Gibbs Measures On Cayley Trees

The purpose of this book is to present systematically all known mathematical results on Gibbs measures on Cayley trees (Bethe lattices).

The Gibbs measure is a probability measure, which has been an important object in many problems of probability theory and statistical mechanics. It is the measure associated with the Hamiltonian of a physical system (a model) and generalizes the notion of a canonical ensemble. More importantly, when the Hamiltonian can be written as a sum of parts, the Gibbs measure has the Markov property (a certain kind of statistical independence), thus leading to its widespread appearance in many problems outside of physics such as biology, Hopfield networks, Markov networks, and Markov logic networks. Moreover, the Gibbs measure is the unique measure that maximizes the entropy for a given expected energy.

The method used for the description of Gibbs measures on Cayley trees is the method of Markov random field theory and recurrent equations of this theory, but the modern theory of Gibbs measures on trees uses new tools such as group theory, information flows on trees, node-weighted random walks, contour methods on trees, and nonlinear analysis. This book discusses all the mentioned methods, which were developed recently.
Contents:

Group Representation of the Cayley Tree
Ising Model on the Cayley Tree
Ising Type Models with Competing Interactions
Information Flow on Trees
The Potts Model
The Solid-on-Solid Model
Models with Hard Constraints
Potts Model with Countable Set of Spin Values
Models with Uncountable Set of Spin Values
Contour Arguments on Cayley Trees
Other Models




Readership: Researchers in mathematical physics, statistical physics, probability and measure theory.Key Features:


The book is for graduate, post-graduate students and researchers. This is the first book concerning Gibbs measures on Cayley trees
It can be used to teach special courses like “Gibbs measures on countable graphs”, “Models of statistical physics”, “Phase transitions and thermodynamics” and many related courses