Markov random fields and their applications.

*(English)*Zbl 1229.60003
Contemporary Mathematics 1. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-5001-6). ix, 142 p. (1980).

This monograph is devoted to a discussion of Markov random fields and the circle of related topics from statistical physics and probability theory. The authors have succeeded in collecting an enormous amount of material from the vast research literature and presenting it at a level accessible to the nonexpert.

The famous Ising model is discussed in Chapter 1. The Peierls contour method is developed and the proof given that the two-dimensional Ising model exhibits a phase transition. Some of the more technical details of this argument are placed in an appendix. A general treatment of Markov random fields and Gibbs measures on finite graphs is presented in Chapter 2. In Chapter 3 the central theme is the limiting behavior, as the graph size tends to infinity, of certain (random) quantities (like the total magnetization in the Ising model). Numerous computer diagrams and simulations are provided which exhibit the qualitative features of these limit theorems. Chapter 4 introduces and discusses briefly a multitude of dynamical systems, including: Russian lamps, voter models, stepping stone model, growth models, percolation models, etc. In Chapter 5 Markov random fields in the Cayley tree are analyzed and Chapter 6 contains some additional applications.

The monograph is an excellent introduction to a fascinating field.

The famous Ising model is discussed in Chapter 1. The Peierls contour method is developed and the proof given that the two-dimensional Ising model exhibits a phase transition. Some of the more technical details of this argument are placed in an appendix. A general treatment of Markov random fields and Gibbs measures on finite graphs is presented in Chapter 2. In Chapter 3 the central theme is the limiting behavior, as the graph size tends to infinity, of certain (random) quantities (like the total magnetization in the Ising model). Numerous computer diagrams and simulations are provided which exhibit the qualitative features of these limit theorems. Chapter 4 introduces and discusses briefly a multitude of dynamical systems, including: Russian lamps, voter models, stepping stone model, growth models, percolation models, etc. In Chapter 5 Markov random fields in the Cayley tree are analyzed and Chapter 6 contains some additional applications.

The monograph is an excellent introduction to a fascinating field.

Reviewer: J. Theodore Cox (MR 2562493)

##### MSC:

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

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

60G60 | Random fields |

82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |