Random-Cluster Dynamics in $\mathbb{Z}^2$

TL;DR

This paper establishes tight bounds on the mixing time of Glauber dynamics for the random-cluster model on a 2D lattice, except at the critical point, using recent phase transition results.

Contribution

It provides the first tight bounds on the mixing time of the random-cluster model's dynamics in two dimensions, extending techniques from spin systems.

Findings

Upper bound of O(n^2 log n) for mixing time away from criticality

Matching lower bound confirming tightness of the result

Analysis leverages recent phase transition breakthroughs in $\

Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a $O(n^2\log n)$ upper bound for the mixing time at all values of the model parameter $p$ except the critical point $p=p_c(q)$, and for all values of the second model parameter $q\ge 1$. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in $\mathbb{Z}^2$. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.