Sharp mixing time asymptotics of Glauber dynamics for the Curie-Weiss-Potts model at low temperatures

TL;DR

This paper provides a precise estimate of the mixing time for Glauber dynamics in the low-temperature Curie-Weiss-Potts model, highlighting metastability's role and showing the absence of cutoff phenomena.

Contribution

It introduces sharp mixing time asymptotics for the low-temperature regime, extending previous high-temperature results and emphasizing metastability's impact.

Findings

Mixing time asymptotics are characterized by metastable transition times.

The system does not exhibit a cutoff phenomenon.

Metastability theory explains slow mixing in low-temperature regimes.

Abstract

In this article, we derive a sharp mixing time estimate of the Glauber dynamics for the Curie-Weiss-Potts model in the low-temperature regime. In contrast to the high-temperature regime studied by Cuff et al. (J. Stat. Phys. 149: 432-477, 2012), in which the Gibbs measure is concentrated around the equiproportional distribution of spins, the Gibbs measure in the low-temperature regime is concentrated on multiple states, each with a dominant number of a single spin. Consequently, global mixing of the system requires sufficiently many transitions between these states. Since these transitions are well explained by the phenomenon of metastability, the theory of metastability plays a central role in the analysis of slow mixing. In particular, the sharp asymptotics for the mixing time is given by the mixing time of the limit Markov chain, which describes the metastable behavior of the dynamics, multiplied by the metastable transition time-scale. As a byproduct, we verify that it does not exhibit a cutoff phenomenon.