Cutoff for the Swendsen-Wang dynamics on the complete graph

TL;DR

This paper proves that the Swendsen-Wang dynamics for the mean-field Potts model on the complete graph exhibits cutoff behavior in the supercritical temperature regime, indicating a sharp transition to equilibrium.

Contribution

It establishes the precise mixing time and confirms the cutoff phenomenon for the Swendsen-Wang dynamics when the inverse temperature exceeds the critical value.

Findings

Mixing time is c(β,q) log n + Θ(1) for β > q.

The dynamics exhibits cutoff in the supercritical regime.

Sharp transition from non-equilibrium to equilibrium within a constant time window.

Abstract

We study the speed of convergence of the Swendsen-Wang (SW) dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all $q \ge 2$ and all values of the inverse temperature parameter $\beta > 0$. In particular, it is known that when $\beta > q$ the mixing time of the SW dynamics is $\Theta(\log n)$. We strengthen this result by showing that for all $\beta > q$, there exists a constant $c(\beta,q) > 0$ such that the mixing time of the SW dynamics is $c(\beta,q) \log n + \Theta(1)$. This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from ''far from stationarity'' to ''well-mixed'' within a narrow $\Theta(1)$ time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.