Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley trees

TL;DR

This paper characterizes boundary conditions that produce specific translation-invariant Gibbs measures for the Potts model on Cayley trees, enhancing understanding of phase behavior at low temperatures.

Contribution

It explicitly describes boundary conditions corresponding to each translation-invariant Gibbs measure for the Potts model on Cayley trees.

Findings

Number of TISGMs at low temperatures is 2^q - 1.

Explicit boundary conditions are provided for each TISGM.

Improves understanding of phase structure in Potts models on Cayley trees.

Abstract

We consider translation-invariant splitting Gibbs measures (TISGMs) for the $q$-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper for each TISGM $\mu$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $\mu$.