Mixing of the Glauber dynamics for the ferromagnetic Potts model

TL;DR

This paper investigates the mixing times of Glauber dynamics for the ferromagnetic Potts model, establishing bounds on the number of colors needed for rapid or slow mixing depending on graph degree and structure.

Contribution

It provides new bounds on the number of colors for rapid or slow mixing of Glauber dynamics in the ferromagnetic Potts model, considering various graph types and degrees.

Findings

Rapid mixing when the number of colors exceeds a lower bound.

Slow mixing on random regular graphs when colors are below an upper bound.

Improved bounds for specific graph classes like toroidal grids.

Abstract

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ($\Delta$) of the underlying graph and the number of colours or spins ($q$) in determining whether the dynamics mixes rapidly or not. We find a lower bound $L$ on the number of colours such that Glauber dynamics is rapidly mixing if at least $L$ colours are used. We give a closely-matching upper bound $U$ on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random $\Delta$-regular graphs when at most $U$ colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree $\Delta$, e.g. toroidal grids for $\Delta = 4$.