gibbs random fields: cluster expansions

Managing Editor:
M. HAZEWINKEL Centre for Mathenuztics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R. S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R. M. C. POL YV ANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Volume 44
by
V. A. MaIyshev and R. A. MinIos Department of Mathematics, Moscow State University, Moscow, U.S.S.R .
.. SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library ofCongress Cataloging-in-Publication Data Malyshev. V. A. (Vadim Aleksandrovich)
[Gibbsovskie sluchalnye polfâ. Englishl Gibbs random fieids : cluster expansions I by V.A. Malyshev and
R.A. MinIos. p. cm. -- (Mathematics and its applications (Soviat Series)
v. 44) Translation of: Gibbsovskie sluchalnye polfâ. Includes blbliographical references and index. ISBN 978-94-010-5649-6 ISBN 978-94-011-3708-9 (eBook) DOI 10.1007/978-94-011-3708-9 1. Random flelds. 2. Cluster analysis. I. MinIos, R. A. (Robert
Adol 'foVich) II. Title. III. Ser ies: Mathellatics and its applicaticns (Kluwer Academic Publishers). Sovlet serles ; 44. CA274.46.M3613 1991 619.2--dc20 91-16172
ISBN 978-94-010-5649-6
Printed on acid-fi-ee paper
AII Rights Reserved © 1991 Springer Science+Business Media Dordrecht OriginaIly published by Kluwer Academic Publishers in 1991 Softcover reprint of the hardcover 1 st edition 1991
CIP
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, incJuding photocopying, recording or by any information storage and retrieval system. without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Et moi, ... , si j' avait su comment en revenir, je n'y serais point aIle.'
Jules Verne
The series is divergent; therefore we may be able 10 do something with it.
O. Heaviside
One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- sense'"
Eric T. Bell
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound_ Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra mathematical sophistication that this requires. For that is where the· rewards are. Linear
vi SERIES EDITOR'S PREFACE
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci ate what I am hinting at: if electronics were linear we would have no fun with transistors and com puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre quently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading - fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
A random field is simply a collection of random variables one for each x E zn or IR n (of finite chunks of these spaces). A free field is one with no interactions and until fairly recently they were the only ones which had been studied in real depth. A Gibbs random field is obtained from a free one by a 'Gibbsean perestroika' (the authors do indeed use this word in the Russian text) which defines a new measure (in various ways) by specifying the Radon-Nikodyn derivative of the new measure, p., with respect to the old free one, /10, in terms of an interaction energy function. This is the main theme of the book: the construction and study of random fields defined by such a finite volume Gibbs specification (modification).
The interest in Gibbs random fields is great because a large number of spatial interaction phenomena can be statistically described by such fields. The subject started in 1925 (with the Ising model) and by now applications include ecology, statistical mechanics, image modelling and analysis. Perhaps I should say potential applications because the underlying deeper theory has only comparatively recently been developed: something to which the present authors have greatly contri buted. That deeper theory is most important; for instance the presence (or absence) of phase transi tions affects statistical inference and identifiability in image processing applications.
The new tool, one of the most powerful methods of contemporary mathematical physics, goes by the name of cluster expansions. This allows one to obtain expressions for local random field characteristics in terms of expansions depending only on a finite set of random field variables (the clustors).
With this book (and its planned sequel) an additional substantial chunk of mathematics is in place and ready to be applied. I am pleased to be able to offer the scientific community the chance to peruse this English language version.
The shortest path between two truths in the
real domain passes through the complex
domain.
La physique ne nous donne pas seulement
l'occasion de resoudre des problemes ... eUe nous fait pressentir la solution.
II. Poincare
them; the only books I have in my library
are books that other folk have lent me.
Anatole France
The function of an expert is not to be more
right than other people, but to be wrong Cor more sophisticated reasons.
David Butler
Michiel Hazewinkel
§1 Gibbs Modifications
Xl
Xlll
1
1
14
Chapter 2. Semi-Invariants and Combinatorics 27
§1 Semi-Invariants and Their Elementary Properties 27
§2 Hermite-Ito-Wick Polynomials. Diagrams. Integration by Parts 35
§3 Estimates on Moments and Semi-Invariants of Functionals
of Gaussian Families
§5 Estimates on Intersection Number
§6 Lattices and Computations of Their Mobius Functions
43
50
59
63
Random Variables
Chapter 3. General Scheme of Cluster Expansion
§ 1 Cluster Representation of Partition Functions and Ensembles
68
73
77
§3 Limit Correlation Function and Cluster Expansion of Measures 86
§4 Cluster Expansion and Asymptotics of Free Energy.
Analyticity of Correlation Functions
§6 Point Ensembles
§ 1 Gibbs Modifications of Independent Fields with
91
95
100
104
Bounded Potential 104
§2 Unbounded Interactions in the Finite-Range Part of a Potential 108
§3 Gibbs Modifications of d-Dependent Fields 110
§4 Gibbs Point Field in R" 111
§5 Models with Continuous Time 115
§6 Expansion of Semi-Invariants. Perturbation of a Gaussian Field 118
§7 Perturbation of a Gaussian Field with Slow Decay
of Correlations 122
of Inverse Covariance) 127
Expansions) 144
§2 Continuous Spin: Unique Ground State 149
§3 Continuous Spin: Two Ground States 155
Chapter 6. Decay or Correlations 163
§1 Hierarchy of the Properties of Decay of Correlations 163
§2 An Analytic Method of Estimation of Semi-Invariants
of Bounded Quasi-Local Functionals
in the Case of Exponentially-Regular Cluster Expansion
§4 Slow (power) Decay of Correlations
§5 Low-Temperature Region
Chapter 7. Supplementary Topics and Applications
§1 Gibbs Quasistates
§4 Gauge Field with Gauge Group Z2
§5 Markov Processes with Local Interaction
167
172
180
187
193
196
196
204
210
214
219
Concluding Remarks
Bibliographic Conunents
226
230
233
237
247
PREFACE
This book is intended for a wide range of readers: specialists in probability theory will see new and fresh topics concerning random fields; specialists in mathematical physics (quantum field theory or statistical physics) will be interested in a thorough presentation of the cluster technique and its various devices; for the remaining mathematicians, the book offers a number of new algebraic, combinatorial, and analytic problems.
We will be occupied with explicit constructions of random fields. They are based on a single method: the so-called Gibbs modification. What is its meaning? The probability distribution of a finite system of random vari ables {6, ... '~n} (of a finite field) is most often given in terms of its density p(Xl, ... , xn) with respect to some (usually Lebesque) measure on Rn. Gibbs modification is a natural generalization of this method to the case of infinite fields. The simplest infinite random fields are independent and Gaussian fields and their functionals (until recently, they remained the only well-studied class of fields). As to the Gibbs method of construction, one first introduces fields with a distribution prescribed by a finite (local) density with respect to in dependent or Gaussian fields (a finite or local Gibbs modification) and then passes to the weak limit of such distributions (a limit Gibbs modification); this limit measure is already singular with respect to the original distribu tion, independent or Gaussian one. Random fields originating in this limit (the thermodynamic limit) actually constitute the main subject of study in the theory of Gibbs fields.
We will present one of the most powerful methods of investigation of Gibbs fields, namely, the method of cluster expansions, and its numerous applica tions. With this method, any local characteristic of a field (finite-dimensional distributions, mean values of local functions, etc.) can be represented in terms of a series where each term depends on a finite group of field variables ("a cluster") and is expressed in terms of them in an explicit way. The idea of the method goes back to the so-called virial expansions in statistical physics and, on the other hand, to the diagram expansion in quantum field theory; in its present-day form the method of cluster expansions represents a math ematicized development of these methods. Unfortunately, to apply a cluster expansion, one needs a "small parameter," i.e., the method appears in the
xi
xii Preface
form of a version of the perturbation theory up to now, even though by its nature it seems to be more universal and, apparently, should have broader applications. We hope that in the future, this limitation will be overcome.
Chapter 1 is an introduction to Gibbs fields; Section 0, in particular, is written in an extremely elementary manner. The reader interested directly in cluster expansions may skip this chapter. In Chapter 2 some subsidiary material is collected: the properties and estimates of semi-invariants, the dia gram technique, and the most important combinatorial lemmas. The general scheme of cluster expansions is presented in Chapter 3, and numerous exam ples of these expansions are given in Chapters 4 and 5. These are the central chapters in the book; they contain the basic methods as well as various re finements of cluster expansions.
Chapters 6 and 7 are devoted to applications, both traditionally probability theoretic ones and also those associated with mathematical physics.
Let us notice that our presentation includes two types of random fields: the point (labelled) ones and those on a countable set, with special emphasis on the latter. This stems, in particular, from the fact that to apply the cluster technique to continuous (functional) fields (that are not included in the book due to its limited extent), one has to reduce them beforehand to fields in a discrete set.
We have not enough space here to indicate explicitly the connection of our constructions with statistical physics and quantum field theory even though their spirit is invisibly present in the book and the terminology betrays their influence.
The limited scope of this book did not allow us to present one of the most important applications of cluster expansions, namely, an investigation of the spectrum of the transfer matrix of a Gibbs field. Also, numerous other meth ods used in the theory of Gibbs fields are not included in this book (cor relation inequalities, reflection positivity, duality transformation, and other special methods). We are planning to write a sequel in which some of these omissions will be rectified.
NOTATIONS
(0, E, I')-a probability space, i.e., a triple consisting of a set 0, au-algebra E of subsets of 0, and a probability measure I' defined on E.
