##
**Gibbs measures and phase transitions.**
*(English)*
Zbl 0657.60122

De Gruyter Studies in Mathematics, 9. Berlin etc.: de Gruyter. xiv, 525 p. DM 178.00 (1988).

This book is much more than an introduction to the subject of its title. It covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and as an up to date reference in its chosen topics it is a work of outstanding scholarship. It is in fact one of the author’s stated aims that this comprehensive monograph should serve both as an introductory text and as a reference for the expert. In its latter function it informs the reader about the state of the art in several directions. It is introductory in the sense that it does not assume any prior knowledge of statistical mechanics and is accessible to a general readership of mathematicians with a basic knowledge of measure theory and probability. As such it should contribute considerably to the further growth of the already lively interest in statistical mechanics on the part of probabilists and other mathematicians.

Before reviewing the contents it may be worth explaining what the book is and what it does not purport to be or do. It is a mathematician’s rather than a physicist’s book. It treats the rigorous mathematical theory of lattice models of classical equilibrium statistical mechanics, i.e. lattice-indexed systems of interacting random spins in statistical equilibrium. It does not deal with non-equilibrium systems or systems evolving in time, particle systems in a continuum, quantum systems or systems with random interactions. Ferromagnetic correlation inequalities feature essentially only in the bibliographical notes, as does the Pirogov-Sinai theory of phase diagrams. However, the topics covered are dealt with systematically, with the development progressing from basics to quite advanced and fascinating material in several subareas.

The book is divided into four parts. Part I (Chapters 1 to 9) deals with the basic theory. The first chapter presents the fundamental concept, going back to Dobrushin, Lanford and Ruelle, of a specification, i.e. of a consistent system of conditional distributions for the spins inside bounded sets given the spins at all external sites, and of a Gibbs measure admitted by a specification. In practice, specifications encountered in this book are defined in terms of densities with respect to the powers of an underlying free measure on the space of spin values. Chapter 2 introduces the physically meaningful Gibbs specifications arising from interaction potentials and their Hamiltonians and defines phase transition as the existence of distinct Gibbs measures admitted by a given specification. Various types of potentials are discussed here and it is also shown that Gibbs specifications are broadly representative. Chapter 3 discusses stationary finite state Markov chains as Gibbs measures - including the one-dimensional Ising model - while the fourth chapter deals with the important question of the existence of Gibbs measures admitted by a given specification, and of conditions under which the former can be obtained as thermodynamic limits of spin systems on finite lattices, subject to appropriate boundary conditions.

To prepare the ground for the later discussion of symmetry breaking Chapter 5 deals with the subject of symmetry preservation: Gibbs measures inheriting the symmetries of the underlying specifications. Chapter 6 presents three examples which exhibit phase transition through symmetry breaking: an inhomogeneous one-dimensional Ising chain breaking the spin- flip symmetry, the Ising ferromagnet in two dimensions, which is shown via the famous Peierls contour argument to break the same symmetry, and the two-dimensional discrete Gaussian model, shown to break the translation invariance. The role of ground state degeneracy (existence of distinct ground states for the potential) is brought out with clarity. Chapter 7 is a mini reference work in its own right. It is devoted to the theory of extremal Gibbs measures and their role in extreme point decompositions. The chapter is self-contained in the sense that it includes a presentation (with proofs) of the required facts from extreme point theory and zero-one laws, including an account of Dynkin’s approach to extreme point decompositions. Chapter 8 presents two standard conditions for the absence of phase transitions: Dobrushin’s “contraction” condition and, in the one-dimensional case, a well established condition on the total interaction between complementary half-lines, which in the case of a pair potential amounts to the familiar condition of faster-than-quadratic decay. Some of the more interesting consequences of Dobrushin’s condition (such as a correlation estimate) are also presented.

The main results of Chapter 9 establish the fact that in the presence of mild decay conditions, symmetries of one-dimensional potentials and continuous symmetries of two-dimensional potentials are necessarily inherited by all admitted Gibbs measures. In the case of dissipative symmetries this leaves the set of Gibbs measures empty. Several concrete examples are discussed, including Ising ferromagnets and antiferromagnets, Heisenberg models etc.

The first two chapters (10 and 11) of Part II treat the subject of Markov (“nearest neighbour”) specifications on the integers. If the set of spin values is infinite, not every admitted Gibbs measure (“Markov random field”) is a Markov chain. Broadly speaking, it is shown in Chapter 10 that extremal or translation invariant Markov random fields are Markov chains and that translation invariant fields are unique. For countably infinite sets of spin values, the Markov chains are characterised in Chapter 11 and criteria and examples given for the presence or absence of phase transition within various subclasses. This chapter also settles some related cardinality questions. Chapter 12 discusses phase transition for Markovian systems on trees, while Chapter 13 develops the theory of Gaussian fields as Gibbs measures.

