Four Competing interactions for models with uncountable set of spin values on a Cayley Tree

TL;DR

This paper analyzes complex spin models with four competing interactions on a Cayley tree, reducing the problem to nonlinear integral equations and exploring Gibbs measures, including their periodicity and non-uniqueness.

Contribution

It introduces a framework for studying uncountable spin models with multiple interactions on Cayley trees, including solutions for Gibbs measures and their properties.

Findings

Periodic Gibbs measures are either translation-invariant or have period two.

Examples of non-uniqueness of translation-invariant Gibbs measures are provided.

The problem reduces to analyzing solutions of nonlinear integral equations.

Abstract

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two. We reduce the problem of describing the "splitting Gibbs measures" of the model to the analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of translation-invariant Gibbs measures.