If 0 is a topological space, E denotes its Borel u-algebra ~(o), i.e., the u-algebra generated by open sets in O.
We· shall usually denote by 0 a set of configurations of a random field. I'o-a ''free'' (nonperturbed) measure on 0 (usually independent or Gaus
sian). For any random variable, i.e., a measurable function e on a probability
space (0, E, 1'), its mean (mathematical expectation) is denoted by
Whenever {el, ... ,en} is a family of random variables,
denotes their semi-invariant, N = {I, ... ,n} (we shall often omit the sub script I')j in addition,
(e~k1, ... ,e~k .. ) = ~, ... '~n'.~. ,en).
k1 times k .. times
G-a graph, i.e., a set V (of vertices) and a collection of unordered (possibly coinciding) pairs of elements of V (edges).
T -a tree, i.e., a connected graph without loops. (Al , ... ,An)-an ordered, and {Al. ... ,An}-an unordered collection of
sets Ai, i = 1, ... ,n (similarly for collections of points). A partition a = {Tl , ... , Tk} of a set A is an unordered collection of
nonempty mutually disjoint subsets n c A, i = 1, ... ,k, whose union is A, U~=ln = A.
xiii
XIV Notations
~N-the lattice of all partitions of the set N = {I, ... ,n}. ~X- C ~N-(with a connected graph G with the set of vertices N) the
lattice of partitions of N whose elements form connected subgraphs of G. j.t".- the Mobius function of the lattice tl.
T-a countable (or finite) set supporting a random field. Q-a space supporting a point field. S-a space of values of a field (a space of "spins" or "charges"). A, A, R-finite (bounded) subsets of T (or Q). (lA-the space of configurations of a field in A(A C T or A C Q). EA-a u-algebra in (lA. UA-a Hamiltonian (energy) in A. UA,y = U(·/y)-a Hamiltonian (energy) in A under the boundary configu-
ration y. j.tA-a finite Gibbs modification (in A). j.tA,y-a Gibbs modification under the boundary configuration y. ZA-a partition function (of a Gibbs modification) in A. ZA,y-a partition function in A under the boundary configuration y. r, "Y,~, ... -"clusters": i.e., connected collections of sets (or "labelled" sets). kA (or kr )-quantities defining cluster representation of partition functions. bR(F), b<;)(F), bR, b<;)-quantities appearing in cluster expansions of
means (correlation functions, free energy, etc.). f~A), fA -correlation functions. :p(~l" .. ,en):-the Wick polynomial (of a Gaussian family of random vari
ables {6, ... ,en}), Let ~ be a family of subsets of T. Then dR(~;AI'''' ,An), where
Ai C T, i = 1, ... ,n, and ReT, denotes the minimal cardinality of a collection {BI , ... ,B.} of sets Bi E ~ such that the collection {BI , ... ,B., AI, ... ,An} is connected and UZ=1 Bi = R.
Further, d(~;AI'''' ,An) = minRdR(~;AI'''' ,An), i.e., it is the mini mal cardinality of a collection {BI , ... ,B.}, Bi E~, i = 1, ... ,s, forming together with {AI,' .. ,An} a connected collection.
Finally dA(~)' where ACT, denotes the minimal cardinality of con nected collections {BI , ... ,B.} of sets Bi E ~, i = 1, ... , s, such that A C Ut=IBi; dA = dA(~) if ~ is the set of pairs of neighbouring sites {t, t'} (i.e., p(t, t') = 1).
For referring to formulae and theorems, the following convention will be used:
(1) denotes the formula (1) in the same section, (2.1) denotes the formula (2) in Section 1 of the same chapter, and (3.2.1) denotes the formula (3) in Section 2 of Chapter 1; similarly for references to theorems and lemmas.
CHAPTER 1
§O First Acquaintance with Gibbs Fields
In this introductory section, we present the basic notions and problems of the theory 6f Gibbs fields and display some typical ways of argumentation by considering a simple and well-examined example of the so-called Ising model.
O.t. Ising Model
Consider the lattice Z" of points t = (t(l), ... ,t(II») E R" of the v-dimen sional real space with integer coordinates. Let AN == A be a "cube" in Z" centered at the origin, i.e., the set of points in Z" whose coordinates have absolute values not greater than N (with an integer N > 0). Each function uA = {Ut, tEA}, defined on the set A and taking values Ut = ±1, is called a configumtion (in the cube A), and the set of all such configurations is denoted by (lA. Obviously, the number of configurations in A equals 21A1 , where IAI stands for the number of lattice sites in A.
We define, on (lA, a function
(1)
called the energy of the configuration u A • The summation in the second sum in (1) is taken over all unordered pairs (t, t'), t, t' E A, such that p(t, t') = 1 (pairs of "nearest neighbours"), with
II p(t,t') = L: It(i) - t'(i) I '
i=l
1
2 Gibbs Fields (Basic Notions) Chapter 1
A physical system with the configuration space OA of configurations in A and a configuration energy of the form (1) is usually called the Ising model. The real numbers h and /3 in (1) are fixed (parameters of the model). We refer to the case /3 > 0, which will be investigated here, as the ferromagnetic Ising model.
We introduce a probability distribution on the space OA defining the prob ability of a configuration u A by
PA(UA) = ZA'l exp {-UA(UA)} . (2)
The normalization factor ZA is defined by the condition
E PA(UA) = 1, f7 AeOA
and thus, ZA = E exp{-UA(UA)}. (3)
f7 AeOA
The quantity ZA is called a partition function and plays a key role in what follows. The probability distribution (2) is called a Gibbs probability distri bution in A corresponding to the Ising model. In general, a model is defined by the choice of a set of values of configurations and of a particular form of energy function (see p. 19).
Having introduced probability distributions on the configuration space, the values Ut of these configurations may be considered as random variables and the formula (2) as the joint probability distribution of these random variables. Later on, the mean (value) of an arbitrary function f on the space OA under the distribution (2) will be denoted by (J)A. The means (UT)A of random variables
UT = II u" u, = 1, (4) 'eT
with TeA being an arbitrary subset of A, are called correlation functions (or moments) of the distribution (2).
For any TeA, we use pf) to denote the joint distribution of the system of random variables {Ut, t E T}, i.e., the collection of probabilities
P(T) (- -) p (- - ) A Utl' ... ,u,,. = r Utl = Utl'· .. , Ut .. = Ut.. , (5)
with T = {tl, ... ,tn } and fUtl' ... ,7ft,,} being an arbitrary collection of values 7f'i = ±1, i = 1,2, ... ,n. The probabilities (5) may be very simply expressed by means of correlation functions (UT)A. Indeed,
pf) (iTtll" .. ,iTt .. ) = 2~ (_1)1: (ft (Uti + iTt;)) 1=1 A
(_1)1: = -n- E CT'(UT')A, (6)
2 T'~T
§O First Acquaintance with Gibbs Fields
with k being the number of values lft; that equal -1, and
0.2. Thermodynamic Limit
3
Now we fix T and let A expand to ZV, A / ZV, i.e., put N -+ 00. If we succeed in proving the existence of the limits
(7)
we may conclude that correlation functions (and finite-dimensional distribu tions) almost do not depend on A for A sufficiently large in comparison with T. Such a passage to the limit is called the thermodynamic limit (the limit of a large number of degrees of freedom O't). The limits (7) are denoted by (O't) and are called limit correlation functions. It follows from (6) that finite dimensional distributions also have limits, which form a compatible family of finite-dimensional distributions. By the Kolmogorov theorem (see [51]), this family defines a system of random variables {O't, t E ZV}, called a (limit) Gibbs random field (for the Ising model), as well as their distribution P (a measure) on the space n = {-1, l}Zv of infinite configurations in the lattice ZV.
The methods developed in this book enable us to establish the existence of the limits (7) for a sufficiently general class of models. However, in the case of the ferromagnetic Ising model, we shall use a special device of the so-called Griffith correlation inequalities to control the passage to the limit (7). The existence of the limit distribution P follows from the following theorem.
Theorem 1. The thermodynamic limit (7) of correlation functions (O'T}A ex ists for f3 ~ 0 and every finite T.
Remark 1. In the case of f3 = 0, it is not difficult to calculate (O't)A:
(8)
Consequently, (O'T)A does not depend on A (for TeA). Thus the thermo dynamic limit (O'T)A exists in this case and equals (8). The random variables 0'( are mutually independent, both with respect to the distributions in finite A and with respect to the limit distribution.
Proof of Theorem 1. Notice that it is sufficient to consider the case h ~ O. This is obvious from the following symmetry property of the Ising model (in
the notations introduced below, (3 and h as subscripts indicate the dependence of Gibbs distributions on these parameters):
(9)
with _uA denoting the configuration whose values have an opposite sign to those of the configuration uA •
It follows from (9) that
() { (UT}A,,8,-h, ITI even, UTA,8h=
, , -(UT}A,,8,-h, ITlodd. (10)
In particular, (11)
for odd ITI. The proof of the relations (9) and (10) is simple, and is left to the reader.
We need two inequalities to prove the theorem. It is suitable to consider a more general situation. Let A be an arbitrary subset of ZV, OA a set of all configurations uA = {Ut, tEA}, Ut = ±1, in A, and the energy UA(UA) of the configuration u A be of the form
(12)
with ht ~ 0 and (3w ~ 0 (in general, the sum in (12) is taken over all pairs of points t, t' E A). The distribution PA on OA is given as before by the formula (2), and ( }A again denotes the mean under this distribution.
