Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $\Delta$-regular tree for large $\Delta$

TL;DR

This paper proves that for large regular trees, the anti-ferromagnetic Potts model with at least five states has a unique Gibbs measure across a range of interaction parameters, confirming a long-standing conjecture.

Contribution

It establishes the uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on infinite regular trees for large degree, for all q ≥ 5 and certain interaction parameters.

Findings

Uniqueness of Gibbs measure for q ≥ 5 on large regular trees.

Confirmation of a longstanding folklore conjecture.

Applicable for all edge interaction parameters in a specified range.

Abstract

In this paper we prove that for any integer $q\geq 5$, the anti-ferromagnetic $q$-state Potts model on the infinite $\Delta$-regular tree has a unique Gibbs measure for all edge interaction parameters $w\in [1-q/\Delta,1)$, provided $\Delta$ is large enough. This confirms a longstanding folklore conjecture.