n

~(qA k=2

Proof

r

k' k-I

)gr

dv .

k-i

This follows in~nediately from Lemma 7.4 on writing

n

Hrn(~,v)

=

HAl(P,v) + Z {Hrk(~'v) - Hrk_l(v,v)} , k=2

and also using the fact that

HA(~,v ) > H r (~,v) .

82

n

Note that so far we have not used the facts that

W

and

v

are translation

121

7.11

lim inf hA(~,~) = 0 . These hypotheses are now brought in to A

invariant, and that

complete the proof.

Lemma 7.6

For all

A e ~

we have

P lim inf J ~(qA,~)g~ ~§

Proof D A

Let

A a ~

=

O

O

> 0 ; we need to show that if

end

e ~

then there exists

d~

d~

Let

with

{An}n> I

~

zd-A

and

A

is any cube with

A~)~--~A , and such that

be a sequence of elements of

~

with the

properties:

(7.8)

the

(7.9)

for any

translate of

An

are disjoint;

n ~ i

An = A2n-lk)A2n

A ; also if

An

is a translate of

is translated back onto

A

A then

and A2n

A2n

is a

will go onto

A ; N~

J

(7.10)

if

N. = jd

then for any

j > 1

~J

Ai

is a cube.

i=l

(This just amounts to partioning appropriate copy of

A

Zd

into cubes congruent to

A , putting the

into each of them, and then enumerating the pieces in the N.

J

right way.) Let

Cj =

~ i=l

Ai ; then using the fact that

HA(I~,~)

is an increas-

122

ing function of

exists

n ~ 1

A , and that

that

such

7.12

lim inf hA(~,v ) = 0 , it is easily seen that $here A

h C (B,V)

< ~/[AI

9 By Lemma 7.5 we have

n

h c (~,~) ~

V,~) +

n

~(qA

ICnl

k=2

r

)gr

k' k-i

a~ k-i

k where as before

rk = ~

A i . But

ICnl NnlZI :

, ~a

thus

i=l

HAI(~'v)

Therefore

+

~_~ W(qA ,I" )gl" dv k=2 q k k-i k-i

there exists at least one

k

<

~N n

with

2 ! k < 2N

n

and with

k

even, such

that

I Let

D

put

~ = D-A

~(q^ r )gr dv "~k'~k-i ~k-i

be the translate

of d

rk

.

<

6 .

that takes

then the Z -invarlance

of

~

Ak

onto

and

v

A ; then

D D A , and if we

implies that

P dv _

The proof of ~ e o r e m assume that the specification

gFk_ I

7.1 is now complete. was zd-invariant,

Note that we did not have to

though in practice we will have to

123

assume this in order to know tbat

G (V) ==o --

7.13

is non-empty,

except for (7.4) the set-up we have considered in this section

Now

includes

the models from Section 6. To see this, just partition

Rd

side; then in a natural way these cubes are indexed by

Z d ~ thus we can write them

as

Z d ) we associate

{Cj}j~zd

set

1

. Then to each

I C. J

of

A e ~ (a finite subset of

up into cubes with unit

R d . It is not difficult to check that (7.4) holds

the sub-

for a hard-

t sA core potential

given by

~

satisfying

potential with finite range (i.e. r >

if

(6.25); and it clear
= 0

for all

0 ); but it will not always hold for a potential

Irl ~ r ~ , for some

given by

@

satisfying

O

(6.24). However,

the only use made of (7.4) is in the proof of Lemma 7.2, and an

examination

of this proof gives us (using the notation of Section 6):

Proposition

7.i

(6.24); let that

in

Let

~ , v ~ Po(~)

lim inf hA(~,v) A R d . Then

Proof

~

be a specification with

given in terms of

~(U ) = v(U ) = i , and with

r

satisfying e G(V)

. Suppose

= 0 , where the lim inf is taken over the set of all cubes

~ e G_(V) .

Looking at the proof of Lemma 7.2 we see that the only time (7.4) is used

is to show that if

A e C

and

F e A

then

A+zd-A

It would thus be sufficient

to show that for each

exists a version,

E (XFIF(A))

g~ , of

with

A s C

0 < g~ < i

with

A C_ zd-A

and such that

there

124

lim

g~(x)

=

WA(X~F)

7.14

for all

x s U

~+zd-A Let

g~(x)

where

=

IWA(XA•

dPA(V)(z) '

A = zd-(Ak)A) , PA

N~ : X(A)•

>R

with respect to

is the projection from

X

onto

X(A) , and

is a version of the Radon-Nikodym derivative of

p~(~)•

. It is easily checked that

E (XFIF(A)) , and since we can assume that

~N~(y,z)

g~

P d (~) Z -A

is a version of

dPA(~)(z) = i

for all

y s X(A) , we have

IWA(X,F) - g~(x)I

II~A(X'F)- ~A(XA• But for any

as

m > I

A--@zd-A

we have, uniformly for

dPA(V)(z) z s U

m

. Thus we would be done if, given any

, that

WA (x;•

)hA (x'F)

6 > 0 , there exists

m~l

such that

lim inf A+zd-A

I U

N~(x~,z) dPA(V)(z)

~

i -

m

We leave the reader to check that, using Puelle's estimates given in Section 6, it is possible to choose

N~

with the above property.

82

125

7.15

We will now look for situations in which the converse of Theorem 7.1 is true; i.e. when we have

(7.11)

if

~ , ~ e G o(~)

then

lim hA(~,~) = 0 A

o

We will be mainly interested in situations which apply to the models of Section 5, though at the end we will consider the set-up of Section 6. The first case we look at is quite simple, but it gives the idea of why (7.11) might hold. We will say that A~

~ = {WA}Ae ~ A s ~

has bounded ranse if there exists

with

Is-tl > r

Z(AUA)-measurable

P~_~oposition 7.2

for all

Let

(7.12)

there exists

(7.13)

~(AU~)

Then

for all

such that if ~A(.,F)

is

F s ~(A) .

such that

= ~{~(A)UZ(~)}

lim hA(~,~ ) = 0 A

s ~ A , t e zd-A , then

~ , v e ~(~) , where

N

r > 0

~

hA(B,v) ~ N

for all

has bounded range. Suppose that

for all

A e ~ ;

A , ~ s ~

(where the limit is taken over the set of all cubes in

~ ).

(Note that we do not assume that the measures or the specification are translation invariant.)

(7.13) is a condition which is satisfied in all the examples we have considered; we will thus assume that it holds for the rest of this section. An examination of the proof of Lemma 7.2 shows that, with (7.13), we can replace the hypothesis that (7.4) holds in Theorem 7.1 by the slightly weaker hypothesis:

7.16

126

(7.14)

if

A s ~ , F ~ ~(A)

~(A)-measurab!e

function

and

8 > 0

g : X

>R

then there exists

such that

, rather than for all

F e ~ .) If

~

and an

]WA(',F) - gl < 8 .

((7.14) differs from (7.4) in that we only have to approximate F e ~(A)

A ~- zd-A

WA(',F)

for

has bounded range then of course

(7.14) holds, and thus the conclusion of Theorem 7.1 is valid. The crux of the proof of Proposition

7.2 (and the key to understanding when (7.11) should hold)

is the next len~r.a.

Lemma 7.7

Let

A , A e ~

Z(A)-measurable

Proof

Since

F e F(A)

for all

with

F g Z(A)

A~A

= @ , and suppose that

. Then for any

HA~j~(N,v) ?_ H~(~,v)

~ , v e ~(X)

we can assume that

~A (. ,F)

is

we have

H~(~,v) < = . Let

, F e F_.(A) ; then

~(Ff~;F) -- I • d~ = [ wA(x,F) d~j(x) = ; ~h(X,F) g~(x) d'o(x)

=

f~

g~ dv

(by Lemma 2.2).

FnP

Thus

P ~(A) =

~ g~ d~

for all

A a ~ , where

A

n

]2 = {

~) k=l

(.~IknFk) :

F k e =F(A) , Fk r F(A)

; (Fkf~Fk)gi(Fjf~F j) = ~ if k#j }.

127

But

D

is a field, and by (7.13) we have

p(A) =

~ g~ dv

for all

A e F(AkJA)

7.17

~(AUA)

; i.e.

= q(D)

. Therefore

~tA~A << VA~)A

and

gAUA

= gA "

A Hence

log g A U l )

d# = ~ ( l o g g~) du = H~(~,v)

The proof of Proposition cube of side

b

(where

bounded range), and let . Then

WA(-,F)

7.2 follows easily from Le~ma 7.7: let

b > 2r , w~th A

.

r

as in the definition of

be the cube of side

is ~(A--A)-measurs~le

for all

b-2r

~

A

having

having the same centre as

F e F(A) , and thus

H~(~,V) = HK_A(~,~ ) . Therefore

I -AI h~(~,v) =

~

I~1

bd-(b-2r) d h~_A(~,v )

2dr =<

b

and hence

h~-A(~'~)

hd

2&r h~_A ) , (~'l.l,v

< ~

-

lim h;(~,v) = 0

'N b

o

We must now find out when (7.12) holds.

Lemma 7.8

(7.15)

Suppose that there exists

w{t)(x,F) ~ ~w{t}(Y,F)

a

for all

>

I

be a

such that

x , y ~ X , F a ~({t})

, t e Zd .

128

Then for any

Proof

p

, v c G_(V) , A c C

First note that for emy

i/~ & g{t} A a ; so

Let

thus

M(F) <__a~{t}(x,F)

=

we have

t e Zd

hA(P,v) ~ l o g

we have

a .

~{t} << v{t}

and

h{t}(P,v) A log ~ . We will show by induction on

t E Z d , A a :=.C with

~(FaF)

7.18

t ~ A . For

for all

F c =F({t}) let

x c X . Let

I w{t}(x,F) d~(x) <_~ I

IAI

that

M(F) = sup ~{t}(x,F) x~X

F c F({t}) , F e F(A) ; then

M(F) dP(x) =

:

M(F)gA(x) d~(x)

~up gA(x) I M(F) d.(x)~ sup gA(x) I ~{t}(x'F) ~(x) xcX

=

and thus

~(A)

sup gA(x) s xe X

IAI

x~X

e sup gA(x) v ( F ~ F ) xg~

~

c sup gA(x) v(A) xgX then

;

for all

B { t } ~ A << v ( t } U A

and

A g ~(tt}UA)

. If

g { t } ~ A Ae(IA[+l)

estimate gives the other inequality.

Combining Proposition 7.2, Lemma 7.8 and Theorem 7.1 we get:

9 A similar

129

Proposition

7.3

some

. Let

if

e > i

Suppose that

lim hA(M,v) A

M , v e P

= 0

~

7.19

has bounded range and that

(Z d)

with

v e G(VD

~

concerning the existence of the Radon-Nikodym

A , A e C

such that for each

with

F e F(A)

A~A

In

so that there are no problems

derivatives

. Our estimates will be based on how close

Let

if and only

doesn't have bounded range.

what follows we will suppose that (7.15) holds,

Le~mla 7.9

M e G(V)

for

(where the limit is tahen over all cubes).

We will now see what c~] be done if

HA(p,v)

. Then

(7.15) holds

~

and the finiteness

of

is to having bounded range.

= r , and suppose

there exists

we can find an F(A)-measiG, able function

GA,A > l h : X----)R

with (~A,~)-'lh

Then for any

]J , v e G(j~)

~AU~.(!J,~))

Proof

Let

<= WA(',F)

<

~A,~h 9

we have

- ix(]J,,))

<__ 2 log (~A,A

F e F=(A) ~ F e F__(A) , and let

h : X

%R

be as in the hypotheses.

Then

~ WA(,,F)dla < c~A,~~h dM = ~A,~.~hg.~ dv

.(F~)

<

((~A,~)2I-~A(. ,F)g~dv

=

(CLA,~)2 I g~ dv

130

Thus

gAUA

~

(~A,A)2

7.20

and a similar estimate gives us

gA

gAUA

>-- (~A,A)-2gA

Therefore HAUA(}J'v) - H~(IJ'v)

Proposition 7.4

~ l o g / g A U ~ } d~~

=

Suppose that

~

<

2 l~

~A, ~ "

82

is such that (7.15) holds for some

e ~ i ;

and that

(7.16)

there exist

such that if

A

71

To :

Z+

is a cube of side

ing the same centre as urahle

,

h : X--~R

> Z

n

A , and if

+

and

with

A

Yl(n)

lim " n-~ n

= O

,

is the cube of side

Y2(n)

lim---7--- = 0 , n~ n n+Tl(n)

hav-

F ~ ~(A) , then we can find an ~(A-A)-meas-

with

{exp(-T2(n))}h ~ ~A(.,F) ~ {exp T2(n)}h .

Then for any

~ , v e G=(~). we have

lim hA(~,v) = 0 A

(with the limit taken over

cubes).

Proof

Let

H~(M,v)

But

A , A

<

HA(~,v)

be as in (8.16). Then by L e n a

H~_A(p,v) + 2T2(n)

< H~(p,v)

<

7.9 we have

2dy](n)nd-llog

c~ + 2T2(n)

, and thus

1 hA(%,,v)

<

--

=

IAI

}[~(~,v)

< =

{2d log ~} Yl(n~ n

+

2 T2(n) nd

.

"

131

7.21

Combining Proposition 7.4 and Theorem 7.1 we get:

Theorem 7.2 any

Suppose that

, ~ s G(V)

v ~ ~(~)

and

~

satisfies

(7.14), (7.15) and (7.16). Then for

lim hA(~,~ ) = 0 . Conversely, if A

lim inf hA(V,v ) = 0 , then A

, v s P (Z d) o

with

v c i(X) , then

p , v c Po(Z d) , with

# s ~(X) 9 In particular, if

lim hA(W,v) = 0 A

~ g(i)

if and only if

9

(As before, the lim and lim inf are taken over the set of all cubes.)

