Cutoff for the Swendsen-Wang dynamics on the lattice

TL;DR

This paper analyzes the mixing time of the Swendsen-Wang dynamics for the Potts model on lattices, proving a sharp cutoff at high temperatures and deriving explicit bounds related to the spectral gap.

Contribution

It establishes the cutoff phenomenon for the Swendsen-Wang dynamics on lattices at high temperatures, providing explicit bounds on mixing times based on spectral gap analysis.

Findings

Proves cutoff phenomenon for Swendsen-Wang dynamics at high temperatures.

Derives explicit mixing time bounds involving spectral gap.

Shows transition from unmixed to well-mixed states at a sharp cutoff.

Abstract

We study the Swendsen-Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a non-local Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work we derive strong enough bounds on the mixing time, proving that the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures exhibits a sharp transition from "unmixed" to "well-mixed," which is called the cutoff phenomenon. In particular, we establish that at high enough temperatures the Swendsen-Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^d$ has cutoff at time $\frac{d}{2} \left( -\log (1-\gamma) \right)^{-1} \log n$, where $\gamma(\beta)$ is the spectral gap of the infinite-volume dynamics.