Translation-invariant and periodic Gibbs measures for Potts model on a Cayley tree

TL;DR

This paper investigates Gibbs measures for the three-state Potts model on a Cayley tree, providing explicit formulas for translation-invariant measures and characterizing the number of periodic measures under certain conditions.

Contribution

It offers explicit formulas for translation-invariant Gibbs measures and determines the exact count of periodic measures for the Potts model on Cayley trees.

Findings

Explicit formulas for translation-invariant Gibbs measures.

Number of periodic Gibbs measures is 2(2^q - 1) under certain conditions.

Characterization of Gibbs measures for ferromagnetic and antiferromagnetic cases.

Abstract

In this paper is studied ferromagnetic three states Potts model on a Cayley tree of order three and we give explicit formulas for translation-invariant Gibbs measures. Furthermore, we show that under some conditions on the parameter of the antiferromagnetic Potts model with q-states with zero external field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic (non translation-invariant) Gibbs measures.