Suppose now that we are in the situation of Section 5, with with

(Yt,F=t) all copies of the same space

of potentials that the

{r

CA

(Yo,Fo) . Let

~

S = Z d , and

be defined in terms

satisfying the hypotheses of Proposition 5.2, and suppose

are zd-invariant in the sense of (5.17); thus

~IIr

I

is

Act tea independent of noted that (7.15) with

~

t s Z d , and we will denote this n ~ e r

by

N . We have already

satisfies (7.14), and a simple calculation shows that a = exp(2N) . Furthermore,

~

satisfies

(7.16) holds, and we can choose

Yl

to

~l(n) be any function with

lira ~ n~ n

Y2 (n)

= 0 ,

lira yl(~i) = ~ ; and then define n~

I leall

=

Y2

by

9

OsAeC diam(A)>Yl(n)

(where

diam(A) = max{ Is-tI : s,t e A }). We leave it to the reader to 'check

what the F(A-A)-measurable

function

h

has to be. Thus for this case the

132

7.22

hypotheses of Theorem 7.2 hold, and so we get the result we stated at the beginning of the section, i.e.

Proposition 7.5

with

/

Let

II@AI I

V

be defined in terms of zd-invariant potentials

finite. Let

~ ' ~ ~ Po (Zd)

with

{r

~ s G(V) . Then

0~A~ lim hA(P,v) = 0

(with the limit over cubes) if and only if

V e ~(~) 9

A

Now consider the set-up of Section 6, with

V

defined in terms of

satisfying (6.24). The converse of Proposition 7.1 is in fact true, i.e. if , ~ ~ G(V)

with

,(U)

= ~(U)

= i

then

lira hA(P,~) = 0 . However, the best A

w&y to show this is to explicitSu write down projections of

p

and

v

hA(~,v) (in terms of

onto the outside of

@

and the

A ), and then estimate directly

how small this quantity is. This is quite hard to do, and requires the use of more estimates from Ruelle (1970). We leave the interested reader to attempt it. Let us now look at a slightly more general situation than that considered in most of this section. We want to include the set-up of Section 5~ but now with S

being any regular lattice (or graph) rather th~n just

have an underlying space , and that for each

S (not necessarily countable), equipped with a G-field

A s ~

we are given a G-field,

(where we make no particular assumptions about ~(A) C ~ ( B )

; let

Z d . Suppose that we

~ = F(S) . We take

C

F(A) , of subsets of

X ). Assume that if

to be some subset of

A C B

if

AI , A 2 E C

then there exists

A s C

with

then

S , ordered by

inclusion, and make the assumptions:

(7.17)

X

AI[)A2~A

;

133 (7.18)

if

(7~

there

for some

For

A I, A 2 s C

exists

with

A

AIC

A2

7.23

then

A2-A I ~ C ;

e C , n = 1,2,... ~

n

such that if

then

A C_A

n

n .

A e ~

let

~A = ~ ( S - A ) .

We will suppose

that

We will need a group of transformations of S=-measurable

hijections

((A) ~ ~ . Suppose transformation

from

T~ : X -

T~Tq = T~q

(7.21)

T (x) = x

(7.22)

T~I(~(B))

(7.24)

m

< ~

H-invariaut

for all

{T~}~eH

. If

hA(~,v)

by

fields,

~ v e P(S)

o~(A)

acting on

such that if

there

1 .

X . Let A e ~

is associated

and

H

be a group ~ e H

then

an F=-measurable

~ , O c H ;

x ~ X , where

for all

a measure

is H-invariant,

random

~=

in such a way that

=F(CI(B))

Again we will write

S

( e H

for all

suppose we have

m(A)

>X

to

for all

e

(7.23)

S

that to each

(7.20)

Finally,

A e C ~

m

A e C

i.e.

P(A)

on

is the identity

B E S,

(S,S)

in

H ;

~ e H .

such that

;

m((-l(B))

= re(B)

for all

( e H , B e S_ .

instead of

P(F(A))

; Pc(S)

will

i.e. those random then we define

fields HA(~,~)

1

h A (]J ,'o ) -

e

,'o )

invariant

denote the

under the group

as before;

we now define

134

7.24

The condition (7.4) still makes sense in the present setting (provided replaced by

Zd

is

S ); thus to give the analogue of Theorem 7.1 we need only find out

what corresponds to "cubes". The property of cubes which was used was that they pack together nicely, and to measure this in our present set-up we define, for

max{ n : there exist

nH(A,A ) =

~i' .... $ n e

H

such that

are disjoint subsets of

Thus

nH(A,A )

is the number of translates of

A

~I(A) ..... ~n(A) A } .

that will pack into

A ; by

m(~) (7.2~) we have

nH(A'A) ~ m(A) " Let

upwards, generates the order on say that

~

is adapted to

H

if there exists

A

~ > 0

~ . (Note that

m(~)

"volume" of

~

which is directed H . We will

such that for all

A e

m(A)nH(A,A) ~

~ ~

be a subset of

~ , and is closed ~ider the action of

m(A)nH(A,~) lira inf

~

-

-

is the fraction of the

m(~) which can be filled by disjoint translates of

set of cubes in the case

S = Zd

then we can take

A . If

~

is the

a = i .) The new form of

Theorem 7.1 now becomes:

Theorem 7.3

Let

_~ = {WA}Ae C= satisfy (7.4); let

and suppose that

adapted to

Proof

H . Then

lim inf hA(P,v) = 0 , where

~

~ , 9 e Po(S) is a subset of

with ~

u e G(V), which is

~ s G(V) .

This is the same as the proof of Theorem 7.1 as far as Lemma 7.6. The proof

of Lemma 7.6 is easily modified, and we leave it as an exercise for the reader.

82

135

7.25

To see when the converse of Theorem 7.3 holds we look at the proof of Proposition 7.4. Let us again suppose that (7.13) holds. Note that although it is not hard to see what the analogue of (7.16) is, there is no real ~ a l o g u e (7.15) unless

S

Propgsition 7.6

(7.25)

is countable. Mimicking the proof of Proposition 7.4 gives:

Let

there exist

~

be a specification with the property that:

Tl ' Y2 : ~

~R

with

= such that if

A c ~

and such that if

F c F(A)

]~ , v ~ G(V)

(7.26)

Then

A e C

with

0 , lim

re(A) A ~ A

Ac~ and

then there is an F( A-A )-measurable

<

WA (-,F)

<

Y2(A) -

m(~-A) < Yl(A) , h : X----~R

w~th

{exp Y2(A)}h .

N

for all

such that

AI , A2 c ~

and

A c C

of the form

A2 C AI .

lira hA(~,v) = 0 9

The above setting gives us the analogue of Proposition 7.5 when replaced by any suitable regular graph. H and

0 ,

re(A)

with

there exists

A = A]-A 2. ~ with

YI(A) =

l i m ~

Ac~

then we c~m find

{exp(-Y2(A))}h

Let

of

~

Zd

is

is then the symmetry group of the graph,

is a set of reg1~Lar shapes for the graph. However, it is worth pointing out

that things can go wrong for some graphs: let ered in Section 5, and let

H

S

be the infinite tree

TN

be the group of all graph automorphisms of

considTN .

(The random fields we explicitly constructed in Section 5 are clearly H-invariant. ) Let

~

be a specification defined in terms of a nearest neighbour, homogeneous

7- 26

136

potential

V ; let

C = { A(n,t)

: n > i

A(n,t) = { s s S : d(s,t) < n ] , and from

s

to

t . Then although

~

t ~ S }

d(s,t)

where

is the length of the shortest path

is not quite adapted to

H , a slight modification

of the proof of Lemma 7.6 still gives us Theorem 7.3. However, shows that the converse of Theorem 7.3 is false in this case: have H-invaria~t

~ , v ~ G(_V)

with

a direct computation it is possible

to

liln hA(~l,v) > 0 . Now it is easily checked

ac_c. that (7.26) holds that fails.

Tnis is not very surprising,

corresponding point in

(since in this case we even have (7.15)), thus it must be (7.25)

A c C

A , thus

the right properties.

must include

because if we are given

]A-A I > 89 I , and so it is not possible to choose The problem is that the boundaries

g_ap~ which has some physical meaning, similar to that which happens n

d-i

for

then the

at least all points that are neighbours

same order of size as the sets themselves.

of order

A s ~

Zd

of sets in

Tais "pathology"

~

Yl

to some with

are of the

will not occur for any

and typically we would expect behaviour where a cube with

nd

points has a boundary

8.1

8. Some thermodynamics

In Section ect to

7 we defined

h(~,u)

, the specific informatioh

~

with resp-

v , but we did not show that the limit defining this quantity

existed. We

now tackle this problem by showing that if under suitable conditions

(8.1)

where

h(~,~)

f(~,~)

ect to

e s p o n d i n g to existence

P(~)

= f(~,~)

is

+ P(~)

free

energy of

a n u m b e r w h i c h can be i n t e r p r e t e d

~ . The existence

of the limit f(~,~)

h(~,~) and

w

with resp-

as the pre s s ure

corr-

can then be got from the

P(~)

.

approach also s11ows us to restate the results of Section 7 as a

form of Gibbs' variational principle: and only if equality

~ c __G_o(V)_ then,

;

t o some k i n d o f s p e c i f i c

of the limits which define

9

and

we have

corresponds

~ , and

~ e Po (Zd)

gain of

~ s (~)_

if and only if

, then from (8.1) we get

h(~,v) ~ 0 , with equality f(~,~) ~ -P(~)

~ s ~ (~) ; i.e. the translation

are exactly those translation free e n e r ~ .

if we have

invariant

again with

invariant

Gibbs states

random fields which minimize the specific

As mentioned in Section 7 this generalizes

result of~ Lanford and Ruelle

if

(Lanford a~_d Ruelle

FS~Lmer's

(1969), F611mer

proof of a (1973)).

Most of this section deals with the set-up of Section

5 (although we will

start as in Section 7), and in particular with specifications

defined in terms of

potentials which satisfy the hypotheses

of Proposition

5.2. In theory the models

of Section 6 could be handled the same way, but the technical overcome are many:

difficulties

to

the reader is invited to try.

Suppose we are in the situation

of Section 7. Thus for each

AC

Zd

we

138

have a o-field

~(A)

8.2

of subsets of a basic space

empty finite subsets of

Z d ) we put

~A = ~(zd-A)

X , and for

A s ~ (the non-

. We make the same assu~#tions

about the o-fields as we did at the beginning of Section 7. Usually in physics the specific free energ~ is defined by is the specific energy of entropy of

~

f(~,~) = e(~,~) - s(~) , where

with respect to

~ , and

s(~)

e(~)

is the specific

p . We will start by looking at what corresponds to

s(N) , and show

that the limit which defines it exists. We define a special kind of random field: if, given any

A I , A2 ~ ~

with

A I ~ A 2 = @ , and

a P(Z d)

is called independen-~

F I s ~(A I) , Y 2 s ~(A2) , then

k ( F l ~ F 2) = k(Fl)k(F 2) . (This says that events depending on disjoint s ~ s e t s Zd

of

are independent.) In the situation of Section 5 the natural (and only) w~j

of getting independent measures is as product measures; the Poisson point processes defined in Section 6 are also independent.

For all the models we are interested

in there is at least one independent measure. We will thus define the specific entropy by putting

s(p) = -h(p,k) for some suitable independent

k e Po(Zd).

The first result of this section shows that this is at least well-defined.

Theorem 8.1

Let

~ , k r P (Z d) o

exists, and equals

with

k

independent. Then

h(~,k) = lim hA(~,k) A

sup hA(P,k ) (with the limit and sup taken over the set of all A

cubes). Note: We must allow

Proof

(8.2)

Define

h(p,k)

to take the value

D : ~

~[0,~]

D(A) ~ D(A)

whenever

by

A C_A

+~ .

D(A) = HA(PSk)

. Then we have

, (using Jensen's inequality, as in Section 7);

8.3

139

(8,3)

D

_ invar{ant,

is trmnslation

Suppose we also knew that

(8.4)

if

A I ....

D

A s C ~ n =

i.e.

=

D(A+t)

was superadditive,

are ~ s j o i n t

Ai

D(A)

for all

,

A e C

Z d

t ~

i.e.

then

>

D(A i) . i=l

Then by a well-known be complete.

Since

*

(see for example

it is so simple,

1 lim --- D(A)

that

result

exists

in Ruelle

let us show how i sup - -

and equals

IAI

(1969))

(8.2),

D(A)

the proof would

(8.3)

and (8.4)

(where the limit

imply

and sup are

A IA I

taken over the set of all cubes). For

A , A g C

let

n(A,A)

be the maximum number

which will fit into

A . If

A

exists

a cube

A

n(A,A)

> (1-6)

--

o

such that

. Let

if

Let

A

A

A with

n = n(A,A)

of

D

~

Ai

above,

, then we can find

a translate

and we are given

is any cube with

--

D(A)

A . Then by

D(A i)

=

nD(A)

and let

~

lim

A

of

A

then there

then

o

over cubes),

D(A)

_>= D

. q~us

--

! --- D(A)

exists

meaning

be any cube with subsets

Ai

smd

and equals

of

if

d = += ).

A O

A

D(A)

l

. Put each

Ai

and (8.4)

> (1-6)--

= d .

o

A , with

; and by (8.3)

D(~) > -

-I . Therefore

~ > 0

(with the sup taken

AI,...,A n , disjoint

(8.2)

translates

I^I

= >__ (l-~)(d-~)

A ~

D(A) > (d-~)iA I (with the obvious

be the cube defined

o

A

d = sup

IAi choose a cube

is a cube,

of disjoint

I^I

D(A)

.

140

8.4

We now need to show that (8.4) holds, and clearly it is sufficient to show i% in the case

n = 2 . This is given by the following lemma.

Lepta 8.1

Let

~ , k c P(Z d)

with

k

independent, and let

A , A ~ C

with

A ~ A = @ . Then

HAO~(~,X)

Proof

We can assume that

easing f'0mction of

A

A ~A(~,~) + ~Z(~,~) 9

HAU~(~,I)

we also have

< ~ , and thus since

HA(P,k)

HA(~,k) < ~ , H~(p,l) < ~ . As in Section 7

dP A let

gA -

. Then we have dk A

HAkj~(p,%) - HA(P,%) - H~(p,%)

=

f l o g (gALl-----~A)dp

gA g%

gAh} A dk

=

I r

dk

gA gA I

~gAg~/ A A dl +

But

I gA~A

dl

=

i gAg~ dl

Therefore

{gAu~-gAg~ } ~

i , and also since

=

IgA

is an incr-

d l l g~ dX

1

=

is independent we have

i .

141

HAU~(~,X)-

8.5

IY (gAUAI dl _>_ 0 .

HA(U,X)--H~(~,X)=

gA g~ !

Suppose now that we have some independent regard as a fixed reference measu.re. cation,

and let

that for each

v c Go(~) A E ~

Let

~

be a translation

. Suppose that for all

there exist

X s P (Z d) , which we will o

~A ' BA s R

A e ~

with

invariant

specifi-

v A << X A , and further

0 < eA ~ 8A

such that

dv A ~A

<

~

BA "

dl A

Then for any

~ s Po(Z d)

HA(U,v)

we have

=

HA(V,X)

HA(U,v)

-

< ~

log

if and only if

--

HA(P,X ) < ~ , and

dv

dXA #

'Thus, from Theorem 8.1 , in order to show that

h(~,~)

exists we need only show

that

lim - -

log

A IAI

--

d~

d~A/

exists; we would also like to identify this limit with e(~,V)

the specific energy of

corresponding

to

~

with respect to

V , and

V . Note that if we could show that

i

lim---

log

--

1

-{e(~,V) P(V)

+ P(V=)} , with the pressure

142

ex•

8.6

with all the terms uniformly bounded, then by the dominated convergence

theorem

h(~,v)

would exist, and the existence of

particular properties of

h(p,v)

would not depend on any

p (other than it being translation invariant).

We will first see when this limit exists in the situation of Section 5. Let

(Yo,~_~,~o) be a probability space, ~ d

a copy of

have

for each

t ~ Zd

let

(Yt~F_t,wt) be

(Yo,~,a~o) . We will use the notation of Section 5; in particular we

X = ~

Yt"

~=

t E Zd (X,~)); thus

~

F----t" Let

~ =

~

t s Zd

~t

(as a measure on

t s Zd

~ s P (Z d) , and it is clear that

~

is independent. Choose

O

bo ~ Yo

and let

b = {b t}

(where

bt

is the appropriate copy of

as a base-point. Suppose we have a specification

~

bo ); use

defined in terms of

b

{r

which are tr~islation invariant (i.e. they satisfy (5.17)), and for which

IleAl I

is finite (i.e. they satisfy the h~otheses of Proposition 5.2).

OEA~

~ider these conditions we will show that

c G (~)), and hence that Let (writing

(8.5)

S

{~}AE~ for

h(p,v)

1 --

EA[

log ( dvA I

\ d% /

exists for all

be defined by (5.13). Then for

converges nicely (for any

e P o ( Z d)

, 9 ~ O~o(V) 9

A ~,,~(__v)

~e

have

Zd )

---(x) =

fA(ys_A• A) a~(y)

d~A

(in that the right-hand side is a version of the Radon-Nikodym derivative, and it

8.7

143

is the version we will always use). For

(8.6)

A c ~

define

JA : X - - - - ~ R

exp V(bs_AXXA )

JA(x) =

l exp VCbs_Axw) d~A(W)

and let

=

WA

li Aff

A~Ar162 AaCs-A)r

The next len~ma gives us an important estimate. Lemma 8.2

Let

v e G(V)

exp(-2WA)

Proof

xsX

exp(2WA)

(Ys_AXXA) dvCy) / JA(X)

9

We have

I Now

~

, A c C . Then for all

exp gA(ys_AXXA)

#~(yS_~•

dr(y) =

I exp gA(ys_A•

exp gA(ys_A•

d~(y)

I

/ exp V(bs_Axw)

=

dZOA(W)

exp {gA(ys_A•

- V(bs_Axw)}

9Thus

A n Aye A ~(S~A)@@

exp(-w A) s

exp gt'(y_R A•

/ ex~ V(bS_A•

<= ex~(WA) .

The result therefore follows by a simple comparison of terms.

82

by

8.8

144

For

B e P "-(Z d) , A e C

define

o

(8.7)

eA(~, Z)

V(bs_A• A)

=

IAI

d~(x)

~,

a~d

(8.8)

PA(__V)

=

--

log

exp V(bs_Axw)

d~A(W )

IAI

Thus

--

].og{JA(X)}

du(x)

=

- eA(U,V) - PA(_u

IAI

=

,

-

and hence by Lemma 8.2 we get

I (

i

(8.9)

d~A~}

d]J +

eA(~,_V)

+

P4(X)

I

<

--W

A .

But:

i lim - -

Lemma 8.3

A

Proof

For any

A e ~

II~AII A DA~ A a (S-A)#r

Let

K

=

~ueh

that

WA

=

0

we have

l z z IAnAIiI%II <=

:

t ~ A

teAcC A r~(S-A):/r

I1%11~given

L

(provided the limit is taken over cubes).

IAI

ileal!

~

Oci~s A n(S-A )~ o

~ > 0

<

t e A

there exists a cube

. For ~ y

t e Zd

let

AO

II~AII-

t~Aes A t~ (S-A)qr

with centre

Ao(t )

0

be the translate

145

of A

e

A

that has centre

o

= A - A

"~ t

t ; for

~

tmAe_C_=

i ~hus

--

let

=

A

m

= { t e A : A (t)C. A } , and let o

. Then

m

~ II~AII a AI'~(S-A)yr

~ A

A E C

8.9

WA

IAI

<

6 + K

(

6 + ~ ~ A

t

~ E A

m

e

IAml . But clearly

lim

--

A

IAI

IAI i lim - -

and thus

WA

II~AII <__ alAml+KIAel .

tEAc~

IAml )

i -

=

over cubes),

t

=

0 .

=

] (with the limit

82

IAI

In order to show that

h(p,~)

Lemma 8.3, Kid ~ h e o r e m 8.1, that

h(p,~)

exists then its value

look at

lira eA(~,~)

(8.1o)

. For

~t(x)

exists we n e e d only show,

l i m eA(P,V) A

and

lim PA(~) A

does not depend on w h i c h

t e Zd

=

define

__

~t : X

CA(XA)

using

exist.

~ c G=o(~)

)R

(8.9), Note that if

is used.

Let us

by

9

IAI

Clearly the sum in (8.10)

function.

of

From (5.17)

converges

absolutely,

it follows that if

and

c p ""~zd) o

r

is a bounded,

then

~

@t dp

F=--measurable

is independent

J

t e Zd . Let us suppose from n o w on that

for the specifications of Gibbs states.

we are considering,

(Yo,F_o)

is a standard Borel space.

there are no problems

Thus,

about the existence

146

Lemma8.4

If

~ E P (Z d) o

then (with the limit taken over cubes)

limA eA(~'V)

=

-

Ir

an 9

V(bs-A• = A~~ . A CA(WA) = t~e~A ~_ 1 ~ t E A ~ A IAU

Proof

Thus

8.1o

I V(bs_A• A) -

~

CA(WA) "

Ct(x) i

tEA

I t EA

! ~A(XA) I = IAI

tEA

I

A n(S-A)~r

- -

A~A~

CA(XA ) I

<

WA 9

IAI

A n (S-A)#~

But

tEA

I eA(~,v) +

~0d~

I =<

i -IAi -W

A ;

and therefore the result follows from Lemma 8.3.

Let us put

e(w,~)

with respect to

Lemma 8.5

P(~)

Proof

=

lim eA(~,~) , and call A

e(~,y)

the specific energy of

~ .

lim PA(~) A

exists (over cubes), and is finite. We denote the limit by

~Id call it the pressure corresponding to

Let

82

p E Po(Z d) , ~ e G ( ~ )

~ .

; then by (8.9)

147

I hA(P,v)

- hA(P,~)

Thus by Theorem

exists

- eA(P,V----) - P A ( ~ ) I

and equals

putting

U = v

though the limit might be

K

l l|

=

hence

PA(V)

~

K

for all

for

entropy

~ ; we have

thus

UA = ~A

Theorem

if

If

(8.11)

0 ~h(~,v)

hypotheses

Proof follows

(8.1i)

as

=

A ~ C .

82

~

s(~)

gives

- h(~,m)

e(p,V_.) - s ( p )

h(B,v)

h(~,v)

~(~)

< ~

= e(~,~)

= 0 , that

then

+ P(~)

hA(~,e)

h(~,v)

because

if'

s(p)

the specific

if and only if

= 0

for all

A ~ ~ ,

exists,

and

.

for some

the previous

if and only if

> -~

exists

ana

s(~) = 0

< ~ , (and also in this

+ P(~)

+~ ).

~ = ~ .)

B e ~o(V~

together

lim PA(~) A

is finite,

, and call

then by Theorem 8.1

s +~ , but if then

hA(V,~)

--

A E ~ , and hence

=

+ PA(~)}

(which could have the value

-~ ~ s(~) ~ 0 . (Note that

~I~is is just collecting because

=

lira {hA(U,v) A

in fact the limit

D e Po(Z d) , v e G ( ~ )

h(p,v)

We have

since

V(bs_^Xw)

s(~) = 0

for all

8.2

WA 9

+ e(w,V)

we get,

~ E Po(Z d) ,

= ~ , because

h(u,~)

+~ . However,

, the

Let us put, of

<

8.1, Le:~aa 8.3 and Lerama 8.4 we have

(over cubes),

In particular,

8.11

=V

satisfying

case

results.

s(~)

the same

> -~ ).

The last part

s(B) > -~ , and putting

v =

in

148

Note

8.12

7~though we have restricted ourselves to looking at limits over c~oes, in

fact the limits exist under much weaker hypotheses;

for example "in the sense of

van Hove": for details the reader should look in Ruelle (1969)~ Gallavotti and Miracle (1967), and Robinson and Huelle (1967).

Combining Theorem 8.2 and Proposition 7.5 we immediately get a version of Gibbs' variational principle:

Proposition 8.1

For

p ~ Po(Z d)

let

specific free energy of

~

equality if and only if

p c G=o(V) .

f(p,~) = e(~,~) - s(p)

with respect to

This variational principle involves varying

(f(p,~)

V ). Then

f(p,V__J> - P ( V )

B s P (Z d) o

with

V

is the , with

fixed; we will

now look at some other variational results in which it is the specification

V

that is ~aried. In particular we will identify Gibbs states with tangent functionais to the pressure. Most of what follows is adapted from Lanford and Ruelle (1969) and Ruelle (]967). If we a~.e going to vary

V

then we need a suitable class of specifications;

the obvious class to use in our present situation is those specifications given by potentials satisfying the hypotheses we have assumed for

set of potentials

V 9 Xo------@R

such that

V . Let

B_

denote the

V(x) = _ _ Z CA(XA ) , with

{r

A~C translation invariant a~id

2

= I ICAII

finite. B

can clearly be regarded as a

OcAcC

real vector space; and in fact as a normed vector spaee~ with the norm of

V ,

149

denoted by

IIVll , given by

~__

8.13

IleAl I . Furthermore, it is not difficult to

0~h~ see that each

B

is complete as a normed vector space, i.e.

V e B

there corresponds a specification

e(~,V) , P(V) , ~(V) , etc. instead of

B

is a Banach space. To

~ ; it will be convenient to write

e(B,~) , P(~) and

G(~) .

We can thus consider the pressure, P , as a function from will show that

P

A

to

R . We

is continuous and convex; and it is therefore natural to look at

the tangent functionals to ionals. If

B

P . Let us recall some basic facts about tangent funct-

is any real Banach space, if

space of continuous linear maps from uous and convex, then

u e A

g(a + &) - g(a) ~ u ( K )

A

to

A

denotes the dual of

R ), and if

g : A

is called a tangent functional to

for all

& e -A- . Let

Ta

A_ (the Banach )R

g

at

is contina e _A if

denote the set of tangent funcw

tionals to

g

at

a ; then

Ta

is a non-empty, compact, convex subset of

A

(the compactness being in the weak topology). Now if

~ ~ Po(Z d)

then it is easily checked that

Ile(~,')II a i , since by Lemma 8.4

~ Go(V)

e(~,v+w)- s(~) hence

Ie(B,V) I = I ~ @ 0 dB I ~ llVII ). Further, if

then Proposition 8.1 tells us that

the pressure at

V , (since if

a-P(V+W)

and

W ~ B

-e(~,-)

e(.,V)

- s(.) = -P(V)

tangent functionals are obtained this way, i.e. if V

then

u = -e(~," )

show that in this representation

B

is a tangent functional to

then

P(V+W) - P(V) ~ -e(~,V+W) + e(p,V) = -e(p,W)

the pressure at

e(~,. ) e B_ (and

for some

;

). We will show that all u

is a tangent functional to

Z e G(V)

. Furthermore, we will

is unique by showing that

-e(~%" ) = -e(v, ")

8.14

150

only if

U = v .