Lenuna 2. The inequalities
(the second Griffith inequality) are valid.
Proof. To prove (13), it is sufficient to verify
L uTexp{-UA(UAn ~ O. "AeOA
(13)
(14)
(15)
By expanding the exponential function exp { - U A (u A)} in the series L::=o( -U A)n In!, by removing the parentheses in each term of this series,
§O First Acquaintance with Gibbs Fields 5
and by taking into account that 0'1 = 1, we express the left-hand side of the inequality (15) in the form of the sum
(16)
with CB ~ o. Since (17)
for any tEA, the sum (16) is equal to C., which proves (13). To prove (14) we investigate two independent samples of the distribution
PA, i.e., a distribution on the space OA x OA of pairs {O'\ iTA} of configurations of the form
PA(O'A, iTA)
= (ZA"1)2 exp {L: ht(O't + iTt) + L: /3t,t' (O'tO't' + (iTt iTt')} . tEA t,t'EA (18)
We introduce new variables
tEA,
taking values (et,7Jt) = (2,0), (-2,0), (0, 2), (0, -2). By means of these vari ables, the probability (18) may be rewritten in the form
Taking into account that
'/i=-2,O,2
for each tEA and each integer k ~ 0, and by repeating the proof of the inequality (13), we get
(eT 7JT' ) A,A ~ 0 (19)
for all T and T', with eT and TJT being defined as in (4) and the mean ( }A,A evaluated under the distribution (18).
Notice that
We shall show that the difference UT - UT and the sum UT + UT can be represented in the form
UT ± UT = L Ci,BeA 'f/B A,B!;;T
(21)
with Ci,B ~ o. The relations (19), (20), and (21) imply the inequality (15). The expression (21) can be proved by induction on ITI if we notice that
UTU{t} + uTU{t} = ~ [(UT + UT){t + (UT - UT)'f/t],
uTU{t} - uTU{t} = ~ [(UT + UT)'f/t + (UT - UT){t]
for t rI. TeA. The lemma is proved.
N ow we return to the proof of the theorem. The derivatives at (UT)A and a{3~'11 (UT)A are equal to
o oht (UT)A = (UTUt)A - (UT)A(Ut)A ~ 0,
o ~f3 (UT)A = (UTUtUtl)A - (UT)A(UtUtl)A ~ 0, U t,t'
(22)
and thus the correlation functions increase when increasing the parameters ht
and f3t,tl. It follows that, in the case of the Ising model,
(23)
for TeAl c A2 • Indeed, the mean (UT)A 1 coincides with the mean under the distribution of the form (12) in A2 , with parameters
and
{ f3 if t, t' are nearest neighbours in A 1,
f3t t' = , 0 otherwise.
U sing the monotonicity of (UT) with respect to the parameters ht and f3t,t l , we get (23). Since I(UT)I ~ 1, the statement of the theorem follows from (23).
0.3. Markov Property
Let A C Z" be a set; its boundary aA is defined to be the set of all lattice sites of distance 1 from A:
aA = {t E Z":p(t,A) = I}. (24)
Let A C Z" be a cube, and let A,B ~ A be such that An B = 0 and aA c B. We use
plA)(iiA /uB ) = Pr{ut = iit, t E A/ut' = Ut', t' E B}
to denote the conditional probability that uA equals iiA = {iit , tEA} on the set A under the condition that its values on the set B equal uB = {Ut', t' E B}.
Lemma 3. The following equalities hold true:
plA)(iiA /uB ) = plA)(iiA /u8A )
= ZA 1(u8A ) exp {-(UA(iiA) + UA,8A(iiA, u8A»} . (25)
Here UA(iiA) is the energy of the configuration iiA defined as in (1), UA,8A(iiA , u8A ) is the energy of the interaction between the configurations iiA and u8A :
UA,8A(iiA,U8A ) = -f3 2: iitut', tEA,t'E8A p(t,t')=l
and ZA(u8A ) is the conditional partition function
(26)
ZA(U8A ) = 2: exp {-(UA(iiA) + UA,8A(iiA , u8A»} . (27) ~A
The first equality in (25) is called the Markov property of the distribu tion PA, and the other equality expresses its Gibbs property: the conditional distribution plA ) is similar in form to the distribution (2), except that the energy U A,8A of the interaction with the "boundary" configuration u8A was added to the energy U A. Usually, the distribution given by the formula on the right-hand side of (25) is called the Gibbs distribution in A with the boundary configuration u8A .
Proof of Lemma 3. It is carried out by a direct computation: The formula (2) implies that
(28)
where (fA = (iTA, UB , (fA\(AUB» and (fA\(AUB) is a configuration in the set A \(A U B). Further,
UA«(fA) = U A(iTA) + U A,B(iTA , uB) + UB(UB) + UA\(AUB)«(fA\(AUB»
+ U B,A\(AUB)(UB , (fA\(AUB»,
with the energies UA,B and UB,A\(AUB) given analogously to (26). Accord ingly, the nominator on the right-hand side of the equalities (28) equals
and the denominator is
with ZA\(AUB)(UB ) and ZA(UB ) defined analogously to (27). Inserting these expressions into (28) and noticing that UA,B(iTA,UB) = UA,8A(iTA,U8A) and ZA(UB ) = ZA(U8A ), we arrive at (25) after obvious cancellations. The lemma is proved.
It is obvious from (25) that the conditional distribution plA)(-fuB ) does not depend on A. This observation lays the foundation for the following definition of the Gibbs random field in ZII.
Definition 1. A probability distribution P on the space n is said to determine a Gibbs random field {(ft, t E ZII} (for the Ising model) if the conditional distribution p(A)(iTA /jjB), generated by the distribution P, coincides with the Gibbs distribution in A, with the boundary configuration U8A (see the second equality in (25» for arbitrary finite subsets A, B C ZII such that AnB=0and8ACB.
Notice that, according to the first equality in (28) and the definition (7), the limit Gibbs distribution constructed above defines a Gibbs random field in ZII also in the sense of Definition 1. Are there still other Gibbs fields in ZII for the Ising model? It turns out that this depends on the dimension of the lattice ZII and on the parameters (h,P). The values of parameters (h,P) for which there exist more than one Gibbs field in ZII define points of the first-order phase transition in the plane (h,P).
Theorem 4. For a ferromagnetic Ising model: 1) for v = I, there is a unique Gibbs field; 2) for v ~ 2 and h :f:. 0, or h = 0 and P sufficiently small, 0 :E;; f3 :E;; f3o(v),
there is a unique Gibbs field; 3) for v ~ 2, the points (O,P) with P sufficiently large, f3 > f31(V), are
points of the first-order phase transition.
We shall prove only the statements 1) and 3) of this theorem.
Before proving the theorem, we investigate the possible ways of construction of Gibbs fields in Z" for the Ising model. Let A C Z" be a cube, (f8A be a configuration in the boundary 8A of the cube A, and let PA,U8A(O'A) denote the Gibbs distribution in A (on the space OA) with the boundary configuration (f8A (see (25)). Let q8A be an arbitrary probability distribution on the set 08A of boundary configurations (f8A. We use the notation P A ,q8A for the distribution
(29)
on the space OA, arising by averaging of distributions PA,U8A over all boundary configurations uBA • The distribution (29) is called the Gibbs distribution in A with a random boundary configuration. Finally, besides PA,fl8A and P A ,q8A,
the Gibbs distribution PXer with the so-called periodic boundary conditions is often considered. It is defined analogously to the distribution PA (see (2)) except for replacing the "cube" A by the "torus" (by identifying the opposite sides of A) and the energy UA in (2) by the energy Urr of the interaction of the nearest neighbours on this torus. The Gibbs distribution (2) is often called the Gibbs distribution in A under the "empty boundary conditions."
Following the proof of Lemma 3, we can easily convince ourselves that the distributions
P P Pper A,fl8A , A,q8A, A (30)
have the Gibbs property (25). As in the case of Gibbs distributions with the empty boundary conditions,
we conclude that the limit P = limA,,/Z" PAra of the sequence PA" of dis tributions of the form (30), with An being an increasing sequence of cubes, Al C A2 C ... C An C ... C UAn = Z", defines a Gibbs field in Z". The converse is also true.
Lemma 5. Every probability distribution P on the space 0 that is a Gibbs random field in Z" is the thermodynamic limit of a sequence PA .. ,q~A .. for
some choice of q~A" •
Proof. For every cube A C Z", we choose q8A to be the probability distribu tion on 08A induced by the distribution P. It is obvious that P A,q8A coincides in this case with the distribution induced by P on OA. Therefore, P A,q8A -+ P
(in the sense (7» as A / Z".
Now we return to the proof of the theorem. Proof of the statement 1) or Theorem 4. Transfer matrix. To simplify the formulae, we put h = O. The 2 x 2 matrix J = lIiuu,1I with matrix elements . {'Juu', ±1
}u,u' = e ,0'0' = , (31)
is called the transfer matrix of the Ising model. Let A = [-N, N] C ZI and PA be the Gibbs distribution in A (under the
empty boundary conditions).
Lemma 6. The following equalities hold:
p {t 1 .... ,t,,}(- - ) A O't1' ••. ,O't.
(e(iJt1), JN1e) (e(iJt2), Jt2-he(iJt1») ... (e, JN2eiJt4) = (33)
(e,J2Ne)
with e = (l,l),e(l) = (l,O),e(-l) = (O,l),N1 = t1+N,N2 = N-tn' -N ~ t1 < t2 < ... < tn ~ N. The proof is obvious.
Now let g(l) and g(2) be two normalized eigenvectors of the transfer matrix J, with eigenvalues A1 and A2,A1 > IA21) O. Using the decompositions
we get
(e(iJt1),JN1e) _ B~iJt1)C1A~\
(e,JN2e(iJt,,») _ B~iJt")C1At\
for large N and fixed {t1,'" , t n }, and thus
Similarly, it can be shown that the probabilities PA{t1"8~:"} have the same .,f ...
limit for any sequence of Gibbs distributions PA 1J8A" , An / Zl. The proof of the first statement is finished. '" II
Remark. From our considerations, one may easily derive that the limit Gibbs field {O't, t E Zl} is a stationary Markov chain with the matrix of transition probabilities
(1) P. J (11 (12 g(12
(110(12 = (1)' A1g(11
and the stationary distribution 1f(1 = (g~1)2,0' = ±l, where gP), g~l are the components of the eigenvector g(l).
Proof of the statement 3) of Theorem 4. We denote the Gibbs distribution in A with the boundary configuration iTt == +1,t E 8A «+)-boundary condi tions) by PA,(+).
Lemma 7. The inequality
PrA,(+)(UO = -1) < 1/3 (34)
holds uniformly with respect to all cubes A C Z", 0 E A, for all sufficiently large {J,P > Pl(V).
First, we shall derive our statement from this lemma. Consider the Gibbs distribution PA,(-) with the boundary configuration iTt == -1, t E 8A (the (-)-boundary conditions). By virtue of the symmetry valid for h = 0, we have
and hence, PrA,(_)(UO = 1) < 1/3
for every A, and consequently,
PrA,(_)(UO = -1) > 2/3. (35)
The inequalities (34) and (35) ensure that there are at least two different Gibbs distributions in Z".
Proof of Lemma 7. To make the proof more transparent, we restrict ourselves to the case v = 2. Let Z2 be the dual lattice obtained from the lattice Z2 by shifting it by the vector (1/2,1/2). For any configuration uA , we use 'Y = 'Y(uA) to denote the collection of those bonds of Z2 that separate two neighbouring sites t,t' E Au8A with Ut:F Uti, (Ut = 1 for t E 8A). It is easy to see that the number of bonds from 'Y(uA) attached to a lattice site from Z2 is always even. Thus, the connected components of 'Y are closed polygons (possibly self-intersecting). We shall denote them by r 1, ... ,r n and call them contours. We shall show that there is a configuration uA with 'Y = 'Y(uA) for each collection 'Y = {rl,'" , r n} of mutually disjoint contours. Indeed, it is enough to put Ut = 1 for tEA that are outside all contours. Further, we put Ut = -1 for the sites that are inside one contour r only, 0', = 1 for the sites that are encircled by two contours, and so on. Thus, there is a one-to-one correspondence between the configurations uA and the collections of contours 'Y. In addition,
UA,(+)(UA) = UA(UA) + UA,8A(uA,iT8A == 1) = 2{Jhl- {JIAI, ZA,(+) = ZA(iTaA == 1) = exp{{JIAI} Ee-21'11I1,
II
where Irl is the number of bonds in r (the length of r) and IAI is the number of bonds from Z2 adjacent to at least one site from A.
Lemma 8. The probability PA,(+)(r) 0/ the event that r is contained in the collection r can be estimated by
Proof. The probability PA,(+)(r) equals
PA,(+)(r) = E PA,(+)(r) 'Y:I'E'Y
E e-2,8hl e-2,8II'I E'e-2,8hl
'Y 'Y
where E~ denotes the sum over all r that do not contain r or intersect it. The lemma is proved.
Further, it is easy to convince ourselves that the number of contours r of the length n encircling a given site to E Z2 is not greater than n23n • Since the event Uo = -1 under the (+ )-boundary conditions implies the existence of at least one contour r encircling the point 0, we have
PrA,(+)(UO = -1) ~ E PA,(+)(r) ~ E n 23n e-2,8n < 1/3 1':1' encircles 0 n;;'4
for f3 large enough. Lemma 7 and, at the same time, the statement 3) of Theorem 4, are proved.
0.4. Other Models
To enrich the reader's idea of what the theory of Gibbs fields is all about, we shall add a few more examples of physical lattice models often met in the literature.
I. Rotator mode I
Let A C Z" be a finite subset, the configurations uA = {Ut, tEA} take values in the n-dimensional sphere, Ut E Rn, IUtl = 1, and the energy of a configuration be given by
UA(UA)= E Jt-t'·(Ut,Uf')+ E(h,ut), f,f'EA tEA
with (. , .) being the scalar product in Rn , and Jt being some finite function of t E Z", t ::I 0, h E Rn. In the case of this model, the Gibbs distribution P is given by
(36)
where PA( (J'A) is the density of the distribution PA with respect to the measure >.~ = >'0 x ... x >'0 of the space OA of configurations, with >'0 a uniform ------IAltimes distribution on the sphere S eRn and ZA the normalization factor.
II. Model with a global symmetry (G-model)
Let G be a group, and let the configuration gA = {gt, tEA}, A C Z", take values from G. The energy of a configuration gA is of the form
UA(gA) = L 1/;(Utg";;l), p(t,t')=1 t,t/eA
where 1/; is some even function on the group G : 1/;(g) = 1/;(g-I). The Gibbs distribution is defined according to the preceding by a formula similar to (2) in the case of the discrete group G, or to (36) in the case of the continuous group G with >'0 standing for the normalized Haar measure on G.
III. Gauge G-model (a mqdel with a local G-symmetry)
In this case, the configurations UA = {gT' TEA} are defined on a finite set of oriented bonds T = (tl,t2), tl, t2 E Z", p(tl,t2) = 1, and take values from G as before. Moreover, gT = (U-T )-1, where -T = (t2' tt) denotes the bond that differs from T by its orientation. The energy of the configuration gA is given in the form
(37) p
with the summation taken over all two-dimensional (nonoriented) plaquettes P of the lattice Z" whose boundary bonds op = (Tl' T2, T3, T4) belong to A, and
Up = gT,UT2gT3gT.·
In addition, the numbering of the bonds Tj and their orientation are chosen so that they form a path around P in some direction. The function 1/; in (37) is a function on the group G so that
1/;(ggOg-l) = 1/;(go). (38)
Under these conditions the value 1/;(gp) does not depend on the orientation and the starting point of the path around p. Most often, the function 1/; is chosen equal to
1/;(g) = ReX,,(g),
with Xa being the character of some irreducible unitary representation of the group G. The Gibbs distribution for this model is introduced analogously to the preceding cases.
IV. P(ip)v-mode1
Configurations zA = {Zt, tEA}, A C ZV, take arbitrary real values, the energy U A (zA) is given by
UA(ZA) = L (Zt - Zt,)2 + L(P(zt) + mzn, (t ,t')E A tE A p(t,t')=l
where m ~ 0 and P(.) is some polynomial of an even degree with a positive leading coefficient. The density of the Gibbs distribution PA on the configu ration space OA = RIAl with respect to the Lebesgue measure (dz )IAI on RIAl is defined by a formula similar to (36). Moreover, it may be easily shown that the normalization factor Z A is finite.
We confine ourselves to this list of models. What kind of questions arise in the theory of Gibbs fields in connection with these (and other) models?
1. The first question is: Does there exist a limit Gibbs field (i.e., a limit probability distribution for infinite configurations)?
2. Second question: Is it unique? This question is very interesting because the existence of several limit distributions (phases) is connected with the so called phase transitions.
3. If the interaction depends on one or more parameters, the question is: Are the various characteristics of the limit field (correlation functions, semi-invariants, and so on) analytic with respect to these parameters (or at least continuous or differentiable)? In addition, the singularities of correlation functions with respect to these parameters (discontinuities, discontinuities of their derivatives, and so on) also indicate phase transitions.
4. Furthermore, questions concerning the ergodic properties of the limit Gibbs fields arise; mixing, decay of correlations, estimates of semi-invariants, limit distributions of sums of local variables, etc.
These are the basic topics of the theory of Gibbs fields touched upon in this book.
Let us now proceed with a systematic presentation of this theory.
§1 Gibbs Modifications
1.1. Random Fields
In this book the following classes of random fields will be studied: (1) Random fields in a countable set T with values in a metric (complete and
separable) space S. The probability space (0, E,J.t) is represented in this case
§1 Gibbs Modifications 15
by the set gr = 0 of functions (also called configurations) x = tXt, t E T} defined on T, with values in S (S is often called the set of "spins"). The space ST is endowed with the (metrizable) Tikhonov (cf. [16]) topology.
We shall investigate probability distributions I' defined on the Borel u algebra ~(S) = E of the space gr. The collection of random variables Xt,
t E T, arising in this way, i.e., the values of the random configuration x at points t E T, forms a random field.
The simplest example of such a field is the field of independent and identi cally distributed variables. In this case, the measure I' on ~(gr) is defined to be the product of countably many identical copies of some probability measure AD on the space S.
(2) Random point fields in a separable metric space Q with values in a space S. The role of the probability space is played by the set 0 of all locally finite subsets x C Q. The subset x (at most countable) is called locally finite if any bounded set A C Q contains only a finite number of points from x. Below, we shall introduce a metrizable topology in O. Every probability measure defined on the Borel (with respect to that topology) u-algebra ~(n) is called a random point field in Q (sometimes we shall say pure point field, emphasizing its distinctness from the labelled point field to be introduced later).
