Gibbs Measures and Phase Transitions, Second Edition

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

Hans-Otto Georgii

Gibbs Measures and Phase Transitions Second Edition

De Gruyter

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

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

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

To my family

Preface

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

VIII

Preface

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

Preface

IX

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

X

Preface

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

Hans-Otto Georgii

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

Hans-Otto Georgii

Contents

Introduction

1

Part I.

9

General theory and basic examples

Chapter 1 Specifications of random fields 1.1 1.2 1.3

Preliminaries Prescribing conditional probabilities ^-specifications

Chapter 2 2.1 2.2 2.3 2.4

Gibbsian specifications

Potentials Quasilocality Gibbs representation of pre-modifications Equivalence of potentials

Chapter 3

Finite state Markov chains as Gibbs measures

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

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

Chapter 5 5.1 5.2

Specifications with symmetries

Transformations of specifications Gibbs measures with symmetries

Chapter 6 6.1

The existence problem

Three examples of symmetry breaking

Inhomogeneous Ising chains

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

XII

Contents

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

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

Chapter 8 8.1 8.2 8.3

Uniqueness

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

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

Discrete symmetries in one dimension Continuous symmetries in two dimensions

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

Part II. Markov chains and Gauss fields as Gibbs measures

189

Chapter 10 Markov fields on the integers I

190

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

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

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

Contents

XIII

Part III. Shift-invariant Gibbs measures

289

Chapter 14 Ergodicity

290

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

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

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

Phase transitions in reflection positive models

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

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

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

383 394 398

Chapter 19 Phase transitions without symmetry breaking

408

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

Continuous symmetry breaking in N-vector models

Some preliminaries Spin wave analysis, and spontaneous magnetization

408 415 420 431 433 434 439

XIV

Contents

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

453 489 495 532 539 541

Introduction

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

2

Introduction

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

Introduction

3

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

H(co) = -

X

JiUficOiCOj -

^

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

n(dm) = Z1 exp [ - jBH (tö)] dœ

4

Introduction

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

Introduction

(0.3)

5

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

Here «A(^)=-

£

J(i,Mitj-Ytti\h+

ji,j)cA

ieA

X 7(U)»/; , [_

J'ES\A

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

fe£A

exp[-j8ff A (&)]

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

6

Introduction

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

Introduction

7

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

plane rotor

discrete Gaussian

Gaussian

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

{-1,1}

S1

Z

U

d= 1 d=2 d>3

1 2 oo

1 1?

0 00

00

00

0 0 oo

Parti General theory and basic examples

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

10

Part I

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

3

4-

•*•

5^9

1

6

Chapter 1 Specifications of random fields

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

1.1

Preliminaries

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

12

Specifications of random fields

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

Q = Es = {m =

(mi)ieS:mieE},

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

& = Ss,

the product E CO - » C O ; ,

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

0(0, &).

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

aA: Q -> EA

denote the projection onto the coordinates in A. Thus ). A subset Sf0 of the index set Sf (which is directed by inclusion) is called cofinal if each A 6 if is contained in some A e y o . For example, if S = Zd for some d ^ 1 then the set ^o = { [ - n , n ] " n S : n ^ 1} of all centered cubes is cofinal. (1.24) Remark. Suppose y is a specification and ^ a random field. Then the following statements are equivalent: (a) ^ e <S(y). (b) [iyA = ii for all A e Sf.

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

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

18

Specifications of random fields

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

1.3

A-specifications

Having introduced the notion of a specification we need to establish that specifications do exist. This section contains some first steps in this direction. Our first remark below will show that it is easy to construct certain trivial specifications. After that we shall discuss the problem of whether non-trivial specifications can be obtained from the trivial ones by means of a suitable perturbation procedure. The above-mentioned trivial specifications are those which specify the independent identically distributed random fields. In the "canonical" setting that was adopted from (1.2) to (1.4), the i.i.d. random fields are nothing but the product measures Xs with X e 0>(E, S). The remark below states that for any given A e SP{E, S) the standard versions of the conditional probabilities Xs (• \STA\ A e £f, constitute a specification. (1.25) Remark. Let X e 0>{E, S) be given. Define XA(A\œ) = XAx Ô^JA) = X*(Ç e £ A : Oo svv e A). Here 0 # A ) which is made up by the energy contributions of all subsystems. These contributions are described by a potential . O^(co) is the excess energy of subsystem A when the energy of all proper subsystems of A is disregarded. The number Z > 0 above is a normalizing factor. The coefficient ß > 0 is related to the absolute temperature T, in that ß = 1/kT, where k is Boltzmann's constant. From a mathematical point of view, there is no loss in setting k = 1. We shall do so throughout this book. In fact, when the temperature is held fixed we can and shall incorporate the factor ß into the Hamiltonian. This amounts to setting ß = 1. As stated before, the Boltzmann-Gibbs distribution does not admit a direct extension to infinite systems. However, when dealing with infinite systems we can still look at finite subsystems provided the "rest of the world" is held fixed. Indeed, starting from a potential we can define for each finite subsystem A a Hamiltonian H® which includes the interaction of A with its fixed environment. The associated Boltzmann-Gibbs distribution can be viewed as a probability kernel which maps each environment to the corresponding equilibrium distribution of subsystem A. The collection of all these kernels constitutes a specification, the Gibbsian specification for O. (Historically, it was this fact which gave rise to the notion of a specification.) Let us turn to the details. As in Chapter 1, we are given a measurable space (£, $), the state space, and a countably infinite set S, the set of sites. We look at the product space (Q,&) = {E,é)s. As in (1.7) we let y be the countably infinite set of all non-empty finite subsets of S. We shall often be concerned with infinite series which are indexed by an infinite subset of 5f. (2.1) Convention. Suppose t?0 is an infinite subset of y and let a: ^0 -> U be any function. We shall say that the infinite sum

exists and equals seU, and we shall write

s = X a ( A )' Ae^o

if the net I

£

\,Aey0,AcA

a(A) I /Aey

converges to s.

Potentials

27

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

HX(a>)=

£

OA(