Gibbs measures on spatial systems on vertices of Cayley trees

TL;DR

This paper develops a general measure-theoretic framework for constructing Gibbs measures on Cayley trees, providing new verifiable conditions for Gibbs specifications applicable to various models.

Contribution

It introduces a new, easily verifiable condition for Gibbs specifications, broadening the theoretical foundation for Gibbs measures on Cayley trees.

Findings

Established a measure-theoretic construction of Gibbs measures

Provided a new condition equivalent to kernel consistency

Results are applicable to multiple models on Cayley trees

Abstract

There are many research works devoted to Gibbs measure for models on Cayley trees. Among these works, there are some works in which the general results are identical, but the considered models are various. In this article, we present the construction of Gibbs measures in the language of measure theory and reply to the question ``When can we construct Gibbs specifications?" Also, we present a new condition (convenient for verification) which is equivalent to the consistency condition of kernels. The obtained results in the article are general, not for a particular model. On the contrary, these results hold for some considered models on Cayley trees.