Gibbs measure for mixed spins and mixed types model

TL;DR

This paper investigates the Gibbs measures of a mixed spins and mixed types model on Cayley trees, deriving recurrence relations and establishing the existence of multiple translation-invariant Gibbs measures, including specific counts for certain cases.

Contribution

It introduces a recurrence equation for the model and proves the existence of multiple translation-invariant Gibbs measures, including explicit counts for specific models.

Findings

Existence of a recurrence equation for the model

At least 3 translation-invariant Gibbs measures for general Cayley trees

At least 8 translation-invariant Gibbs measures for the binary Cayley tree

Abstract

In the present paper, we study the $(2,q)$-Ising-Potts model on the Cayley tree. We have derived a recurrence equation that shows the existence of a splitting Gibbs measure for this model. Furthermore, we have proven that for the $(2,q)$-Ising-Potts model on the Cayley tree of order $k\geq2$, there are at least 3 translation-invariant splitting Gibbs measures. We also prove that for the $(2,3)$-Ising-Potts model on the Cayley tree, specifically the binary tree, under certain conditions, there are at least 8 translation-invariant splitting Gibbs measures.