Part III (Chapters 14 to 16) is devoted to translation invariant spin systems on \({\mathbb{Z}}^ d\). Chapter 14 deals systematically with ergodicity and Chapter 15 with the variational characterization of translation invariant Gibbs measures through their property of minimising the specific free energy. Entropy and pressure play an important role here. Chapter 16 establishes the convexity of pressure as a function of the potential and proceeds to geometric considerations relevant to the study of the phase diagram: the classification of potentials according to the number of pure phases (extreme points of the set of translation invariant Gibbs measures) they admit. Translation invariant random fields with finite specific entropy can be identified (via the specific energy functional) with hyperplanes tangent or asymptotic to the graph of the pressure; in the former case they are Gibbs measures for the potential corresponding to the point of contact. Other results centre on the existence of potentials exhibiting phase transitions with prescribed order parameters and the fact that potentials with non-unique pure phases are in some sense exceptional.

A common theme linking the chapters of Part IV is the inequality known as the chessboard estimate. This is proved in Chapter 17, leading to a refinement of the Cauchy-Schwarz inequality valid for rotation invariant and reflection positive random fields on discrete tori. (Reflection positivity is a condition of positive definiteness to do with the reflection symmetry). Chapter 18 takes up the Peierls argument once more. It is shown that for short range potentials on \({\mathbb{Z}}^ d\) with stable ground state degeneracies subject to appropriate symmetry conditions, Gibbs measures obtained as thermodynamic limits under periodic boundary conditions present a familiar percolation phenomenon at low temperatures: infinite clusters, resembling oceans surrounding small islands. The possible “states” of the oceans correspond to the different ground states, implying symmetry breaking. These considerations cover several familiar models. There is an extension of this, covered in Chapter 19, to potentials with ground state degeneracies not displaying symmetries, but this involves a perturbation of the potential. This extension is sufficiently close in spirit to the Pirogov-Sinai theory to provide the author with the opportunity of outlining very concisely the main result of this theory in the bibliographical notes to this chapter. The chapter also deals with the interesting phenomenon of phase transition arising at a particular value of the temperature, not from a ground state degeneracy but from the competition between the free distribution and the interaction energy superposed on it.

The Fourier analytic methods of the last (20th) chapter establish conditions for the breaking of rotational symmetries by vector spins on \({\mathbb{Z}}^ d\). The proof of the existence of long range order here hinges on controlling low frequency spatial spin waves by means of the infrared bound. The resulting criterion is applied to models such as the classical Heisenberg model and even the one-dimensional long-range model with pair potential displaying slower-than-quadratic power decay. (The case of quadratic decay is mentioned but not treated in the book).

The excellent and extensive bibliographical notes which follow are as valuable a feature of the book as the mathematical text, and there is a long list of references to the literature, including the very recent one.

Before reviewing the contents it may be worth explaining what the book is and what it does not purport to be or do. It is a mathematician’s rather than a physicist’s book. It treats the rigorous mathematical theory of lattice models of classical equilibrium statistical mechanics, i.e. lattice-indexed systems of interacting random spins in statistical equilibrium. It does not deal with non-equilibrium systems or systems evolving in time, particle systems in a continuum, quantum systems or systems with random interactions. Ferromagnetic correlation inequalities feature essentially only in the bibliographical notes, as does the Pirogov-Sinai theory of phase diagrams. However, the topics covered are dealt with systematically, with the development progressing from basics to quite advanced and fascinating material in several subareas.

The book is divided into four parts. Part I (Chapters 1 to 9) deals with the basic theory. The first chapter presents the fundamental concept, going back to Dobrushin, Lanford and Ruelle, of a specification, i.e. of a consistent system of conditional distributions for the spins inside bounded sets given the spins at all external sites, and of a Gibbs measure admitted by a specification. In practice, specifications encountered in this book are defined in terms of densities with respect to the powers of an underlying free measure on the space of spin values. Chapter 2 introduces the physically meaningful Gibbs specifications arising from interaction potentials and their Hamiltonians and defines phase transition as the existence of distinct Gibbs measures admitted by a given specification. Various types of potentials are discussed here and it is also shown that Gibbs specifications are broadly representative. Chapter 3 discusses stationary finite state Markov chains as Gibbs measures - including the one-dimensional Ising model - while the fourth chapter deals with the important question of the existence of Gibbs measures admitted by a given specification, and of conditions under which the former can be obtained as thermodynamic limits of spin systems on finite lattices, subject to appropriate boundary conditions.

