Random-cluster dynamics on random regular graphs in tree uniqueness

TL;DR

This paper proves rapid mixing of the random-cluster Glauber dynamics on random regular graphs below a certain phase transition threshold, with implications for the efficiency of related algorithms in statistical physics models.

Contribution

It establishes the mixing time bounds for the Glauber dynamics on random regular graphs in the tree uniqueness region, extending understanding of phase transition effects.

Findings

Glauber dynamics mixes in Θ(n log n) time below the threshold.

Fast mixing of Swendsen–Wang dynamics for Potts model on random regular graphs.

Introduces a novel iterative scheme to analyze cluster formation and shattering time.

Abstract

We establish rapid mixing of the random-cluster Glauber dynamics on random $\Delta$-regular graphs for all $q\ge 1$ and $p<p_u(q,\Delta)$, where the threshold $p_u(q,\Delta)$ corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $\Delta$-regular tree. It is expected that this threshold is sharp, and for $q>2$ the Glauber dynamics on random $\Delta$-regular graphs undergoes an exponential slowdown at $p_u(q,\Delta)$. More precisely, we show that for every $q\ge 1$, $\Delta\ge 3$, and $p<p_u(q,\Delta)$, with probability $1-o(1)$ over the choice of a random $\Delta$-regular graph on $n$ vertices, the Glauber dynamics for the random-cluster model has $\Theta(n \log n)$ mixing time. As a corollary, we deduce fast mixing of the Swendsen--Wang dynamics for the Potts model on random $\Delta$-regular graphs for every $q\ge 2$, in the tree uniqueness region. Our proof relies on a sharp bound on the "shattering time", i.e., the number of steps required to break up any configuration into $O(\log n)$ sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.