Translation-invariance of periodic Gibbs measures for the Potts model on the Cayley tree

TL;DR

This paper proves that all periodic Gibbs measures for the Potts model on a Cayley tree are translation-invariant, extending previous work and covering both ferromagnetic and antiferromagnetic cases.

Contribution

It demonstrates that for the Potts model on Cayley trees, all periodic Gibbs measures are necessarily translation-invariant, regardless of the model's parameters.

Findings

Periodic Gibbs measures are translation-invariant for the antiferromagnetic Potts model.

Periodic Gibbs measures are translation-invariant for the ferromagnetic Potts model.

Result holds for Cayley trees of order two and arbitrary order k.

Abstract

Work is a continuation of work TMPh, 175(2), 2013. We study the Potts model with zero external field on the Cayley tree. It is shown that for any values of the parameter all periodic Gibbs measures are translation-invariant for the q-state antiferromagnetic Potts model on the Cayley tree of order two and for the q-state ferromagnetic Potts model on the Cayley tree of order k.