Stochastic processes on a lattice and Gibbs measures. Transl. from the French by Bertram Eugene Schwarzbach
type: Item
instance of: scholarly article
title: Stochastic processes on a lattice and Gibbs measures. Transl. from the French by Bertram Eugene Schwarzbach
zbMATH Open document ID: 0726.60050
author: Bernard Prum
author: Jean-Claude Fort
published in: Mathematical Physics Studies
publication date: 1992-09-17T00:00:00Z
review text: [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 …
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 49343
zbMATH Keywords: theory of Gibbs measures
zbMATH Keywords: Markov fields
zbMATH Keywords: problem of phase transition
zbMATH Keywords: ferromagnetic Ising model
zbMATH Keywords: Pirogov-Sinai theory
zbMATH Keywords: positive reflexivity
zbMATH Keywords: Lee-Yang theorem
zbMATH Keywords: image processing
reviewed by: S. Poghosyan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 49343