To prepare the ground for the later discussion of symmetry breaking Chapter 5 deals with the subject of symmetry preservation: Gibbs measures inheriting the symmetries of the underlying specifications. Chapter 6 presents three examples which exhibit phase transition through symmetry breaking: an inhomogeneous one-dimensional Ising chain breaking the spin- flip symmetry, the Ising ferromagnet in two dimensions, which is shown via the famous Peierls contour argument to break the same symmetry, and the two-dimensional discrete Gaussian model, shown to break the translation invariance. The role of ground state degeneracy (existence of distinct ground states for the potential) is brought out with clarity. Chapter 7 is a mini reference work in its own right. It is devoted to the theory of extremal Gibbs measures and their role in extreme point decompositions. The chapter is self-contained in the sense that it includes a presentation (with proofs) of the required facts from extreme point theory and zero-one laws, including an account of Dynkin’s approach to extreme point decompositions. Chapter 8 presents two standard conditions for the absence of phase transitions: Dobrushin’s “contraction” condition and, in the one-dimensional case, a well established condition on the total interaction between complementary half-lines, which in the case of a pair potential amounts to the familiar condition of faster-than-quadratic decay. Some of the more interesting consequences of Dobrushin’s condition (such as a correlation estimate) are also presented.

The main results of Chapter 9 establish the fact that in the presence of mild decay conditions, symmetries of one-dimensional potentials and continuous symmetries of two-dimensional potentials are necessarily inherited by all admitted Gibbs measures. In the case of dissipative symmetries this leaves the set of Gibbs measures empty. Several concrete examples are discussed, including Ising ferromagnets and antiferromagnets, Heisenberg models etc.

The first two chapters (10 and 11) of Part II treat the subject of Markov (“nearest neighbour”) specifications on the integers. If the set of spin values is infinite, not every admitted Gibbs measure (“Markov random field”) is a Markov chain. Broadly speaking, it is shown in Chapter 10 that extremal or translation invariant Markov random fields are Markov chains and that translation invariant fields are unique. For countably infinite sets of spin values, the Markov chains are characterised in Chapter 11 and criteria and examples given for the presence or absence of phase transition within various subclasses. This chapter also settles some related cardinality questions. Chapter 12 discusses phase transition for Markovian systems on trees, while Chapter 13 develops the theory of Gaussian fields as Gibbs measures.

Part III (Chapters 14 to 16) is devoted to translation invariant spin systems on \({\mathbb{Z}}^ d\). Chapter 14 deals systematically with ergodicity and Chapter 15 with the variational characterization of translation invariant Gibbs measures through their property of minimising the specific free energy. Entropy and pressure play an important role here. Chapter 16 establishes the convexity of pressure as a function of the potential and proceeds to geometric considerations relevant to the study of the phase diagram: the classification of potentials according to the number of pure phases (extreme points of the set of translation invariant Gibbs measures) they admit. Translation invariant random fields with finite specific entropy can be identified (via the specific energy functional) with hyperplanes tangent or asymptotic to the graph of the pressure; in the former case they are Gibbs measures for the potential corresponding to the point of contact. Other results centre on the existence of potentials exhibiting phase transitions with prescribed order parameters and the fact that potentials with non-unique pure phases are in some sense exceptional.

A common theme linking the chapters of Part IV is the inequality known as the chessboard estimate. This is proved in Chapter 17, leading to a refinement of the Cauchy-Schwarz inequality valid for rotation invariant and reflection positive random fields on discrete tori. (Reflection positivity is a condition of positive definiteness to do with the reflection symmetry). Chapter 18 takes up the Peierls argument once more. It is shown that for short range potentials on \({\mathbb{Z}}^ d\) with stable ground state degeneracies subject to appropriate symmetry conditions, Gibbs measures obtained as thermodynamic limits under periodic boundary conditions present a familiar percolation phenomenon at low temperatures: infinite clusters, resembling oceans surrounding small islands. The possible “states” of the oceans correspond to the different ground states, implying symmetry breaking. These considerations cover several familiar models. There is an extension of this, covered in Chapter 19, to potentials with ground state degeneracies not displaying symmetries, but this involves a perturbation of the potential. This extension is sufficiently close in spirit to the Pirogov-Sinai theory to provide the author with the opportunity of outlining very concisely the main result of this theory in the bibliographical notes to this chapter. The chapter also deals with the interesting phenomenon of phase transition arising at a particular value of the temperature, not from a ground state degeneracy but from the competition between the free distribution and the interaction energy superposed on it.

The Fourier analytic methods of the last (20th) chapter establish conditions for the breaking of rotational symmetries by vector spins on \({\mathbb{Z}}^ d\). The proof of the existence of long range order here hinges on controlling low frequency spatial spin waves by means of the infrared bound. The resulting criterion is applied to models such as the classical Heisenberg model and even the one-dimensional long-range model with pair potential displaying slower-than-quadratic power decay. (The case of quadratic decay is mentioned but not treated in the book).

The excellent and extensive bibliographical notes which follow are as valuable a feature of the book as the mathematical text, and there is a long list of references to the literature, including the very recent one.

Reviewer: F.Papangelou

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |