By Utkir A Rozikov

The function of this booklet is to provide systematically all identified mathematical effects on Gibbs measures on Cayley timber (Bethe lattices).

The Gibbs degree is a likelihood degree, which has been an enormous item in lots of difficulties of likelihood thought and statistical mechanics. it's the degree linked to the Hamiltonian of a actual process (a version) and generalizes the proposal of a canonical ensemble. extra importantly, while the Hamiltonian should be written as a sum of components, the Gibbs degree has the Markov estate (a definite type of statistical independence), therefore resulting in its common visual appeal in lots of difficulties open air of physics akin to biology, Hopfield networks, Markov networks, and Markov good judgment networks. in addition, the Gibbs degree is the original degree that maximizes the entropy for a given anticipated energy.

The process used for the outline of Gibbs measures on Cayley bushes is the strategy of Markov random box concept and recurrent equations of this idea, however the sleek thought of Gibbs measures on bushes makes use of new instruments akin to staff conception, details flows on bushes, node-weighted random walks, contour tools on bushes, and nonlinear research. This booklet discusses all of the pointed out equipment, which have been built recently.

**Contents:**

- Group illustration of the Cayley Tree
- Ising version at the Cayley Tree
- Ising sort versions with Competing Interactions
- Information circulation on Trees
- The Potts Model
- The Solid-on-Solid Model
- Models with demanding Constraints
- Potts version 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, likelihood and degree theory.**Key Features:**

- The ebook is for graduate, post-graduate scholars and researchers. this can be the 1st booklet pertaining to Gibbs measures on Cayley trees
- It can be utilized to educate particular classes like “Gibbs measures on countable graphs”, “Models of statistical physics”, “Phase transitions and thermodynamics” and lots of similar courses