The aggregate path coupling method for the Potts model on bipartite graph

TL;DR

This paper extends the aggregate path coupling method to analyze the Potts model on bipartite graphs, deriving large deviations and identifying phase transition points for mixing times.

Contribution

It introduces an extension of the aggregate path coupling technique to bipartite graphs and applies it to determine the phase transition in mixing times for the Potts model.

Findings

Derived large deviations principle for the Potts model on bipartite graphs.

Extended aggregate path coupling method to bipartite graph setting.

Identified the critical interface value $eta_s$ for mixing regimes.

Abstract

In this paper, we derive the large deviations principle for the Potts model on the complete bipartite graph $K_{n,n}$ as $n$ increases to infinity. Next, for the Potts model on $K_{n,n}$, we provide an extension of the method of aggregate path coupling that was originally developed in Kovchegov et al 2011 for the mean-field Blume-Capel model and in Kovchegov and Otto 2015 for a general mean-field setting that included the Generalized Curie-Weiss-Potts model analyzed in Cuff et al 2012. We use the aggregate path coupling method to identify and prove the interface value $\beta_s$ separating the rapid and slow mixing regimes for the Glauber dynamics of the Potts model on $K_{n,n}$.