Stochastic processes on a lattice and Gibbs measures. Transl. from the French by Bertram Eugene Schwarzbach
[For the Fench original (1986) see Zbl 0655.60104.] This book presents a survey of recent developments in the theory of Gibbs measures and Markov fields. It consists of ten chapters. Chapter 1 describes some basic notions of the Ising model with a historical perspective. In Chapter 2 the DLR problem of the description of all probability measures \(\mu\) with the given specification \(\pi\) is stated. These measures \(\mu\) can be obtained as solutions of corresponding equations in terms of conditional expectation with respect to \(\mu\). The investigation of the structure of the set \({\mathfrak a}(\pi)\) of the solutions is beginning in Chapter 2 and continuing in Chapter 3. There are introduced the notions of Gibbs specifications and Gibbs measures and the existence of Gibbs measures is investigated first in the case when the set \(\Phi\) of the states of the system is finite (Dobrushin's theorem), then in the case when \(\Phi\) is a Polish space (Preston's theorem and Dobrushin's theorem). The last paragraph of this chapter considers the models of P(\(\phi\)) type and Gaussian Gibbs measures. The main topic of this book is of course the problem of phase transition. Chapters 4-8 contain various methods of solving this problem. The main result of Chapter 4 is Aizenman's theorem which asserts that in the case of ferromagnetic Ising model on \({\mathbb{Z}}^ 2\), \({\mathfrak a}\) is precisely the segment \(\{\mu =\lambda \mu_++(1-\lambda)\mu:\;\lambda \in (0,1)\}\) with the laws \(\mu_+\) and \(\mu_ -\) corresponding to \(+1\) and -1 exterior conditions. Chapter 5 gives several inequalities such as GKS, GHS, Simon-Lieb etc. Chapter 6 presents the Pirogov-Sinai theory which is based on the generalization of the so-called Peierls' contour method. Chapter 7 is devoted to another method of solving the problem of phase transition based on ``positive reflexivity''. It contains the proof of the theorem of Fröhlich, Simon, Spencer about the existence of the long-range order in d-dimensional, \(d\geq 3\), classical Heisenberg model at low temperature. Chapter 8 gives the proof of Dobrushin-Schlosman theorem which asserts that for two-dimensional systems with \(\Phi\) being the unit circle, under general enough conditions any Gibbs state is invariant under the group of rotations. There are also presented Onsager's method of calculation of the exact value of the specific Gibbs free energy and the Lee-Yang theorem about the zeros of the partition function. Chapter 9 describes rather briefly spatial-temporal processes which are Markovian with respect to time and introduces elements of the theory of Markovian semigroups and their infinitesimal generators. The last Chapter 10 deals with statistical problems of the following type: how to infer the parameters of the model from the observation of a configuration upon the finite set of states. Some applications of the theory such as image processing and artificial intelligence also are discussed. At the end of the book there is a ``thematic catalogue'' which permits to find information about particular ideas whether in this book or in the literature.
scientific article; zbMATH DE number 49343