Now suppose that a metrizable space S, also called the space of "charges" (or "labels"), is given. We use ns to denote the space of pairs {x,sx}, with x En and Sx being a function on x taking values from S. Such pairs will be called configurations. In the space Os, as well as in n, a metrizable topology can be introduced. Every probability measure on ~(nS) determines a labelled random field in Q with values in the space S of charges.
The theory of Gibbs fields includes the investigation of: (3) Ordinary or generalized fields in R". The probability space is, in this
case, a topological vector (locally convex) space n of functions or distributions (in the sense of Schwartz) defined on R". As before, a random field is given by a definition of a probability measure on the Borel u-algebra ~(n).
In what follows, we emphasize that, by mentioning any random field, we implicitly understand a measure on the space of its configurations.
1.2 Method of Gibbs Modifications!
Gibbs modification (of a random field) is an important device for the con struction of new measures from an originally given measure 1'0 (or from a family of measures). We shall first describe a general scheme of dealing with
1 Translators remark: The authors stress, by introducing a new term "gibbsovskaya pere stroika," the fact that the new measure arises by modifying the original one. To reflect this stress, and since the notion is crucial in the whole book, we avoided the term "specification" sometimes used in the literature, and follow the authors in introducing a new term. In the present context the translation "Gibbs modification" is preferable to, also possible, "Gibbs reconstruction."
Gibbs modifications. Finite Gibbs modifications. Let (0, E, 1'0) be a measurable space with a
finite or u-finite measure 1'0 (called usually a "free" measure), and let U(x), x E 0, be a real function on 0 (taking possibly the value +00), often called the interaction energy (or Hamiltonian).
The measure I' with the density
ddl' (x) = Z-I exp{-U(x)} 1'0
(1)
with respect to the measure 1'0 will be called the Gibbs modification of the measure 1'0 by means of the interaction U.
It is moreover assumed that the normalization factor Z (called the partition function) fulfills the stability condition
Z = 10 exp{-U(x)}dl'o(x):I 0,00. (2)
Using finite Gibbs modifications, the measures absolutely continuous with respect to Ilo arise. A more interesting class of measures, already singular with respect to the original measure Ilo, arises when passing to the weak limit of finite Gibbs modifications.
1.3. Weak Convergence of Measures
Let 0 be a topological space, 2J = 2J(0) its Borel u-algebra, and E C 2J some of its sub-u-algebras.
Definition 1. Let a directed family :F = {A} of indices be given. We call a measure Il, defined on the u-algebra E C 2J, the weak limit of the sequence of measures IlA, A E :F, defined on E if
(3)
for any bounded continuous E-measurable function f given on O.
A more general situation may be examined. Let a complete family {EA' A E :F}, EAl C EA2' Al < A2 , of sub-u-algebras of the u-algebra 2J (i.e., such that 2J coincides with the smallest u-algebra containing the algebra I]{ = UAEFEA) be given; the u-algebras EA will be called local u-algebras and any function f, defined on 0 and measurable with respect to some of the local algebras, will be called a locaP function (function f, measurable with respect to au-algebra EA, A E:F, will often be denoted by fA)'
2The notion cylinder function is commonly used for local functions (translators remark).
A finitely additive measure I' defined on the algebra ~ so that its restriction JlII:A to any O'-algebra EA is a O'-additive measure on EA is called a cylinder measure. If 0 is a complete separable metric space, and the cylinder measure I' is a probability (Jl(A) ~ 0, A E ~, 1'(0) = 1), I' can be extended to a 0'
additive (probability) measure defined on the O'-algebra ~ (the Kolmogorov theorem; cf. [51]).
The following definition generalizes Definition 1.
Definition 2. Let a finite or O'-finite measure be given on each O'-algebra EA. A cylinder measure I' on ~ will be called the weak local limit of the measures JlA if
lim f f(z)dJlA = f f(z)dJl A 10 10 (4)
for any bounded continuous local function f defined on O.
In other words, a cylinder measure (or its extension to a measure on the O'-algebra ~) is the weak local limit of measures {JlA, A E .1"} if, for each Ao E.1", the restrictions JlAII:Ao = Jl~, Ao < A, A E .1", of the measures JlA to the O'-algebra EAo weakly converge to JlII:Ao = JlAo.
In the case 0 = sr (with T a countable set and S a metric space; see Section 1), the index A runs over finite subsets ofT, and EA = II"Al(~(SA», where II"A : ST -+ SA is the restriction mapping (cf. p. 19), the convergence (4) is called the weak convergence of finite-dimensional distributions if, moreover, JlA are probability measures.
The relationship between Definitions 1 and 2 is established by:
Proposition 1. Let a family {EA A E .1"} of O'-algebms be such that the set Co(O) of bounded continuous local functions is dense everywhere in the space C(O) of all bounded continuous functions defined on 0 (in the uniform metric in C(O»). Then the necessary and sufficient condition for a measure I' on ~(O) to be the local limit of probability measures {JlA} (defined each on the O'-algebm EA) is that their arbitmry extensions jJA to probability measures on the O'-algebm ~(O) weakly converge to 1'.
The proof is obvious.
1.4. Limit Gibbs Modifications
Let a complete directed family {EA' A E.1"} ofsub-O'-algebras of the O'-algebra ~(O) be given, and let a free measure Jl~ and a Hamiltonian U A be defined for each A so that the stability condition (2) is satisfied. A cylinder measure I' on the algebra ~ = UEA (or its O'-additive extension to the O'-algebra ~(O» is called a limit Gibbs measure (or a limit Gibbs modification) if it is the weak local limit of the Gibbs modifications JlA of the measures Jl~ (by means of the energies UA).
We note that the theory of Gibbs measures becomes meaningful only with a special choice of O'-algebras EA , measures Jl~, and Hamiltonians UA. Now
we are going to describe the respective ways of such a choice of EA, It~, and U A in connection with the three types of random fields listed above.
(1) Gibbs modifications of fields in a countable set T. We introduce a set of configurations SA = {zA = (Zt, tEA)} defined in finite ACT, and endow it with a Tikhonov topology and Borel IT-algebra 8(SA). The restriction mapping CPA : Z t-+ zA = zlA defines a IT-algebra EA = CPA1(~(SA» C ~(sT) that will be often identified with ~(SA). Obviously, {EA' ACT} is complete in ~(sT).
Remark 1. The set Co(sT) C C(sT) of bounded continuous local functions on sT is, according to Stone's theorem (cf.[16]), dense everywhere in C(sT), and hence Proposition 1 applies in the case considered.
Hamiltonians U A are usually defined with the help of a potential {c) A;
ACT, IAI < oo}, i.e., a family of functions c)A on 0 that are measurable with respect to IT-algebras EA (Le., c) A can be viewed as a function defined on the space SA).
For any finite A, we put UA = L: c)A. (5)
A!;A
Often, instead of the explicit formula for the potential, the formal Hamil tonian (formal sum)
(6)
is used.
Remark 2. In many cases, the free measures I'~ are restrictions of some probability measure 1'0 defined on sT to the respective IT-algebras EA C ~. In such cases, instead of a Gibbs modification I'A defined by the formula (1), the measure fJ.A given on the IT-algebra ~(sT) by
ddfJ.A(z) = ZA'lexp{-UA(Z)} 1'0
(7)
is investigated. The measure fJ.A is a "natural" extension of the measure itA to the whole IT-algebra ~(ST). This measure is also called a finite (i.e., given in a finite volume) Gibbs modification of the measure 1'0. By virtue of Remark 1, a limit Gibbs measure I' on the space ST (i.e., the weak limit of the measures itA) is the weak limit of the measures fJ.A, A / T.
(2) Gibbs modifications of point fields. Let A C Q be a domain in Q (cf. (2) in Section 1), OS(A,n) C (A x s)n/fIn be the set of sequences of pairs
factorized with respect to the group fIn of permutations of n elements (two sequences (7') are considered to be equivalent if one arises from the other by
means of some permutation). In this way, OS(A, n) is endowed with a metriz able topology. We use the notation OS(A) = U~=o OS(A, n), OS(A,O) = 0, and we introduce on OS (A) the topology of the direct sum of topological spaces. By means of the restriction mapping
CPA: (z,sz) 1-+ (znA,szlznA) E OS(A), (7") ,
where A is a bounded domain in Q, the topology on OS is defined as the weakest topology making all mappings CPA continuous. We define, for any bounded domain A C Q, the sub-oo-algebra of the Borel oo-algebra ~(OS) by
The family of local oo-algebras EA generates the whole Borel oo-algebra ~(OS), and the set Co(OS) of bounded continuous local functions is dense everywhere in C(OS) (due to Stone's Theorem; cf. Remark 1).
Poisson field. In the role of the free measure 1'0 on Os, the distribu tion of the s<rcalled labelled Poisson field in Q is chosen. It is defined in the following way. Let a positive oo-finite (or finite) measure d>.o such that >'o(A) < 00 for each bounded domain, A be given on the space Q, and let a probability measure ds be given on the space S. The measure (d>.o x ds)n, defined on the space (Q x s)n, induces, on the space OS (Q, n) == O~, the factor-measure
dVn = (d>'o x ds)n In!, n > 0, vo(0) = 1. (8)
We shall consider a measure v on the space O~n = Un~oO~ of finite configu rations in Q, coinciding on each set O~ with the measure Vn , n = 0, 1, ....
N ow let A C Q be a bounded domain and I'~ be a probability measure on OS(A) equal to
I'~ = e->'o(A)v (9)
(since OS(A) C ogn' the measure v is defined on the space OS(A), too). Notice that 1'~(OS(A,n», i.e., the probability of the occurrence of exactly n points of the labelled field in A (with arbitrary values of charges), equals >'8 (A)e->'o(A) In!. Each measure I'~ can be considered as defined on the 00-
algebra EA. Furthermore, one may easily verify that there is a unique measure J.t0 on the space OS such that its restrictions to sub-oo-algebras EA coincide with the measures J.t~. The labelled point field in Q generated by this measure is called the Poisson field with independent charges.
Any function ~[(z, sz)] defined on the set O~n of finite configurations (x, sz) is called a potential. For each bounded domain A C Q, we put
UA[(Z,Sz)] = :E ~[(y,s!l)]' !I~znA
with s" = sill" being the restriction of the function Sll to y C z. The Gibbs modification I'A of the Poisson field 1'0 is defined by the density
(10)
Primarily, the case of the proper point field in the space R" is investigated. At the same time, the Poisson measure 1'0 is defined with the help of the Lebesgue measure d>.o = d" z on R", and the energies U A are defined by means of a two-point (or two-particle) translation-invariant potential 41, i.e.,
(11)
where jJ. E Rl (so-called chemical potential), cp is an even function defined on the space R", and (J > O.
We formulate here one stability condition for Z A in the case of a two-particle potential of the form (11).
Theorem 2. Let cp be a real even upper semi-continuous function on R". Then the following conditions are equivalent:
(a) the inequality n n
LLCP(qi - qj) ~ 0 i=1 ;=1
is fulfilled for any nand qi E R", i = 1, ... ,n; (b) there is a B ~ 0 such that
for any z E Ofln and A C R". (c) the partition functions ZA are finite for all bounded domains A.
(12)
(13)
For proof see [60]. An example when the condition (a) implies the stability of Z A is the case of positive definite function cp( q), q E R".
(3) Gibbs modifications of measures on function spaces. Let 0 be some locally convex space of functions z(t) = {Z1(t), ... ,zn(t)}, t E R", defined on the space R", with values in Rn.
Suppose that the topology on 0 is such that the functionals of the form Fto(z) = Z.l:(to), to E R", k = 1,2, ... ,n, are continuous with respect to it (i.e., the convergence of a sequence of functions in 0 implies their pointwise convergence). For each bounded open or closed set A C R", we define the u-algebra EA to be the smallest sub-u-algebra of the Borel u-algebra 2J(O) making all the functionals {Pt o• to E A} measurable. Suppose that the family of u-algebras EA is generating for the u-algebra 2J(O).
We suppose that a probability measure Jlo (free measure) is defined on the Borel u-algebra !B(n), and a functional U A(Z) is given to each bounded open or closed set A C R", so that:
1) UA = 0 if IAI = 0, where IAI is the Lebesgue measure of Aj 2) UA1UA2 = UA 1 + UA2 if IAl nA21 = OJ 3) UA is EA-measurable. A family {UA} of functionals fulfilling conditions 1), 2), and 3) is called a
local additive functional. Suppose that the stability condition
0< 10 exp{-UA(z)}dJlO < 00 (14)
is fulfilled for each bounded domain A C R" and define a Gibbs modification JlA of the measure Jlo with the help of the formula (7). In what follows, a limit Gibbs modification of the measure Jlo is defined as before.
Here is a typical example of a local additive functional (in the case when the space n contains only smooth locally bounded functions z(t)):
t = (t(l), ... ,t(II»),
with ~ a real function of n( v + 1) variables that is bounded from below.
Remark 1. In some cases, local additive functionals may also be defined on the space of distributions (called the Schwartz space V'(R")) such that they fulfill the stability condition (14) (Jlo is a probability measure on V'(R)), and the Gibbs modifications JlII and the limit Gibbs modification J.t may be defined with the help of them.
Remark 2. We have examined Gibbs modifications of measures on function spaces with the help of additive local functionals in such detail only because this case covers most of the known examples of such modifications. Of course, it is also possible to investigate nonlocal functionals U A, e.g., functionals of the form
UA(z) = 11 ~[z(t),z(t')]d"td"t', AA
with ~ being a bounded real function of 2n variables. Discretization. The discretization method is used when applying the theory
of cluster expansions to functional fields. By discretization a reduction to a field in a countable set is to be understood. Namely, the space R" is broken up into congruent cubes, and the field in Z" (the centers of those cubes), with values in the space S of spins coinciding with some space of functions defined on a cube, is investigated. At the same time, the energy is rewritten in terms of the newly introduced field in Z" with values in S, and the free
measure of the original field determines a measure on SZ". Even though we do not investigate fields in this book, many statements may be immediately, and in an obvious way, transferred to the case of functional fields. Notice that no method of cluster expansion that would exclude discretization exists at present (except for the expansion in a series of semi-invariants; see Section 6.1V).
1.5. Weak Compactness of Measures. The Concept of Cluster Expansion
Let A be some collection of measures defined on the whole Borel (i-algebra ~(O) of a topological space 0 or on its sub-(i-algebra E C ~(O). By weak compactness of the set A, its sequential compactness is always understood, i.e., there is a weakly converging sequence J.l.n -+ J.I., n -+ 00, J.l.n E B, in any infinite subset B C A.
Lenuna 3. In the case of a complete separable metric space 0 and E = ~(O), each of the conditions stated below is sufficient for weak compactness of the set A.
1) Each measure J.l E A is a probability measure, and there is a compact function h > 0 defined on 0 such that
10 h(x)dJ.l < C
for any measure J.l E A where C does not depend on J.l. A function h on 0 is called compact if the set {x E 0, h(x) < a} is compact for any a > O.
2) There are a nonnegative measure J.lo on ~(O) and a J.lo-integrable func tion 't'(x) ~ 0 such that any measure J.l E A is absolutely continuous with respect to J.lo and
x EO.
The statement 1) is a simple corollary of the known weak compactness criterion of Prokhorov (cf. [6]). A derivation of the statement 2) can be found in [16].
Definition 3. Let {J.lA, A E F} be a family of measures, each of which is defined on the (i-algebra EA from a complete family {EA' A E F} of sub (i-algebras of the (i-algebra ~(O) (here, as above, :F is a directed family of indices). A family {J.lA, A E :F} is called weakly locally compact if the set {J.l~o, Ao < A} of restrictions of measures {J.lA} to the (i-algebra EAo is weakly compact for any Ao E:F.
Lenuna 4. Let a family {J.lA, A E :F} of measures be locally compact. Then, in any increasing sequence Ai < A2 < ... < An < ... of indices ensur- ing that the sequence of (i-algebras EAR' n = 1,2, ... , is complete, there
is a subsequence possessing the same property and a cylinder measure J.l on ~ = UEA such that
(J.ln = J.lA,.). (15)
The proof is obvious. Now we shall introduce the concept of cluster expansion of measures in a
way applicable to the case of fields in a countable set T (with values in a space S).
Let G c Co(sT) be some set of bounded continuous local functions whose linear hull is dense everywhere in the space C(sT) of all bounded continuous functions. Let the mean {F)II of an arbitrary function F E G under a measure (or cylinder measure) I' be expanded in the form
(F)II = (16) RCT,IRI<oo
with bR(F) being some quantities depending on F and finite subsets ReT. Every such expansion is usually called a cluster expansion of the measure 1'. Of course, the extent of "constructiveness" of definitions of bR(F) determines the significance of such an expansion. In the following, we shall construct the expansions (16) explicitly each time, and the statement that this or that measure admits a cluster expansion is to be understood only in the sense of (16). Definition 4. Let {J.lA, ACT} be a family of measures defined each on the o--algebra EA = ~(SA)(A C T, IAI < 00). The family {J.lA} is said to admit a cluster expansion if
1) it is weakly locally compact; 2) there is a set G C Co(sT) of bounded continuous functions whose linear
hull is dense everywhere in the space C(sT) such that the mean (F)IIA == (F)A of any function F E G admits an expansion
(F)A = I: b~)(F) R!;A
with the quantities b~)(F) fulfilling the following conditions: a) there is a majorant
b) there are limits
Lemma 5. Let a family {IlA} of measures admit a cluster expansion. Then the weak local limit
II = lim IlA A/T
(20)
In the case of probability measures {IlA}, the cylinder measure II is also a probability, hence it can be extended to a probability measure on the tT-algebra ~(n).
Proo£ The condition 1) and Lemma 4 imply the existence of at least one weak limit point of the set {IlA}. Its uniqueness, and (20), follow from the condition 2). The cluster expansion (16) of the limit measure is obvious.
The cluster expansion of measures in the case of point fields is defined in Section 6.3.
§2 Gibbs Modifications under Boundary COIlditions and Definition of Gibbs Fields by Means of Conditional Distributions
The definition of limit Gibbs modifications presented above does not include all interesting cases of fields of that kind, and we offer a more general definition of the limit Gibbs field. We confine ourselves only to the case of fields in a countable set T (on which a metric p is given) with values in a (metric) space S. Further, we suppose that a finite or tT-finite measure is defined on S, and we choose, for any finite set ACT, the measure Il~ = A~, i.e., the product of IAI copies of the measure AO, as the free measure Il~ on the space SA.
