Swendsen-Wang is faster than single-bond dynamics

TL;DR

This paper proves that the Swendsen-Wang dynamics has a larger spectral gap than single-bond dynamics for the random-cluster model, indicating faster mixing, with bounds on mixing times on high-dimensional tori at the Potts transition.

Contribution

It establishes a spectral gap comparison between Swendsen-Wang and single-bond dynamics and provides bounds on mixing times at the phase transition for large q.

Findings

Swendsen-Wang has a larger spectral gap than single-bond dynamics.

Upper and lower bounds on mixing times are exponential in L^{d-1}.

Results apply to high-dimensional tori at the Potts transition for large q.

Abstract

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, that updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the single-bond dynamics on the discrete $d$-dimensional torus of side length $L$ at the Potts transition temperature for $q$ large enough that are exponential in $L^{d-1}$, complementing a result of Borgs, Chayes and Tetali.