Theorem 8.3

Define

-e : P ( z d ) - - - - ~ B

by

(-e)(u) = -e(~,,)

. Then

-e

is

O

injective,

and for any

of tangent ft~ctionals

V s ~ to

P

the image of at

C_~o(V) under

-e

is exactly the set

V .

We start the proof of the theorem by looking at some of the properties the pressure.

Before doing this,

proof of Theorem 8.3 is that if subset of

U = { V e B : I ~ ( V ) I = i } , then

U

of the

is a dense

B . (This will be proved as Theorem 8.4.)

Lemma 8.6

P : B ------*R

IP(h) - P(v2)l

Proof

let us just mention that one consequence

of

is continuous,

~

A simple estimation,

IPA(VI) -

Lermna 8.7

P~(v2)I

P : B_-------~ R

then

P(tV I + (l-t)V2)

Proof

We will show that,

and exponentiating,

~id in fact

ilvl- v211

which is left to the reader,

a l l v I - v211

~r

is convex;

0 < t <__ i

i.e. if

shows that for any

and

VI , V2 e B

tP(u I) + (l-t)P(V 2) .

for any

A e C_ ,

PA

is convex. Multiplying by

we need to show

l exp{tVl(bs_Axw)+ (l-t)V2(bs_AxW)}dmA(W)

(8.12) <

Ace

[~ e~ vI(bs-A• d~A(w)It [I expV2(bS-A• ~A(~) ](l-t)

IAI

151

But if we put

p

i/t , q = ] A l - t )

=

t

~ then

8.15

p

and

q

are conjugate exponents,

emd thus (8.12) holds by applying Holder's inequality to the functions

fl

and

f2 ' -where

fl(w) = exp{tVl(bs_A•

P~o~Oosition 8.2 -

Proof

,

f2(w) = exp{(l-t)V2(bs_AxW)}

The mapping

-e : F (Z d)

-

O

We want to show that

~i = P2 " For

V e B

let

V ; then by Lemna 8.4

{r

{,~V}VsB

to be able to s e p a r a t e the elements of

show that

is injective.

for all

V s B

ilrLDlies

denote the interaction potentials determining

-e(~,V) = I C Y dp , where

for some

82

--

-e(Pl,V) = -e(P2,V)

The problem is thus to show that

and F(A)-measurable

)B

9

implies

=

0 c A s ~ - IAI CA(xA)V .

is a sufficiently rich set of functions

P (zd). Let 0

A e ~ ) with

-e(Pl," ) = -e(p2 , ')

r

f(x) = 0

f ~ B(~) ( i . e . f whenever

x0

: f dp I = I f dp2 " Let

is bounded

= b0

. We will

CA : X(A) ----->R

be given by

CA(w) = [A I ~

(-l)IA-Alf(wAXhB_ A) ;

A= A

then

f(x) =

0cA•c C - -1

SA(XA)

(with only finitely many non-zero terms), and

IAl

CA(W) = O

whenever

wt = bt

for some

t e A 9 Take

A e ~

with

O s A

and

8.16

152

define

{r

by

r

if

A = A+t

for some

t e Zd

cA(W)= { 0

Let

V

otherwise.

be the potential corresponding to

-i-

A~F

~A(XA ) ~ where

F

~I

and

; then

V e B_ and

is the set of translates of

IFI--IAI ). By hypothesis we have translation invariance of

{r

U2

A

that contain

~ V d ~ l - - ~ V d~2" and using the this gives us

I%~(xa)d~1(~)= ~(xa) d~e(~). Doing this for all

ation gives ~

A e ~

with

0 e A , and taking the appropriate linear combin-

~ f dw1 = I f d~2 " In particular, again using translation invar-

lance, this will hold if

n -(8.13)

f(x) = - - ~ ft.(xt. ) i=l i l ~.-measurable, and 1

,

where

ft. : Yt.-i l

ft.(bt.) = 0 1 1

for some

R

is bounded,

i .

n But if

and

gi

ft.(bt.) @ 0 l l satis•

for all

i , then we can write

(8.13). ~nerefore

f = c +~ gi ' with i= I

I f d~l = I f d~ 2

for all

f

c ~ R

of the form

~s3

8.17

n

f(x)

= 7~

ft.(Xt.

i=l hence

m

) ' with m

ft. : -ft.:= m m

9R

F=b.-meastLrable and bomqded; a~,d i

DI = ~2 "

The technical result

(due to Gallavotti and Y~racle (1967), and Lanford

and Robinson

(1968)) which enables ~s to deduce that all tangent functionals

of the form

-e(~,*)

Prol~osition 8.3

is the following:

Let

A

be a real separable Banach space, and let

be continuous and convex. For als to

g

at

are

a ; let

a s -A

let

Ta

g :A

.~R

denote the set of tangent function-

U = { a c _A : ITal = l } . Then

U =

~

U n , where

n>l each

U

is s~l open dense subset of

A ; in particular

U

is a dense subset of

n

A . Further,

--

an ~ U then

with u

Proof

if

n

a s _A an--~a

~u

smd

u

is an extreme point of

such that if

un

Lanford and Robinson (1968).

4

is the tangent functional to

~nd ~ r a c l e

~

an

(1967), for the second see

be a closed separable subspace of

B . P -e

restricted to

as a mapping from

~0

is

Po(Z d)

(though this mapping need not be injective m~ymore).

Proposition

Then

at

82

still continuous and convex, and we can consider to

g

(weakly).

For the first part see Gallavotti

Let

then there exist

Ta

8.4

u=-e(~,')

Let for

V c ~O sor~e

and .

u c ~0

e ~(V)

.

be a tangent functional to

P

at

V .

154

Proof

Let

written as

'1'

8.18

be the set of t~ugent functionals to

-e(~,.)

for some

u(W) = I r W dIJ for all

p s G(V)

W c 4

. If

u ~ T

P

at

with

V

which can be

u = -e(p,.)

" But is is easily seen that for any

W e B

B(A) , the bounded ftm_ctions from

are ~(A)-measurab]e for some

A e C . Thus by Theorem 3.2 (and the separability of

) we have

T

is (weakly) closed. T

u

if

u

Pn e ~o(Vn)

then

-e(Pn,')

u n = -e(lJn," ) . Thus

{nj}j> I =

and

u(W) =

D c Go(V )

that

is sn extreme tangent functional.

is extreme then by Proposition 8.3 there exists 4n e ~

a unique tangent functional, u n , to

R

is clearly convex, therefore it is suffic-

ient to prove the proposition in the case when If

to

@W

is in the uniform closure of

4

X

then

P

at

V n , and

such that ~here is

Vn - - - ~ V , U n - - - ~ u

is a tangent functional to

P

at

. But

V n , and hence

lim I @ W dw n . Suppose we could find a subsequence

such that

p(F) = lira Pn (F) j~ j

for all

F e A ; then we

would have P lira ~ @ W dPn. j+~ J j

P

-e(#,W)

=

\ ~ W dp

=

=

u(W)

(for a l l

W e ~O )

and the proof would be complete. ~nis fact is given by the next lemma.

Lemma 8.8

Let

V

exists a subsequence

, V e B_ with {nj}jk 1 _

all

Proof

and

Vn

~V

~ e G(V) =

snd let

82

Pn ~ =O(Vn) " Then there

such that

~(F) = lim ~n.(F) j~

for

J

FE~.

Let

{ITin)}Ac C (resp. {WA}AeC ) be the specification corresponding to

Vn

155

8.19

(resp. V ). It is not hard to show that for any ITA(n)(',F) ~

WA(" ,F)

if we have

uniformlJ~ as

~ s Po(Z d)

with

n--~

A s C

and

F e A

we have

Thus, as in the proof of Le~na 3.4,

p(F) = lim ~n.(F) j+~ J

for all

F e A , then we have

s G(V) . The rest of the proof is like Lemmas 3.1 and 3.2 (using estimates similar to those in the proof of Proposition 5.3), and is left to the reader.

82

We can now complete the proof of Theorem 8.3 by finding a nice separable subspace of

B . For st~pose there exists a closed separable subspace -e : P (Zd)

such that the mapping

~

is injective. Let

V s B

~0

of

B

and

u

be

O

a tangent f~uctional to by

.~

and

V . ~l

P

u(W) = -e(~,W)

injective

~ ~2

that

g c ~i " Since

for all

-e : Po(Z d)

u(W) = -e(D,W)

W e ~2

~I

>~

W e 4

~F = o ( A )

and

for some

~ = ~ ; thus in particular

There exists a closed separable suhspace

We need to find

for all

for all

u(W) = -e(~,W)

-e : Po(Z d)

Proof

be the closed suhspace of

is the closed subspace spanned by

we must have

Le~ma 8.9

~i

is unique. But we must have

able, and so --B2~ ~0

V ; let

B

spanned

is separable and so by Proposition 8.3 there exists

such that

because if

at

>~

~ e --o ~ (V)

is

for all

W s B ;

W

~2

then

is separ-

~ E C_~o(V) ; and as

u(W) = -e(p,W) .

~O

of

B

such that

is injective.

~0

such that

BI " ~2 e Po (Zd)

and

~ @W d~l = ~ r

imply

~i = ~2 " There exists a countable field

; for

A e ==C let

_~A__ C

d~2

==oF such

~ ( A ) C. =-~(A) be the field generated by

156 p r o d u c t s o f elements o f B --OO

~

, thus

8.20

~o(A)__ i s c o u n t a b l e and

consist of those potentials in

B

=oF(A) = o ( A ( A ) ) .

which are determined by

{r

Let

with

n

CA : X(A)------~R

of the form

~

akXAk , with

ak

rational and

Ak e A_o(A) ,

k=l and such that there exists then

CA = 0

~i = ~2

. B

--OO

whenever

A E ~

so that if

A e ~

with

0 ~ A

and

A - A ~ @ ,

is countable, and the proof of Proposition 8.2 shows that

~W

be the closed span of

d~l = ~ , W

B --00

d~2

for all

W g --oo B . We can thus let

B --o

82

The proof of Theorem 8.3 is now complete. It is worth noting that if is finite then Let

B

Proposition 8.3 it follows that

U

B

is separable then from Theorem 8.3 and

is dense in

B ~ and furthermore that

dense G6 (i.e. a countable intersection of open sets). If

Theorem 8.h space of

Proof

B

U

U then

O

is separable, ~Id in this case the above proofs can be simplified.

U = { V e B : IG_o(V) I = 1 } ; if

we still have

Y

is dense in

~ . If

is a dense G~ in

Clearly the second part implies

separable suhspaee of

_B with

~

~

then by Proposition 8.4 and Ler~aa 8.9 tar,gent functional to

P

at

is a

B i s not separable then

B .

is a dense subset of Uf%B_I

U

V (in

4

U

is any closed separable sub-

~i

B1 .

is dense in (with

~

I_G_o(V)I = 1

B . Let

~

be any closed

as in Lemma 8.9). If

V E

if and only if there is one

w B_2 ). Thus by Proposition 8.3

U~B2

is a

157

dense G 6 in

~2 " Now if

apply the above with

~2

~i

8.21

is any closed separable subspace of

the closed span of

~

and

~i "

B

we can just

82

We have seen that the pressure is a convex function; let us now see to what extent it is strict]@ convex. Suppose there exist

VI , V2 E B

and

t ~ (0,i)

such that

(8.14)

P(tV I + (l-t)V2)

=

By convexity (8.14) holds for all VI

and

tP(Vl) + (l-t)P(V2) .

t

with

0 ~ t ~ i , i.e.

P

is linear between

V 2 . One would thus expect that the tangent functionals to

tV I + (l-t)V 2

do not depend on

G ( t V 1 + (l-t)V 2)

P

at

t e (0,i) ; and thus by Theorem 8.3 that

does not depend on

t e (0,i) . We will show that this is true,

but will deduce it from Proposition 8.1.

Le~ma 8.10 let

Let

V 1 , V2 e B

and suooose that (8.14) holds for some

~ e G3(tV I + (l-t)V2) . Then

Proof

Write

U e G_o(VI)~Go(V2)

t E (O,l) ;

.

V = tV I + (l-t)V 2 ; then by Proposition 8.1 we have

t{e(~,V I) - s(~)} + (l-t){e(p,V2) - s(~)}

=

e(B,V) - s(~)

=

-P(V)

=

-tP(VI) - (I-t)P(V 2) .

thus t{e(~,V I) - s(u) + P(V1)} + (l-t){e(u,V 2) - s(p) + P(V2)}

But for

i = 1,2

we have

e(~,V i) - s(u) + P(V i)

~

=

0 .

0 ; and hence

158

e(~,Vi) - s(u)

=

-P(Vi)

. ~ ~ ( V l ) n C = o ( V 2) 9

Le~a

8.11

Let

(8.15)

~I

if

Proposition

_Go(tV I + (l-t)V2)

be a s~ospace of

V1 , V2 s B

8.5

If

strictly convex on

Proof

is independent of

with

B

V1 # V2

then

is a subspace of

B_I

t s (0,i) .

82

with the property

B

~ ( V I ) ~ A ( V 2) = @ .

satisfying (8.15)

then

is finite and

~

o

gives positive weight to each point of

not hard to see that

Y

8.6

a-field on

Yo ' mld that

Suppose that

G ). Let {r

then it is o

satisfies

Proposition

determined by

is

82

o

empty open set

P

~i "

Immediate from Lemma 8.11.

Y

8.1, we get

and suppose that (8.14) holds for some (and thus all)

This follows easily from Lemma 8.10.

Let

If

agsin using Proposition

82

VI , V2 e ~

t E (0,I) . Then

Proof

. Therefore,

8.22

~o ~l

Yo

is a topological space~ that

=oF

is everywhere dense (i.e. So(G) > 0 be the subspace of

for which

given the product topolo~).

(8.15). More generally we have:

CA : X(A)

TT1en B1

satisfies

B ;R

is the Borel

for every non-

consisting of those potentials is continuous

(8.15).

(where

X(A)

is

159

Proof

Let

V e ~i

easily checked that Wrlting of

~

S

for

and

Zd

: X if

with respect to

X(A) . Now for any

if

G C.X

~

to

A

pA(B)

fA

is the projection of

is everywhere dense, and ~

, thus

p~(p)

x).

is the Radon-Nikodym

P~(H)

p

onto

has a strictly

is everywhere dense. Tmerefore

is non-empty and open then there exists a non-empty open

i = 1,2

fl

by (5.13). It is

is everywhere dense, since from the definition of the product topology

let

~ ~(VI)~(V2) both

A e ~ , then

0~AXPs_A(W) , where

Go a _A 9 Suppose there exist For

and

p e ~(V)

A c ~

V

is continuous (with the product topology on

~R

positive density with respect to mAXPs_A(~)

be given in terms of

{fA)As C

?

8.23

and

(f~)A~

V I ' V2 ~ ~i

A

A ~ f2 A fl

for some

Vi

and

A A fl = f2

x e X : f~(x) @ f~(x) }

with

~ ( V I ) ~ ( V 2) r @ .

by (5.13); let

A e C (since

are versions of the Radon-Nikodym derivative of

~AxPs_A(D) , and so

everywhere dense.

VI @ V2

be defined in terms of

. We must have f2

with

Go C_G

V I @ V 2 ); but p

with respect

~AxPs_A(H)-a.e. This is a contradiction because

is non-empty and open, and we know that

~AXPs_~(p)

82

The strict convexity proofs given here have been adapted from Griffiths and Ruelle (1971) (where the case

Y

finite is considered). O

is

9.1

9. Attractive specifications

If we are given a specification

~

(and let us suppose that

G(V)

is non-empty)

then one of the most important problems to solve is to determine whether or not there iS exactly one Gibbs state. Even in what would appear to be very simple cases this is extremely hard (as the results in Section 5 show). The situation in which there has been most success in solving this problem is with the Ising model of a ferromagnet (as described in Section 5). In this case the existence of certain inequalities

(the FKG, GHS and Griffiths inequalities, see Fortuin,

Kastelyn and Ginibre (1971), Griffiths, Hurst and Sherman (1970), and Ginibre (1970)) help to simplify things. ~hese inequalities come from the fact that there is an "attractive interactmon " " involved with a ferromagnet. In this section we will generalize this by introducing the notion of attractiveness in terms of an order structure on the underlying space

X . We will look at specifications that

have proprrties with respect to the order which can be thought of as corresponding to attractive interactions. Let us first consider conditions for there only to be one Gibbs state in the general set-up of Section 2. Let simplicity we will assume that

~ = {~A}A ~

RA = X

for all

be a specification,

and for

A a ~ (though no problems arise

if this assumption is not made); we are thus really in the situation of Section i. As before, let to

R . For

B(~)

denote the set of bounded F__-measurable functions from

f ~ B(~) , A e ~

recall that we defined

wAf : X

~R

by

(~Af)(x) = If(Y) ~A(X'~) ; it will be con~zenient to sometimes write

WA(x,f~

instead of

(wAf)(x) . Let

X

9.2

161

~A(f) =

sup ~A(x,f) x~X

Lemma 9.1 tion of

Proof

~ (f)

, ~A(f) =

inf ~A(x,f) x~X

is a decreasing function of

A ; ~A(f)

is an increasing func-

A .

Let

A ~ A E ~

with

A ~ A ~ then

x~X

<__ sup xgX

Similarly

Let

-

~(f)

w+(f) =

Lemma 9.2

xaX

I +~A(f;' ~ ( x , d y )

> wA(f)

lim ~ ( f )

If

f e B(~)

.

=

w (f) .

A

82

,

~-(f) =

and

lim hA(f) .

p E ~(V)

then

~-(f) < If d~ <= ~+(f) .

Proof

Hence

If

A c C

then

~ = ~WA ' and thus

Ifd=f(y)IA(x0y)dCxl=IA(x,fl(x) ! wA(f)

<

f dp

<

§ wA(f;

.

82

162

A subset

pl,

W

~2 g P ( F ) a n d

I,e.Tm~a 9.3

Let

of

B(F)

will be called a deter~ining class if

I f d~l

W

9.3

=

i'f dp 2

for all

be a determining class. If

f ~ W

imply

pI

=

P2

"

fsW

w+(f) = ~.-(f) for all

then there is at most one Gibbs state.

Proof

Immediate from Lemma 9.2.

82

Now suppose that we are in the set-up of Section 3 ; so we have a family {-~)SaN

of sub-o-fields of

let

F

B(A)

satisfying (3.1),

(3.2),

be the functions in

(3.3) and ( 3 . 4 ) . Let

B(_F_) that are _~-measurahle

ee__N for some

0 ~: N 9

Lemma 9...4

Suppose

f e B(A) . Then

V

(3.8) and (3.11); and

w-(f) < ~+(f)

IG(_~)I > 1 . (Note that, since we are assuming

just says that for any

B(~)

satisfies

A s C , F e A

~A(',F)

for some

R A = X , (3.8)

is in the uniform closure of

.)

Proof

We can find

x n ~ X , such that

{An}n>=l ~+(f) =

increasing and generating the order on

_C_ , and

lira w A (Xn,f) . By a slight modification of the n-~= n +

proof of Theorem 3.2 there exists

that

p (A) = +

lim wA (Xn.'A) j-~o n. j J

~

c G_(_V) and a subsequence

for all

A ~ =~ . Hence

I

{nj}j> 1

such

h d~ + = lim ~A (Xn.,h) j-x~ n. O J

163

for all

exists

h e B(A) , and in particular

p- e G(~)

Proposition 9.1

with

Suppose

determining class. Then

Proof

z-(f) =

~

9.4

~+(f) = I f d~+ " In the same way there

I f d~- , and thus

satisfies

I~(~)I = i

(3.8) and (3.11), and

if and only if

Combine Lemmas 9.3 and 9.4.

I~(~)I > i .

82

W~B(~)

~+(f) = w-(f)

is a

for all

f s W .

82

To give our definition of an attractive specification we need an order on

X . Let

~

be a partial order on

such that

(9.1)

~

is measura01e,

(9.2)

~

is directed both upwards and downwards,

there exist

Let where

= {WA}As ~

Proposition

{(Xl,X2) ~ XxX : x I ~ x2} s F xF ;

x3 , x4 s X

BI,(Z) (resp. B+(s f : X--~R

i.e.

X

such that

is increasing if

Suppose

(3.11). Then there exists

xI , x 2 ~ X

f(x) ~ f(y)

~

if

whenever

B(~) (resp. B(A__j ),

x ~ y . A specification

WA f c B+(_F_J for all

is an attractive specification

~+ c ~(~)

then

x 3 ~ x I ~ x4 , x 3 ~ x2 ~ x 4 .

) denote the increasing functions in

will be called attractive

9.2

i.e. given

such that

w+(f) = ~ f J

f s B+(F)

, A s ~ 9

satisfying (3.8) and dB +

for all

164

for all

f r B+(~) . Similarly there exists

for aZl

f s B+(A) .

Proof

Let

9.5 ~- a G(V)

~n(F)---~(F)

of

for all

F r A

where

A

F = ~(A=o) . By the proof of ~ e o r e m M(X~F)

w-(f) =

M(X,~_) be the space of finite signed measures on

ized (as in the proof of Theorem 4.3) so that

and

with

. For each

f e B@(~)

we are trying to show that

~n---~B

I

(X,_F) , topolog-

if and only if

is some countable field with 4.3

G(V~

f d~-

A~C

A

is a non-empty compact subset

let

~

U(f)

is non-empty. Each

U(f)

is closed

f E ~+(A) (since

f e B(~) ), so by the compactness of

finite intersection is non-empty. ating the order on

G(V)

we need only show that any

Choose an increasing sequence

C ; then for any

(since the limit is monotone). Let

f r B+(A)

fl ....

we have

fme

{An}n> I

~+(f) = lira w A (f) n -~eo n

B+(A) ; we can find =

n

1 , 1 < j < m ~ such that

for each

n > i

choose

w+( f~ ) =

xn s X

so that

n

done by (9.2)). Since

thus for

V

is attractive we have

i ~ j <= m

~+(f.)~

=

lira ~A (Xn'f~) " n+~

n

< x =

for n

x (~) s X n

lira WAn (xn( j ),f~) n-~ x~'

gener-

for

'

i =< j =< m . Now

1 < j < m (which can be =

=

wA (Xn'fj) >----~A t'x(J),fj)n and n n

9.6

165

Just as in the proof of Lemma 9.4 we can find such that

]J(A) =

lim w A (Xn.,A) j-~= n. O J

for all

e G(V)

~nd a subsequence

{nj}j~ 1

A e A . We thus have for

m

that

~+(fj) =

~ fj dz , and thus

~ e ~

U(fj) .

82

j=l

Theorem 9.1

Suppose

(3.11), and that

~

B+(~)

e G_(_V) such that

~- ~ G_(~) with

is am. attractive specification is a determining

w+(f) =

~-(f) =

f dz +

I f d,-

satisfying (3.8) and

class. Then there exists a unique

for all

for all

f ~ B+(A) , and a unique

f e B+(A)

. Furthermore,

IG(~)I = i

+

if and only if

Proof

~

= ~- .

The existence of

~+

and

~-

is given by Proposition 9.2; their unique-

ness follows from the definition of a determining class, and the last part follows from Proposition

9.1.

82

We will assume from now on that we have an attractive specification

V

B@(~)

is a determining class, and that

satisfying (3.8) and (3.11). ~+ (resp. ~-

will be c~'~lled the high density

(resp. low density) Gibbs state. It is clear from

the uniqueness of

that they are extreme points of

~+

and

B-

_G(V) .

In order to check if there is only one Gibbs state it would appear that we would have to see whether

w+(f) = w-(f)

for all

f c B+(~)

. However,

in most

9- 7

166

cases we only have to check this equality over a much smaller set of functions. This follows from the next lemma, which in the case of the Ising model is due to Lebowitz ~nd Martin-Lbf

Lemma

Proof

9.5

Let

E = { f s B+(A)

We first rescale

> 0 , such that if

~+(h) = w-(h)

du+ :

]

(l-f~176

f

: w+(f) : ~r (f) } . If

and

g : there exist ~

fo = ~f + Y ' go = 6g + 6

is the same as

on the other hand

(f~176

(1972).

l h d~ + = l h dz-

~'og~ : (l-fo)(l-go)

I (f~176

dP+ ~

d~-

(since

I (l-f~176

, and so

fg

W all

Let

is a determining f s W .

W

be a subset of class. Then

then

(for

with

fg ~ E .

~ > 0 ,

0 ~ f o ~ go ~ i . Since

h ~ B+(A) ) it is clear

' go r E ), and also

O

(since

-(l-fo)(l-g o) s ~+(A)

fog ~ a E . Therefore

= (~6)-l(fogo - ~fo - Ygo + Y~) "

•neorem 9.2

6, y, 6 r R

then

+ (fo+go) - i , and

f

dp-

f , g e E

)" Thus

fg e E , since

82

Bt(_A)

IG__(~)I = i

such that the algebra generated by if and only if

~+(f) = ~-(f)

for

167

Proof B

Let

W~

be the algebra generated by

is the vector space spanned by

B§

O

9.8

W ; let

Bo = { f-g : f ' g e B+(A)_ }.

, and it is not h a r d to see that =-

Wo C

Bo . Let

space of

B

Eo = { f E Bo : ~ f

. If

f , g ~ E

O

with

dp-),

so

Eo

is clearly a sub-

f = fl - f2 ' g = gl - g2

and

O

fl' f2 ' gl" g2 e B+(~) Lemma 9-5

d~ + = I f

fg ~ E

, then

. Thus

E

o

and this implies

fg = flg I + f2g 2 - flg 2 - f2g I , and so by is an algebra, hence if

W C_ E

0

~

= ~

then

W

0

C_ E 0

0

82

Before looking at examples of attractive specifications let us consider what happens when the attractive specification is H-invariant, where group of ~ m e a s u r a b l e

bijections from

H-invariant if, given any r

~

~

Le~:a 9.6

and

If

~f+(f) = ~ + ( f ~ )

Proof

Let

(r

~

~ ~ H

= ~A

X and

to

A ~ ~ , ~hen there exists

and

~w~

@ c H , f e B(~)

.

A c ~ . Choose

~nus

But by the definition of

A E ~

"

= ~-(fo~)

r ~ H , f E B(~)

is a

X . As in Section 4 we will call

is H-invariant, then for any

, ~-(f)

H

we h a v e

A e ~

so t h a t

we have

with

9.9

168

+" ~A(i) ~ w (fo~) ~ ~+(fo~)

and therefore have

w+(f) ~ ~+(for

w+(f) ~ ~+(fo@) same way.

. Replacing

, and hence

f

by

for all

serves the order on

all

and

r

by

r

A ~

we

gives us

. The other equality is got the

82

fo~ s B(~)

for all

fo~

~+(f) = w+(fo~)

We now add the assumption that that

. Since this holds for all

f s B(~)

X , in that if

@(A) C_

~

for all

, @ s H . We will also assume that x , y s X

with

@ c H . From this assumption we immediately

f s B@(~)

@ e H . This implies

x ~ y get that

then

r

H

pre-

~ r

fog s B@(~)

for

, @ E H .

+

Theorem 9.3

If

~

is H-invariant

I~(~)I = 1 if and only if

then

l~(~)I = 1

are H-invariant.

and (where

Also

C-o(=V) is the set of H-invariant

Gibbs states).

The second part follows immediately

Proof For s_ny

f ~ B+(A)_ , @ ~ H

f d~ I'+

Since

B%(A)

=

~+(f)

=

from Theorem 9.1 and the first part.

we have

fo~J ~ B+(A)

~+(fo~)

=

is a determining

Ifo~

, and thus by Lermna 9.6

d~ +

class, this implies

=

f d(r +)

+

=

@~+

.

; and hence

~

+

is

H-invariant.

Finally,

note that if

V

is H-invariant,

Theorem 9.2 in that the hypotheses H-invariant

algebra generated by

about W

W

then we can improve on

can be reduced to having the

a determining

class.

(In the most important

169

9.10

examples we will look at this will mean that

W

can consist of a single f ~ e -

tion. )

We will now look at attractive specifications in the setting of Section 5. Let

Y

be a Polish space (i.e.

o

metric space), let order on

Y

(9.3)

be the Borel ~-field on

is closed, i.e.

<= thus satisfies (9.1), and each

t e S (where

be a copy of A C

S

S

X(A)

o

, and let

<

be a partial

{ (Yi'Y2) ~ Yo•

(Yo,%)

: Yl =< Y2 }

is closed in

Yo • o

is clearly a standard Borel space. For

is any countable set, not necessarily

Z d ) let

(Yt,F__t)

the product order, i.e.

{xt}te A <= {Yt}teA

if

xt < Yt

t e A . This order again satisfies (9.1), (9.2) and (9.3) (with the

product topology on

A =

Y

(Yo,F=o) . We again use the notation introduced in Section 5. For

we give

for all

can be topologized as a complete, separable

o

satisfying (9.2) and

o

~

F

Y

U

F(A)

, and

X(A) ). We take

B(A)

N = __C and let

B A = F(A)

; thus

consists of the bounded functions that are

AEC

~(A)-measurable for some

A s _C .

One more assumption about

(9.4)

given f

YI' Y2 e Yo

with

Y

o

and the order

Yl @ Y2

<

will be made; this is

then there exists

bounded, continuous, increasing and with

f : Yo

)R

with

f(yl ) @ f(y2 ) .

The most important cases we will be concerned with are when

Y

o

is either

R

or

170

some closed subset of

R (with the usual ordering);

all the above conditions.

and 48.) Also note that if (9.2),

Lepta 9.7 functions

generated by

Proof

holds.

{Y } n

(This follows

B§

class.

that are ~({t})--measurable is a determining

W

and let

C

o

X

then by the Stone-Weierstrass are uniformly

f e W F

o

of

. Take arly X

with

F

Let

~

dense in

by

Bf(A) :-

denote the continuous

algebra containing the constants, is a compact subset of

C(F)

(1965)~ pages

30

these conditions.

W

consists

of those

t ~ S ,-then the algebra

the algebra generated by o Let

W

functions

and

C(F)

in

W

denotes the continuous

theorem the restrictions . Let

B , ~ c P(~)

X

with

o

. C

o

is an

functions

of the functions ?f

d~ =

~f

in d9

X . If on

F , C

o

for all

is Polish there exists a compact subset

~(F) > i - ~ , v(F) > i - ~ 9 For any

X-F

(resp. ~ ) denote the restriction

W

be the algebra gen-

o

and (by (9.4)) separating the points of

8 > 0 , then since

F

is com-

o

sequence of spaces

if

for some

Y

class.

is contained in the vector space s p u m e d

F

Furthez~ore,

The second part implies the first because

erated by

clearly satisfy

o

from Nachbin

is a finite or countable

is a determining

W

Y

(9.3) 8nd (9.4) then their product satisfies

B§ in

these

It is worth noting that if (9.3) holds and

pact then (9.4) automatically

satisfying

9.11

f E W

o

we have

X-F

of

p

(resp. v ) to

F

(as a meastu-e

9,12

171

on

F ); then the above application of the Stone--Weierstrass theorem gives us

t ]~ f dp -

If

IIp-~li

~ 26

and thus

II~-~ll

d~ I

~ 48 , and hence

Let

(where

281]f11~

[Ik[]

> = ~ .

for all

f e C(F) ,

i s the total variation o f

X ). Therefore

82

WI be any subset of

separate the points of ated by

<

W

such that the continuous functions in

Wt

X ; the proof of Lemma 9.7 sho~rs that the algebra gener-

W l is a determining class.

In order to give examples of attractive specifications in this setting it will be convenient to start with a special case and assume (until further notice) that :

(9.5)

Yo

is totally ordered, i.e. if

YI' Y2 E Yo

then either

Yl =< Y2

or

Y2 ~ Yl ;

(which really amounts to assuming that x , y r X(A)

(xvY)t

Thus

with

=

Y

o

is a subset of

x = {xt)ta A , y = {Yt}tcA , then define

max{xt~Yt} ,

(x~Y) t

=

R ). If

A ~

S

and

xVy , x^y r X(A)

by

min{xt,Yt} 9

X(A) , with the product order, is a lattice. As usual~ we will only look at specifications given in terms of interaction

potentials { @ A } A e ~

such that

~

IleAl I

is finite for all

t e S (and thus

taAa~ (3.8) and (3.11) hold). The inequality which gives us examples of attractive

172

9.13

specifications is the following:

Theorem 9.4

~ f l d~A

Let

=

A a C

I f2 d~A

and

=

fl' f2 : X(A)

for all

fl.>_ 0 , f2 >= 0 , and

x , y ~ X(A) .

h : X(A)---'~R

Then for any bounded, measurable, increasing

Proof

with

i . Suppose that

f:(xvY)f2(xAy) h fl(x)f2(Y)

(9.6)

~R

we have

We defer this to the end of the section. The proof we will give is taken

from Preston (197hc). The inequality is a generalization of a result of Holley (1974), which in turn was used to give another proof of the FKG inequalities (Fortuin, Kaste3@n and Ginibre (1971)).

Let

V(x)

=

{r

~

82

V : X

be as above and let

)R

be given, as always, by

0

CA(XA ) "

Ac~

Theorem 9.5

(9.7)

Suppose that

V(xVy) + V(xAy) ~ V(x) + V(y)

Then the specification corresponding to

Proof

Let

{~}Ae~

for all

V

x , y e X

O

is attractive.

be defined by (5.13). If

h

B§

ana

AE~

then

9.14

173

~rA(X,h)

Let

=

x , y a X

J h(z) WA(X,dz)

with

f~(w) = fA(xs_~• Then

g

y ~ x

=

h(xS_A•

and define

~ f2(w) = fA(ys_A•

d~A(W) .

fl' f2' g : X(A)

~R

by

, g(w) = h(Xs_AXw) .

is bounded and increasing~ and

~A(x,h)

= ~ g(W)fl(W) ~^(w) ,

~A(y,h) : ~hCYs_A•

(w) d~A(w) ~ ~g(w)f2(w)~A(w)

Thus the proof would be complete if we could show that

fl

and

f2

satisfy (9.6).

But this amounts to showing that

(9.8)

where

exp gl(UVv) exp g2(u^v)

~

exp gl(u) exp g2(v)

for all

gl(u) = gA(Xs_A• ) , g2(u) = gA(ys_Axu ) , and where

gA

u , v e X(A) ,

is defined by

(5.12). (This is because the terms which occur from the denominator in (5.13) c~ncel out.) (9.8) is of course equivalent to

(9.9)

and ~f

gl(uVv) - g](u)

x , y e X

o

~

g2(v) - g2(u^v)

for all

u ~ v e x(A) ;

then

g~(uvv) - gl(u)

= V(Xs_A•

g2(v) - g2(u^v)

= V(Ys_a•

- V(xs_A• - V(Ys_a•

Thus from (9.7) we have (9-9) holds provided

x ~ y ~ X

, 9 9 It therefore holds in o

174

general~ since

gA(x~Xhs_~)~

~ gA(x)

9.15

as

A--~S

.

In practice we would like conditions on the V

satisfies (9.7). But note that

the sense that for each

(9.lO) If

~a(uv~)

IAI

= 1

+

V

CA

which would ensure that

will satisDy (9.7) if each

CA

does, in

A s

~a(u^~)

CA(U) + ~A(v)

then (9.10) always holds; for

for all

[A I > i

u , v ~ X(A) .

a useful sufficient condition

is given by

Proposition 9.3.

Let

A s

s~%8 for esch

negative ~Id increasing; let

~A(y)

then

CA

Proof

let

=

c §

~ tea

let

CA : X(A)----+R

:

Yt-----~R

be non-

is defined by

ht(Yt) ,

c = 0 , and will proceed by induction on

then there is nothing to prove~ so assume

B=A-

ht

satisfies (9.10).

We can assume that

[A I = i

e s R . If

t e A

{t) . Define

~B : X ( B ) - - ~

by

IA[ > i , take

~B(Z) =

]~

[A[ . If t e A

and

hs(z s) , so by

sgB the induction hypothesis

~B

satisfies (9.10). Let

loss of generality we can assume that

~A(xvz) + ~A(xAy) =

x , y E X(A) , and without

Yt ~ xt " Then

~A(x) - ~A(y)

ht(xt){ C B ( ~ V Y B ) - CB(:~B)} - ht(Yt){$B(YB) - CB(XBAYB)}

9.16

175

Now as

CB CB

is increasing, thus

r

satisfies (9.10) we have

Therefore since we also have

) ~ CB(XB)

and

CB(YB ) ~ CB(XBAYB ) ; and

CB(XBVYB) - CB(XB) ~ CB(YB ) - CB(r~AYB)

0 ~ ht(Y t) ~ ht(xt)

CA(xvy) + CA(xay) - CA(X) - CA(y)

.

we get

~

0 .

82

Oi" course, the hypotheses of Proposition 9.3 are by no me,is necessary for

CA

is when

to satisfy (9.]0). One case that is worth looking at a bit more closely IAI = 2 . For this it is easily seen that the condition

removed from the hypotheses of Proposition 9.3. Further, if an open inter~al, and

CA : Yo • o

)R

ht ~ 0

IAI = 2

and

can be Yo

is

is differentiable, then it is not diff-

icult to check that (9.10) is equivalent to having

DI2@ A ~ 0

(where

DI2

denotes

the mixed partial derivative).

Let

p : Y

o

~R

be bounded, continuous and strictly increasing (i.e.

~(yl ) < p(y2) if Yl < Y2 )" and for t e S

let

Pt : X

)R

be given by

Pt(x) = P(Xt) . The proof of Lemma 9.7 shows that the algebra generated by is a determining class, and thus, by Theorem 9.2, w+(pt ) = w-(p t)

for all

IG(V__)I = i

t E S (or, equivalently, for all

if and only if

t c S

Pt d~+ = ~ Pt d~'- ). An important way of looking at this is as follows:

If

~. ~ P(F)

then we can regard

Yo-valued stochastic process prob.{Xtl ~ F 1 ,..., X t

B

{Xt}te S

as defining the joint distribution of a indexed by

~ F n} = ~({x e X : x n

S , (with

e F I ,..., x t tl

{Pt}teS

e Fn}) ). n

176 Let

{Xt}tc + S (resp. {Xt}tE S-

9.17

) be the stochastic process given by

~+

(resp. U- );

then we can tell whether these processes are the same just by comparing the distrX"+ t

ibutions of the individual random variables

and

X

~

for each

t ~ S

(some-

thing which is far from true for two arbitrary processes).

Now suppose that

S = Z d , and that

{r

in that they satisfy (5.17) 9 By ~ e o r e m 9.3

D

are translation invariant, +

end

~-

are translation invar-

iant; further, since the smallest zd-invariant algebra containing as the algebra generated by if

{Pt}tezd , it follows that

P0

I~(~)l = i

is the same if and only

w+(p 0) = w-(p 0) . A consequence of this is that the variational principle

considered in ~heorem 8.3 is much simpler in the present sit,~ation. We will now look at this. We are assuming that instead of

S = Z d (thongh for convenience we will write

Z d ) and that we have a specification

satisfying (5.17), with

~

IleAl I

~

defined in terms of

finite, ~nd with

V

S {r

satisfying (9.7).

OEAE~ Let

p : Yo

(where For

b

~ a R

>R

be bo~]ded, continuous, strictly increasing, and with

is the base point); for define a potential

v~(x)

t E S

V~ : X ~

= V(x) + ~ ~ _

define ~R

Pt : X

"~R

by

by

~t(x) ;

tgS

thus

VX

corresponds to the interaction potentials

{r

, where

p(b) = 0

Pt(X) = P(Xt)-

177 X CA

r

It is clear that

VX

=

CA

=

r

if

9.18

IAI ~ 2 ,

+ Xp(E)

.

satisfies all the conditions we have assumed for

can be thought of as adding en external field to field proportional to the pressure,

V

(with the strength of the

k ). We are going to prove that

as a function of

X , is differentigole

V ; VX

i~(~)i = i at

if and only if

X = 0 . Thus unlike the

general case considered in Section 8, where we had to look at all the tangent functionals to the pressure in the Banaeh space the one-dimensional

subspace

{ VX : X e R }

A s C , x E X , X s R

For

=

exp gXtxs_Axw)

^

(where

g~

is defined in terms of

PA'x(V~ )

(thus

PA(~)

Lemma 9.8

_l_

IAI

log

~_ PA,x(V~)

Calculus.

82

=

V X ).

I

B .

{r

dLOA(W) ,

X

by (5.12)); and let

ZA'x(V~) >

defined in (8.8) is the same as

kernels corresponding to

Proof

=

of

let

A,

I ZA,x(V X)

~ , here we only have to look at

~_~

PA,b(V)).

WA,x(x,Pt)

(where

OrA,X }AE___

are the

9.19

178

Lcm~a 9.9

For each

A ~ ~

choose

x(A) s X . Then

P(Vk) =

lim PA~x(A)(V)) A

,

provided the limit is taken over cubes.

Proof

From the definition of

only show that

P

we have

p

+

, p-

~+(~)

where

p~

; so we need

: R ~-~R

=

easily enough

II

by

Po d.~ ,

p-(~)

=

PO d~X

;

(resp. ~X ) is the high density (resp. low density) Gibbs state corres-

ponding to and o n l y

lim PA,b(Vx) A

lim {PA,xtA~(Vx) ~ i - PA, b(Vk)} = 0 , and this fo!l~r A

from the estimates used in Section 8.

Define

P(V k) =

V x . Then

if

p-(X)

Proposition 9.4

p- < p+

= p+(k)

If

and, as we have already noted,

IG=(Vx)I = i

if

.

P(V~)

is differentiable, as a function of

X , at

X = ko

I ~ ( V x 11 = 1 .

then

0

Proof

Without loss of generality we can assume that

~

O

= 0 . For each

+

c~n choose

x(A) s X

such that

WA(X(A),Pt ) ~ WA(Pt) - I/IA I

Then by Lemma 9.8 we have

amaXPA'x(*;(vx) Ix=o >_ IAI

) - z/IAI}

for all

Aes teA.

we

179

= IAI

l/l~l}

{'~ (%1

Choose a sequence of cubes

and 9.8

f

~,+(%1

Am + S

and let

9.20

I/IAI

o+(o1

fro(X) = P A m , x(Am)(Vx)

is convex and differentiable; by Lemma 9.9

m

a -- P(Vx) aX

.

(If

fm(y)~--)f(y) f

p

P(Vx)

I~

as

R

is

m---) ~

is differentiable at

- 1/IAml

< p-(O) , and since

an

open interval and

for all

are differentiable at

fA(O) ~ p + ( O )

. By Len~mas 8.7

~P(Vx)

as

X = 0 . Let

IX=O ; then by a standard fact about convex functions we have

p : lim f~(0)

and

f (l)

9

m

m---~ ~ , and by hypcthesis

P =

I/IAI

: I ---OR

y r I ; and if for some

Yo ' then f'(yo ) = lim f'(yo ) m

, and hence p

f

< p+

p ~p+(0)

we must have

are convex, with

Yo r I

all the

f

m

") But

. The same proof also gives us p (0) = p+(0) . Therefore

l~(v) l = 1 .

An immediate consequence o f Proposition 9.4 is that { is at most countable; since

P(Vx)

, being a convex function of

to have a derivative at a countable number of points. of Propostion 9.4, and show that if

X ~ R : I~(Vx)l > 1 }

I~(V X )I > 1 o

We now look at the converse

then

P(V l)

is not different-

-

iable at

X = I

. This will be done by identifying

i , can only fail

p

+

and

p

as respectively

O

the left- and right-hand derivatives of

-

Lemma 9. I0

p

P(V X) .

+

, p

-

are increasing functions, p

is upper semi-continuous.

+

is lower semi-continuous,

p

180

Proof

Since

PA,x(Vx)

9.21

is convex, Lenlma 9.8 shows that

9

p

+

WA,k(x,Pt)

IAI

is increasing (as a function of

and

--

k ), and from this it easily follows that

p

are increasing. Now

p+(1)

I

=

+ P0 d~x

Therefore to show that for any

A e C

so we have

p

=

+

+

f (~) x

=

AEC

it is sufficient to show that

is a continuous function of

_

=

sup xEX fx(X)

=

the increasing directed set defined by

~X

+ inf ~A,x(p0) .

=

is upper semi-continuous

WA,x(po )

+ k(p O) WA,

+ wx(P0)

~

lira x fx(X)

k . Let

fx(k) = ~A,l(X,Po);

(where the limit is taken over

<= ). Some elementary calculus shows

{~A,x(X'PoPt) - ~A,~(X,Po)~A,~(x,Pt)}

tEA

and thus

I~-- fx(X)I ~ 21A i Ilpil~ . Therefore {fx}xEX 8l

family, and thus

+

WA,x(p 0)

lower semi-continuous.

is an equicontinuous

is continuous. The same method gives us that

p-

is

~I

P(Vk) , being convex, has a left-hand and a right-hand derivative at each point. Let us denote them by

Pr_r~o~osition 9.5

Proof'

P

=

p(1)

It is easi]~u seen that

p(1)(X)

,

P+

p(1)

and

=

p(r)(x)

respectively.

P(r)

and

p(r)

are increasing, and

p(l)

is

181

lower semi-continuous,

p(r)

exists a countable set

E ~

9.22

is upper semi-continuous. R

such that if

I @ E

Theorem 9 . 6

]~(V% )I : 1

if and only if

9.h there

then

p-(~.) = p(1)(1) = P (r)(1) = +(~)

From this ~nd Lemma 9.10 the result follows.

By Proposition

.

82

P(VI)

is differentiable

at

I = 1~

0

Proof

This is just Propositions

I f we consider

p+

and

9.4 and 9.5.

p

82

as derivatives of the pressure with respect

to an external field, then we can interpret them as "densities", says that

I~(Vl) I > i

discontinuity

occurs at exactly those points 9 I

in the density.

and Theorem 9.6

where there is a jump

Thus, in physical language, we have more than one

Gibbs state at exactly those places where there is a first order phase transition.

For

and

A e ~

let

+ p (I) = WA,l(p0)

,

pA(1) = WA,l(p0) -

; thus

p~ + p+

pA + p-

Theorem 9.7

12(Vl )I = i

if and only if the functions { P ~ } A e ~

are equicont-

O

inuous at

1 0

continuous at given any all

G

; or, equivalently,

1~

~ > 0

whenever

(Functions

if and only if the functions

gs

then there exists Ix - Yl < Y -)

are called equicontinuous y > 0

such that

{PA}AeC at

x e R

are equiif,

Iga(x) - g~(Y)l < ~

for

182

Proof

+

Since

PA

p

+

is continuous at

~

o

+ p_ A

~

R

the fl~ctions

PA

X

at

PA +

p+

, it is easy to

is the exact condition to ensure

o

. The result thus follows from Theorem 9.6.

A particular case of Theorem 9.7 is: I

+

is continuous and increasing: and

check that the equicontinuity of the that

9.23

~i

Suppose that on some interval

are all either convex or concave (or that the

PA

are

all either convex or concave). Then the hypotheses of Theorem 9.7 hold for any ~o

I , so we must have

, -.I~(V~)I = i

for all

~ c I . We will now look at an

important example of this. (This example will include the Ising model for a ferromag~et.) Let and let

tO o

Y

o

= R

with the usual order, let

be an even probability measure on

under the reflection {r

~'-~-~

). Take

0

F =o

be the Borel q-field on

(Yo,~) -

(i.e.

tO o

R ,

is invariant

as the base point. Consider a potential

given by

r

= 0

if

IAI @ 2 ,

Js-t = Jt-s ' and

~

r

(~'q) = Js-t ~q , where

IJtl

Js-t e R

finite. Then, although we do not have

t @ o

(-N,N)

for some

~_

IIr

0~A~

finite, it is quite easy to check that if and (3.1l) hold. (If in fact

with

~

o

tOo

dies off quickly enough then (3.8)

has compact support, then we can replace

N > 0 , and in this case we would have

~

IIr

I

R

by

finite.)

0~AE~ We will assume that p : Y

o

----~R

by

Jt ~ 0 , because this implies that (9.7) holds. Define

p(~) = ~

( p

is not bounded, but ot~ proofs still work if

tO

O

183

dies off quickly enough.)

9.24

It now follows from a result of Ellis

(1975) that there +

exists a class of even probability measures

Z , such that if

is concave on

me s Z

{ ~ : k > 0 } . Therefore if

then

~o e Z

then

l~(Vk)l = 1

for all

k > 0 . But we leave the reader to check that performing the reflection at each site amounts to changing IG(Vk) I =

= i

for a].l

IG(v~)i= > l

is if

,

Z mass at

Vl

to

I < 0 . Thus if

pA

6~-~

V_k , and thus we would also have

~o s Z

then the only possibility for

~=0

includes the case when

~o = ~2{ 6 §

~ . In this case we can take

Y

+ ~ - 1 } ' where

= {-i,+i}

8~

is the point

, and this gives us the Ising

O

model. The above result is then the one described in (5.24), and in this situation Ellis' result becomes the GHS inequalities Other measures in

Z

(1970)).

include measures which have densities with respect to

Lebesgue measure of the form zing constant, ~id

(Griffiths, Hurst and Sherman

U(z)

U(z) =

ek z

e exp(-U)

, where

c

is the appropriate normali-

is an entire f~iction with expansion

2k

, with

ak ~ 0

for

k ~ 2 , and

aI

real .

k > i

We have assumed that

Y

is totally ordered, but it should be clear that O

everything can be adapted to the case when

Yo

is a finite product of totally

ordered spaces. The condition

(9.7) on a potential

V

ification attractive.

is the product of

factors then Theorem 9.6 has

If

Y

n

will still make the spec-

O

to be changed to looking at the derivative of the pressure in an appropriate n-dimensional

space. We leave the reader to sort out the details.

184

9.25

Let us return for a moment to the general situation considered at the beginning of the sectibn. Suppose that (9.1) and (9.2), and

~

+l

has a partial order

~

satisfying

is an attractive specification satis~'ing (3.8) and

(3.11). Suppose also that we will denote by

X

and

X actually has a maximum and a minimmn element, which -1

respectively.

what we have just considered whenever

Y

(For example, this will happen in

is compact; in particular with the O

Ising model.) Then for

~(f)

f E B@(~)

we have

: ~A(+l,f) , ~(f)

: ~A(-1,f).

In most reasonable cases this will imply that for all

~+(A)

=

lim WA(4.I,A) ,

Ac~

B-(A)

=

A e

lim ~A(-I,A)

-

Aes

(In order to prove this we would need some density property of

B (A) O

where

Bo(~)

is the space spanned by

Bo(~) = B(~) .) In the above situation we thus have

limits over all of

~

B(~)

~

--

and

~

as actual

(rather thsr~ just over subsequences) of measures of the

form ~A(x,.)

We will end the section by giving a proof of Theorem 9.4. Instead of proving this result, we will in fact prove the following: assigning

Y

(recall that we are

is totally ordered) O

Theorem 9.8

Let

Theorem 9.4; for

A s C i = i, 2

and let

fl" f2 : X(A) -----+R ~i

,

Bi(A) . For the Ising model we in fact +

have

in

"-

satisfy the hypotheses of

be the probability measure that has density

185

fi

with respect to

9.26

~A " Then there exists a probability measure

(X(A)•215

on

such that

(9.~)

v(F•

= ~l(F)

for all

F ~ F (A) ;

(9.12)

~(X(A)•

= ~2(F)

for ~ l

F s ~(A)

(9.13)

~(~ (x,y)

: y i x )) = 1 .

(9.11) and (9.12) say that the projections of factors are because if

~i

and

~2

h : x(A)

v

;

onto the first and second

respectively. Theorem 9.8 easily implies Theorem 9.h )R

is bounded and increasing, then putting

E = { (x~y) : y ~ x ) we have

I hlxl

hIyl d I ,yl = I m(xl -

d Ix,yl

0

E

(since

h(x) - h(y) ~ 0

if

(x,y) ~ E ). In fact the existence of

v

satis-

fying (9.11), (9.12) and (9.13) is equivalent to having

l h dB I

~

I h d~ 2

for all increasing bounded

h .

This follows from Strassen (1965) , Theorem ii.

We prove Theorem 9.8 by induction on IAI = i . Let

gl' g2 : Yo - 9

R

IAI . First consider the case when

be non-negative, measurable, and with

186

I

gl da~o

=

by projecting

I ~

g2 d~o

=

i . Let

o(B)

=

y xy , thus for O 0

~o ( { Y ~ Yo : ( Y ' Y )

Define a probability measure

6

on

(y xy ~F xF ) o o~o~o

be the measure on

o

onto the diagonal of

O

9.27

B E F xF

=0 ~0

.

E B })

(YoXYo,FoX~o)

got

by

-1 =

,,,in{glI./.g

{yl %

+

,

+

where

gl(x)

=. {gl(x)

(Note that since

- g2(x)}

+

,

g2(y ) = {g2(y ) - gl(y)}

gl + g2 = g2 + gl

I ~2(z) d~ o (z)

and thus the definition of

~

=

we have

I-

d~~

gl (z)

is symmetric in

gl

'

and

g2 " If the integral is

zero then we leave out the second term in the definition of

Lem~a 9.11

Let

g

be as above, and for

measure having density

gi

with respect to

onto the first and second factors are

Proof

Y1

This is a simple computation.

Lemma 9.12

Suppose for all

i = 1, 2

x , y e Yo

let

6 .)

Yi

be the probability

o~~ . Then the projections of and

Y2

respectively.

82

with

gl(x)g2(y) ~ gz(y)g2(x) .

y ~ x

we have

187

Ther~

~({ (x,~) : . ~ , s

Proof

It

is

sufficient

suppose there exist gl(y) > g2(y)

})

=

to

show t h a t

x , y

.

with

, g2(x) > gl(x)

gl(x)g2(y)

which contradicts

1

9.28

gl(x)g2(y)

y < x

a~d

= 0

unless

gl(y)g2(x)

y < x . Thus

> 0 9 Then we have

, and hence

< gl(y)g2(x)

the hypotheses

,

of the lemma.