Finally, we suppose that we are given a potential {ciA; ACT, IAI < oo} of a finite range, i.e., ciA == 0 if diamA == max'I,'2EA p(it, t2) > d for some constant d > 0, and that the Hamiltonian U A = EACA ciA determined by it fulfills the stability condition -
for any finite ACT. Let IlA be a Gibbs modification of the measure A~, and for any Ao C A, we use Il~O(IiA\Ao) to denote the conditional probability distribution on the set of configurations zAo E SAo under the condition that a configuration iA\Ao E SA\Ao, in the set A\Ao, is fixed. An elementary com putation (cf. Section 0) shows that the density of the measure ll~oUiA\Ao) with respect to the measure A~o equals
(1)
with
ZAo (iA\AO) = lAo exp {-UAo (zAo /iA\Ao)} d"~O,
UAo (zAoliA\Ao) = UAo (zAo) + L ~A (zAo UiA\Ao). (1') A:AnAo¢'
An(A\Ao}¢'
Here, zAo UiA\Ao denotes the configuration in A whose restrictions to Ao and A \Ao are equal to zAo and i A \Ao, respectively. The second summand (1') is called the energy of the interaction with an external (boundary) configuration ("the boundary term").
Notice that, for a fixed Ao and a sufficiently large A :::> Ao (so that p(Ao, T\A) > d), the energy UAo(zAoliA\Ao) does not depend on the whole configuration i A \Ao, but only on its restriction i 8dAo to the d-neighbourhood of Ao, i.e., 8dAO = {t E T\Ao, pet, Ao) ~ d}.
We denote this energy by
UAo (zAo/ i 8dAo), (I")
and let l'~g4AO denote the Gibbs modification of the measure ,,~o by means of
the Hamiltonian (I"). The measure l'~gdAo is called the Gibbs distribution on Ao with the boundary configuration i 8dAo in the neighbourhood 8dAO'
The formula (1) suggests the following: Definition. A probability measure I' on the space SF is called a Gibbs dis tribution in T (for a given potential {~A}) if, for any finite ACT and any configuration i E SF\A, the conditional distribution 1'(./ zT\A = i) on the set SA coincides, under the condition that the external configuration zT\A is fixed and equal to i, with the measure I'~BdA given by
I' (/zT\A = i) = I'~BdA' (2)
with i 84A being a restriction of i to 8dA. This definition is due to Dobrushin, Lanford, and Ruelle (DLR), and equa
tion (2) representing it is sometimes called the DLR equation. From the definition (2), in particular, the so-called d-Markov property of
a Gibbs measure JJ follows. Namely, the conditional measure JJ(./xT\A = i) depends only on the values of the configuration i in the set 8dA for arbitrary ACT and i E SF\A.
Let ACT be a finite set and let some probability distribution q = q8d A on the set S8dA of boundary configurations i = i 8dA be given. The measure
JJ~ = J I'~dq(i) (3)
SBd A
on SA is called a Gibbs distribution with a q-random boundary configuration in A.
Proposition 1. For a measure p on the space ST to be Gibbsian, it is nec essary that, for any increasing sequence An / T, n -+ 00, of finite sets An, there is a sequence of distributions qn = q8d A .. defined each on the set S8d A .. of boundary configurations, so that the weak local limit of measures p~,," coincides withp, i.e.,
lim pA .. = p, (4) n-+oo fa
and it is sufficient that the condition (4) is fulfilled for some increasing se quence An / T.
Corollary. Let a family {p~} of Gibbs modifications be such that there is a unique limit
/L = lim /LA r A/Tr$
for any sequence A / T and any choice of boundary configurations x E S8d A.
Then p is the unique Gibbs measure on sT. Proof of Proposition 1. Necessity. We choose a distribution q = plS8dA on the set S8d A induced by the measure p for any ACT. Then it is obvious that P~ = pISA' and (4) is fulfilled. Sufficiency. For any Ao C A such that p(Ao, T-A) > d and any distribution q of boundary configurations x E S8d A, the conditional distribution p~UxA\Ao) on SAo generated by the Gibbs measure P~ on A with random boundary configuration coincides (as is easily seen from (1) and (3)) with the measure P~gdA' where x8dAo is the restriction of the configuration xA\Ao to odAo, i.e.,
P~ (-/xA\Ao) = P~gdAO' (5)
Now let An / T and qn be sequences such that (4) is fulfilled. Since the equality (5) is satisfied for any fixed Ao C T and all sufficiently large An, it is also valid for the limit measure p.
Remark 1. A Hamiltonian UA of the form (5.1), and the Gibbs modification PA of the measure )..~ generated by it, are sometimes called, for uniformity, the energy (and, accordingly, the Gibbs measure) with the "empty" boundary conditions. Moreover, their weak local limit, the limit Gibbs modification, is the Gibbs distribution in the sense of the definition (2) as easily follows from the proved proposition.
Remark 2. Definition (2), presented here for the case of a finite-range po tential and the free measure p~ = )..~, can be applied as well (with some refinements) to the case of an arbitrary "rapidly decreasing" potential {~A}' and also to the case of any d-Markov free measure pO (or even a measure J-l 0
with a rapid "decrease of memory"). In addition, a definition of the Gibbs field in an infinite space, analogous to the definition (2), is possible both for the case of point fields and for the case of fields in RV(with ordinary or generalized configurations). However, they are rarely studied at present.
CHAPTER 2
§1 Semi-Invariants and Their Elementary Properties
Let a family {6, ... ,en} of random variables be given (repeated occurrence of identical ones among them is not excluded). In the following, we shall suppose that all moments of these variables are finite:
(Iejl) < 00 (1)
for each j and k = 1,2, .... For any nonempty subset TeN = Nn = {I, ... , n}, we use the notation
eT = II ei, eT = {ei : i E T}, iET
i.e., eT is a subsystem of the system {6, ... ,en}. 1fT = 0, we put e. = 1 and introduce the symbol e •. For any random variable e, we denote the system of k samples of the random variable e by {k, and {o = ee. An analogous meaning is given to the notation
n
elK. _ {elkl elk" } 'oN - '01 , ••. ''on ,
with K. = (k1' ... , kn ), ki ~ 0, a multiindex. We shall consider the function
defined for all purely imaginary A1, ... , An. It coincides with the characteristic function of the family {6, ... ,en} with real variables tj = Aj/i.
27
28 Semi-Invariants and Combinatorics Chapter 2
Lenuna 1. The function f(>"1, ... , An) is infinitely differentiable for all purely imaginary Aj. Mixed moments ({,}) exist for an arbitrary multiindex K.. More- over,
Ok 1 + .. ·+k" I (eN) = k k f(A1,"" An)
OA11 •• , OAn" A1=A2=.,,=A .. =0 (2)
(we put 00 /oA8f = f).
The proof can be found in [58]. It follows from Lemma 1 that In f is an infinitely differentiable function in
a neighbourhood of the point (0, ... ,0).
Definition 1. The number
(3)
is called the semi-invariant of the family {6, ... , en} of random variables. The abbreviated notation for it is (e~). Accordingly, the semi-invariant of the family e::- is denoted by
{ell'} {elk1 elk,,} "/If = "1 , •.. , "n .
In addition, let us agree that
Properties of semi-invariants. A.
We have
with Ai = E:~1 Aij for ki > 0 and Ai = 0 for ki = O. Using the definition (2) we get (4). B. Symmetry and multilinearity.
for any permutation
(4)
and
(a'e~ + a"er,6, ... ,en) = a'(e~,6, ... ,en) + a" (er, 6, ... ,en)'
The symmetry is obvious and the multilinearity follows from the equality
(exp{Al (a'e~ + a"en + A2e2 + ... + Anen}) = l(a'AI, a" Al, A2,' .. , An)
by a direct application of (3). Here, I(A~, A~, A2,' .. , An) is the characteristic function of the family {e~, ef ,6, ... , en }.
C. If N can be split up into two nonempty nonintersecting subsets, N = A U B, so that the family e~ does not depend on e~, then
(6,· .. ,en) = o. (5)
Proof. For AI, ... , An sufficiently small, we have
In(exp(AI6 + ... + Anen) = In ( exp (~Aiei) ) + In ( exp (~Aiei) ) .
By applying the definition (3), we get (5).
In particular, even if only one variable ej = const., the semi-invariant (el,'" ,en) equals zero.
Property C suggests that the semi-invariant characterizes the degree of de pendence of random variables. Notice also that, for n = 2, the semi-invariant coincides with the covariance
as easily follows from the definition. D. Expression 01 moments by means 01 semi-invariants.
(6) a={T1 , •.• ,T,,}
with the sum taken over all partitions of the set N, i.e., over all unordered collections a = {T1 , •.• , Tie} of mutually disjoint non empty subsets (blocks) 11 c N such that N is their union.
Proof. Consider formal Taylor seriesl for functions 1 and In 1 in the point Al = ... = An = 0:
(7)
with the sum taken over all multiindices "; here,
,,! = k1!. .. kn!, AN- = A~l . .. A~".
(8)
We fix k and compute the coefficient of A1 ... An in the formal series of the k-th summand of the right-hand side of (8). It is easy to see that this is the sum of expressions of the form
over all partitions {Tl, ... , Tk}. E. Expression of semi-invariants by means of moments.