82

Together Le~nas 9.11 and 9.12 give a proof of Theorem 9.8 lot the case IAI ~ i ; the construction Suppose now that i = i, 2

let

has density

p(~i ) gi

of

~

will also be useful for the general case.

IAI > 2 , let

t E A

denote the projection

with respect to

gi (x)

=

I

and put

of

~A ' where

fi (xx~) d~t(()

Di

onto

A = A - {t} . For X(A)

gi : X(A)----)R

. Then

p(~i)

is given by

"

Yt

~e

proof of &~eorem 9.8 can proceed by induction because of the next lemma.

Len~na 9.13

If

g](uVv)g2(uAv)

Proof

Let

fl(xVy)f2(xAy)

> gl(u)g2(v)

t'or all

G = { (~,n) e YtxYt

L = { ((,q) ~ YtxYt

> fl(x)f2(y)

u , v e X(A)

: q < ~ } ,

: ~ < q } , Then

for all

x , y e X(A)

then

.

E = { (~,q) e YtxYt

: $ = q } ,

188

gl(Uvv)g2 (uAv)

=

I I

9.29

fl(UVv'~)f2 (uAv''l) d~t(~)dmt(q)

G~E~L

=

I I fl(UVv'~)i'2 (uAv'~) d~t(~) dmt(n) E + I I {fi(~vv'~)f2 (u^v'~) + fl(uvv"~)f2 (u^v'~)} ~ t (~) ~ t (~) "

G Similarly

gl(u) g2 (v) = ~I fl(u'~)f2(v'q) E +

{fl(u~)f2(v,q) li G

d~t(~) @~'t(~)

* fl(u,n)f2(v,~)}

d~t(~) dmt(q) 9

But by hypothesis ~e have

f1(uVv,~)f2(uav,~) ~ fl(u,~)f2(v,~)

,

and thus we can ignore the terms involving integrations over

E . The proof of

the lemma would therefore be complete if we could show that

fl(uvv,~)f2(u~v,n)

+ fl(uvv,n)%(~v,~)

fz(u,~)f2(v,n)

whenever

+ fl(U,n)f2(v,~)

q < ~ . Let us write

a

=

f l C u V v , ~ ) f-2 C 1 .~ v , q l . . ,

e

=

f1(u,~)f2(v,n)

,

d

b

=

=

f z C u v v , q ) f_2 ( ~ _v , ~ ) _

fl(u,n)f2(v,~)

.

189

It is easily c h e c k e d that if 8nd

ab ~ cd . W e want,

f r o m L e m m a 9.1k.

L e m m a 9.14 and

ab ~

Proof

w e get

of course, to show that

a, b, c, d

cd . Then

a > 0 . Now

n < ~ then by hypothesis we have

a = 0

b e n o n - n e g a t i v e real numbers w i t h

then

c = d = 0

w h i c h gives

aa + ab ~ ac + a d . D i v i d i n g by

what m o d i f i c a t i o n s

aa + cd ~ ac + ad , and since a

82

i = i, 2

define

by

Let

=

fi(u,~)/gi(u)

u , v a X(A)

Fl(U,~Vr,)F2(v,~^n)

Clear.

cd ~ ab

gi > 0 . (We leave the reader to see

9

is thu~s the R a d o n - N i k o d y m d e r i v a t i v e of

Proof

gives the result.

are n e e d e d if this is not the case.) For

Fi(u,~ )

L e ~ n a 9.15

a >__ c , a ,> =d

and the result is true; w e can thus assume

For s i m p l i c i t y let us assume that

F.I

a + b ~ c + d , a n d this follows

a + b >_ c + d .

(a-c)(a-d) ~ 0

F.i : X(A)---~ R

a ~ c , a ~ d ,

82

Let

If

9.3o

with

~i

w i t h r e s p e c t to

v ~ u . Then for all

>__ Fm(u,~)F2(v,n)

p(~i)•

.

~ , ~ e Yt

9

82

N o w suppose that T h e o r e m 9.8 is true for all sets w i t h c a r d i n a l i t y less than

]A I . Then b y L e p t a 9 . 1 3 there exists a p r o b a b i l i t y m e a s u r e

~

on

190

(X(A)•

such that the projections of

second factors are

Define

9.31

p(~l )

~nd

p(p2 )

Q~ R : X(A)xX(A)xYtxY ~

Q(u,v,~,n)

=

'~R

onto the first ~ d

respectively, and

~({ (u,v)

: v ~

}) = :

by

min{Fl(U,$),F2(v,q )}

;

+

R(u,v,$,n)

whcre

=

S(U,V)

S(u,v)-l{Fl(~,~)

= I{F2(v'q)

- F2(~,n )} {F2(v,n)

FI(U'~))+ dwt(q) " Let

-

(YtxYt,~,xF__--t) got by projecting probability measure

=

Lemma 9.!6

Proof

v

v

on

Q ~x~ t

- Fl(U0~)} + ,

~t

onto the diagonal of

(X(A)•

+

~t

(A)xK(A))

be the measure on

YtxYt . Define a

by

R 0•

satisfies (9.11) and (9.12).

This is a straightforward calculation.

Finally, the proof of Theorem 9.8 is completed with:

Lemma 9.17

Proof

~

satisfies (9.13).

We need only show that

follows from Lemmas 9.12 ~ d

R(u,v,~,q) = 0

9.15.

82

if

v < u

and

~ < q , and this

191

9.32

It is worth noting that Theorem 9.8 gives the following generalization of the FKG inequalities:

Theorem 9.9

that

Let

f : X(A)---@E

f(xVy)f(xay) ~ f(x)f(y)

measure that has density

be non-negative, with

for all

J f d~A = i , and s u c h

x , y ~ X(A) ; let

f with respect to

~A " If

~

be the probability

g , h : X(A) ----~R are

bounded, _Fo(A)-measurable and increasing, then

~ g h dP >_~ ~ g d~ l h d~

Proof

If

c ~ R

then ~ g h a11 -

is unchanged if f2 = f ' and

g

I g dlJ l h ~

is replaced by

g + c ; we can thus assume that

g > 0 . Let

fl = gf{ ~gf d~A}-I ; then

fl(xvY)f2(x^y) = { ~ gf do~n}-ig(xvy)f(xvy)f(x^y)

T h ~ by Theorem 9.8 we bare

Igh d~

For

Y

=

I g d~ Ibfl d~A

~

I g d~lhf2 d~A

=

~ g d ~ l h d~ 9

82

finite, Theorem 9-9 was first proved by Fortuin, Kastelyn e~ud Ginibre 0

(197].). In the generality given here it can be found in Cartier (1972).

R.I

Refere~ees

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to appear.

Dunford, N. and Schwartz~ J.T. (1958). Linear Operators I. New York: Interseience. Dynkin, E.B. (1971a). Entrance and exit spaces for a Markov process. Actes Congres Intern. Math. 1970, Tome 2, 507-512.

198

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Dynkin, E~B. (1971b). The initial and final behavio~r of trajectories of a Markov process. Russiarl Math. Surveys, 26, 4, 165-185. Dyson~ F.J. (1969). Existence of a phase transition in a one-dimensional ising ferromagnet. Comm. Math. P.hys., 12, 91-107, 212-215. Ellis~ R.S. (1975). Concavity of magnetization for a class of even ferromagnets. Bull. American Math. Soc., 81, 925-929. F~llmer, H. (1973). On entropy and information gain in random fields. ,Z. Wahrs. verw. Geb., 26, 207-217. FBllmer, H. (1975a). Phase transition and Martin boundary. Sem. Probabilites Strasbourg, IX,

to appear.

FDllmer, H. (1975-b). On the potential theory of stochastic fields. Invited address: 40th. Session, ISI, Warsaw, Poland. Fortuin, C.M., Kasteleyn, P.W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys., 22, 89-]03. Gallavotti, G. (1972). Instabilities and phase transitions in the Ising model. A review. Rivista del Nuovo Cimento, 2. Gallavotti, G. and Miracle-Sole, S. (1967). Statistical mechanics of lattice systems. Comm. Math. Phys., 5, 317-323. Georgii, H.-O. (1972). Phasen~bergang i. Art bei Gittergasmodellen. Springer Lecture Notes in Physics, Vol. 16. Georgii, H.-O. (1973). Two remarks on extremal equilibrium states. Comm. Math. Phys., 32, 107-118. Georgii, H.-O. (1975). Stochastiche Felder t~id ihre Andwendung auf Inter~_ktions systeme,

to appear.

194

R. 3

Ginibre, J. (1970). General formUlation of Griffi~hs inequalities. Comm. Math. Phys., 16, 310-328. Griffitbs, R.B. (1964). Peierls' proof of spontaneous magnetization in twodimensional Ising ferromagnets. Phys. Rev. A136, 437-438. Griffiths, R.B., Hurst, C.A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys., ii, 790-'95. Griffiths, R.B. and Ruelle, D. (1971). Strict convexity ("continuity") of the pressure in lattice systems. Coram. Math. Phys., 23, 169-175. Grim~nett, G.R. (1973). A theorem about random fields. BUll. Lond. Math. Soc., 5, 81-84. Hammersley, J.M. and Clifford, P. (1971). Markov fields on finite graphs and lattices,

in,published.

Holley, R. (1971). Free energy in a Markovian model of a lattice spin system. Comm. Math. Phys., 23, 87-99. Holley, R. (1974). Remarks on the FKG inequalities. Corm. Math. Phys., 36, 227-31. Kesten, H. (1975). Existence and uniqueness of col~mtable one-dimensional Markov random fields,

to appear.

Lanford, O.E. (1973). Entropy and equilibrium states in classical statistical mechanics,

in Statistical Mechanics sm.d Mathematical Problems, Springer

Lecture Notes in Physics, Vol. 20. Lsnford~ O.E. (1975). Time evolutions of large classical systems,

in Dynamical

Systems, Theory and Applications, Springer Lecture Notes in Physics, Vol. 38. Ls-uford, 0.E. and Robinson, D.W. (]968). Mean entropy of states in quantum statistical mechanics. J. Math. Phys., 9, 1120-1125.

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Lanford, O.E. and Ruelle, D. (1969). Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys., 13, 194-215. Lebowitz, J.Lo and Martin-LSf, A. (1972). On the uniqueness of the equilibrium state for Ising spin systems.

Comm. Math. Phys., 25, 276-282.

Marchioro, C., Pellegrinotti, A. and Presutti, E. (1975). Evolution for v-dimensional statistical mechanics. Cozm~. Math. Phys., ~0, 175-186. Meyer, P.-A.

(1966). _robablllte~ P 9 9 t et Potentiel. Paris: Hermann.

Nachbin, L. (1965). Topology and Order. Van Nostrand Mathematical Studies, No. 4, Princeton, N.J. : D. Van Nostr~id Company, Inc. Nelson, E. (1973). The free Markov field. J. Funct. Anal., 12, 211-227. Nguyen, X.X. and Zessin. H.N. (1975). Punktprozesse mit Weehselwirkuug. 0nsager, L. (1944).

to s~ppear.

Crystal Statistics L. A two dimensional model with an order-

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unpublished.

Gibbs states on countable sets. Cambridge Tracts in

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Math.

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I.l

Index

anti-ferromagnet

5.20

base point

5.9

boundary

5.15

Choquet simplex

2. ii

countably generated q-field

2.13

cube

7.4

determining class

9.3

DLR-equations

1.3

dynamical system

1.4

entrance boundary

1.8,2.6,2.14

equilibrium state

1.3,1.4

external field

5.19

extreme point

2.6,2.11,4.2 5.18,5.19

ferromagnet first

order

phase

transition

FKG-inequalities

9.22 9.1,9.13,9.32

GHS-inequalities

9.1,9.24

Gibbs state

1.7,2.2,2.7

high density

9.6

invariant

4.2

low density

9.6

Gibbs' variational principle

7.1,8.1

Griffiths inequalities

9.1

Hamiltonian

1.2, 1.4

H-invari an ce

4.4

1.2

199

infinite tree model

5.21

interaction attractive

5.18,5.22,9.1

repulsive

5.20, 5.23

superstable

6.24

Ising model

5.17

lattice model

1.2,1.3,5.1

local observables

3.3

Markov chain

3.1, 5.23

measures independent

6.5, 6.8

tempered

6.24

neighbour

5.16

observables at infinity

2.5

point process

6.1

Poisson

6.9

Polish space

9.10

potential

5.9, 6.9

finite range

5.13

hard-core

6.16

interaction

1.4, 5.9, 6.9

nearest neighbour

5.16, 5.21

pair

5.19, 6.15

stable

6.i0

pressure convexity of, strict convexity of,

8.i, 8.5, 8.i0 8.14 8.22

probability kernel

1.5

projective limit of standard Borel spaces

3.2

1.3

200

quantum field theory

i.i0

random field

2.1

H-ergodic

4.4

independent

8.2

Markov

5.16

random walk

3.1, 5.27

separable o-field

6.2

short range correlations

2.13

simplex

5.16

specific energy

8.2, 8.5, 8.10

specific entropy

8.2, 8.11

specific free energy

7.2, 8.1, 8.12

specific information gain

7.2, 7.4, 8.1

specification

1.7, 2.2, 2.6

attractive

9.1, 9.4, 9.13

bounded range

7.15

finite range

5.13

nearest neighbour

5.15, 5.23

spontaneous magnetization

5.19

standard Borel space

2.5, 2.14

stochastic field

2.1

stochastic process

1.7

symmetry breakdown

4.1, 5.20

tail-field

2.5, 2.11, 2.13

trivial

2.5, 2.9

tangent functional

8.13

thermodynamic limit

1.5

van Hove convergence

8.12