(9) or
with the summation taken over partitions a = (T1 , ••• , Tk ) of the set N. The proof can be carried out analogously to the proof of property D. It will
be proved below as a particular case of the Mobius inversion formula. F. Characterization of Gaussian families. A system is Gaussian if and only
if (ei1, ... ,ei.)=O, s>2,
for any (possibly coinciding) indices i1 , ••• ,i3 EN. This property follows from the fact that the logarithm of the characteristic
function of Gaussian families (and only of such) is a quadratic polynomial. The set of all partitions of N is denoted by ~N =~. We define in ~ an
ordering by putting a < 13 if each element of the partition a is contained in some element of the partition 13 (a "refines" 13). We denote the partition to points (the minimal partition) by Q, and the partition consisting of the only element N (the maximal partition) by 1. The smallest partition 6 E ~ such that a < 6 and 13 < 6 is denoted by a V 13 (for a more explicit account on the lattice of partitions, see Section 6).
A partition 13 is called connected with respect to the partition a if a V 13 = 1. This obviously means that N cannot be split up into two subsets, each being a union of elements of 13 as well as of a.
G. Generalized expansion over connected groups. For any partition a = {T1 , ••• , Td E ~N' it is
(eTw .. ,eTk) = l: (eQl)·· . (eQm) (10) (j:orV(j=1
§1 Properties of Semi-Invariants 31
with the sum taken over partitions {3 = {Ql, ... , Qm} connected with respect to a.
Proof. Consider first the particular case when ITll = ... = ITII-il = 1, ITIII = 2. In other words, we shall prove that (n = k + 1)
(6, ... ,en-2,en-l . en)
= (6, ... ,en) + E (en-l,eT}(en,eN .. _2\T)· T!;;{1, ... ,n-2}=,N' .. _2 (11)
We shall proceed by induction on n. Denoting TJ = en-I' en and using the property D, we have
(ell, .. en) = (ell, .. , en) + E (eTJ ... (eT,.>, {T1 ..... T.}
1I~2
(6 ... en-2TJ) = (6, ... , en-2, TJ)
+ E [(e~J ... (e~.,TJ}+(e~l} .. ·(e~.}(TJ)] {Sl ..... S.}
1I~2
with {11} a partition of Nn and {Sd a partition of Nn - 2 • Hence,
(6, ... , en-2, TJ) - (ell, .. , en)
= E [(eh} .. · (eh>l - E [(e~l}'" (e~.,TJ) + (es.}(TJ)] .
Since ISIII < n - 2, we infer from the induction hypothesis that
(e~., TJ) = (eS",en-lIen) + E (e~"en-l}(e~lI,en). S'US"=S"
Moreover, (TJ) = (en-l,en) + (en-l}(en)'
(12)
Inserting the last two formulas into (12), we can easily convince ourselves that only the terms
will remain on the right-hand side of (12).
We pass to the general case. Let the statement (10) be proved for all partitions of the sets Nil with a number of elements less than n and for all partitions a' E ~,N' less than the partition a. We shall prove its validity for the partition a. Let n E Til and
Using (11) we get
+ E ('1s,en)('1Nlr _1\S,eT,,\{n})' Sr;,N"_l (12a)
By the induction hypothesis, each semi-invariant on the right-hand side can be expanded over connected groups. Inserting these expansions into (12a), we may convince ourselves that one gets a sum of terms of the form (eQ1 ) ... (eQm ) for each partition {3 connected with respect to a.
H. Cauchy inequalities. Suppose that the function (exp(Alel + ... + Anen) may be analytically continued to a closed polydisc IAII ( rl,···, IAn I ( rn and differs from zero on it. Denote r = {AI,'" ,An; IAil = ri, i = 1, ... , n}. Then
(13)
with e = SUPr Iln(exp(Alel + ... + Anen)} I. This property is a reformulation of the Cauchy inequality (cf. [66]) and
can be derived from the integral Cauchy representation of a function which is analytic in a polydisc.
The conditions of Proposition H are clearly fulfilled in the case when all ei are bounded.
Even for one random variable, the estimate (13) cannot, in general, be improved: it is not difficult to find examples so that 1({n)1 ~ en . n! for any constant e and infinitely many values of n.
Remark. In many problems of probability theory and mathematical sta- tis tics, an important role is played by asymptotic estimates of semi-invariants (e?l, ... , e~k .. ) for fixed nand kl , ... , kn -+ 00. These estimates are usu ally obtained by standard methods of the asymptotic analysis and theory of functions of several complex variables. Notice that the specific character of estimates obtained in this book is related to the case n -+ 00.
I. Expression of semi-invariants in the form of moments. This useful device consists of the following. Let a family {6, ... , en} of random vari ables be given. We consider n independent collections of random variables: {eP), ... ,ell)}, ... ,{e~n), ... ,eln)}, each of them having the same distribution as the system {6, ... ,en}' Let w = exp{211"i/n} and ('" = L:i=l wje~). Then
1 - - (eI. .. ·,en) = -(el ... en).
n (14)
c(;) ~ c(;+I) . 1 1 .. '" ~"1: , J = , ... , n - ,
c(n) c(1) .. '" 1-+ .. ",
§1 Properties of Semi-Invariants 33
preserves the probability distributions, and hence the families (el,"" en), (WeI,'" ,wen) have the same distribution. It follows from this, in particular, that
for TeN, and thus
for T if; N. From this and property D, one may conclude that
(15)
Properties Band C imply that (eN) = n{eN), and this, together with (15), gives (14).
J. Formal semi-invariants. Let for each TeN a number f(T) be given. These numbers will be called (mixed) moments of a virtual field (on N).
Define the semi-invariants g(T), TeN of a virtual field by the inductive formula
g(T) = f(T) for ITI = 1
and
g(T) = f(T) - I: g(Bt) .. , 9(Bk) for ITI > 1, (16) {B1 •...• Bk}
with the sum taken over all partitions {Bt, ... , Bk} of the set T, with k ~ 2. Notice that formula (16) coincides exactly with formula (6). For the numbers g(T) the formula
(16') a
analogous to formula (9) (see the property E), is valid with the sum taken over all partitions a = {TI , . •. , Tk} of the set N. Formula (16'), as well as formula (9), can be obtained with the help of the comparison of the expansions of the generating functions
and
with
We shall need an analogy of the suitably reformulated property C. Let N be the set of vertices of some graph G. We denote the subgraph of the graph G spanned by a set A of vertices by GA for A eN. For B C A we call the graph G B a component of G A if, in G A, there are no edges connecting B with A\B.
A system (virtual field) U(T), TeN} is called independent with respect to the graph G if
(17)
for any A eN and all partitions {Al' A 2 } of the set A such that GA" i = 1,2, are components of GA.
Lenuna 2. If the graph G is not connected and the system f(T) is independent with respect to G, then
g(N) = O. (18)
The proof is carried out by induction on n. Let { = {Al' A 2 } be a partition of N such that GAl and GA~ are components of the graph G = GN ({ exists since the graph is disconnected). From (16) we have
g(N) = f(N) - E~ .. - E~~., (19) where we sum over all a < { in E' and over a :/: 1 in E". All the summands in E" equal 0 according to the induction hypothesis (since in each such a there is an element intersecting Al and A 2 ). At the same time, according to (17),
E' = f(At}f(A 2),
which implies the lemma. K. Semi-invariants and modifications of measures. Let (0, E, Jlo) be a prob
ability space, {l, ... ,{k be random variables with finite moments, and U be a bounded random variable. Consider a new measure (the Gibbs modification of Jlo)
(20)
with z a real parameter, and XA the indicator of the set A. Then the semi invariant (6, ... ,6)1' evaluated with respect to the measure Jl is an analytic function of z in a small neighbourhood of zero in the z-plane, and
00 n
({l, ... ,{k)j.1 = L ;(6, ... ,{k, UIn )j.Io· (21) n:O n.
Proof.
({l, ... ,{k)j.1 = O~l ~k. O~k In(exp{Al6 + ... + Ak{d)j.Il>'l: ... :>'k:O
= O~l ~k. O~k In(exp{Al6 + ... + ~k{k + zU})j.Io l>'l: ... =>'k:O•
Expanding the last logarithm in z, we obtain (21).
§2 H ermite-/to-Wick Polynomials. Diagrams. Integration by Parts 35
§2 Hermite-Ita-Wick Polynomials. Diagrams. Integration by Parts
1. Wick polynomials
Let e be a Gaussian random variable defined on some probability space (0, E, Jl). The polynomials in e (for brevity called Wick polynomials2 and denoted by the notation : : introduced by him) arise by orthogonalizing the system of monomials l,e,e2, ... in L2(0,E,Jl). Definition 1 (Wick exponential Cunction). We put
( def exp(ae) {( () 1 2( 2)} :exp ae): = (exp(ae») = exp a e - e - 2"a e , (1)
with a being an arbitrary complex number. Expanding the right-hand side of (1) in a (converging) power series in a, we denote the coefficient of an In! by :en :, i.e.,
(2)
It is easy to see that :en : is a polynomial of the n-th degree in e with the leading coefficient 1, i.e.,
(3)
with Q being a polynomial of the degree not higher than n-l. The polynomial :en : is called the Wick power. For example,
If (e) = 0, then
:e: = e - (e), :e: = e - 6e(e}, :e4 : = e4 - 6e(e} + 3«(e})2. (4)
For any polynomial p(e) = an en + ... + ao in the random variable e, we define the polynomial :p(e): by the linear relation
We emphasize that the operation P 1-+ :P: depends on the distribution of e (more exactly, on (e) and (e».
In the following, we shall consider the case of (e) = O.
2These polynomials, as we shall see below, coincide with the known Hennite polynomials

gibbs random fields: cluster expansions

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