Comparison of Swendsen-Wang and Heat-Bath Dynamics

TL;DR

This paper proves that the Swendsen-Wang process mixes rapidly for the Potts model on bounded degree graphs and introduces a modified algorithm with similar guarantees for planar graphs.

Contribution

It establishes spectral gap bounds linking Swendsen-Wang and single-spin dynamics and introduces a modified Swendsen-Wang algorithm with proven rapid mixing.

Findings

Spectral gap of Swendsen-Wang is bounded below by a constant times that of single-spin dynamics.

Rapid mixing of Swendsen-Wang for 2D Potts model above critical temperature.

Modified Swendsen-Wang algorithm achieves rapid mixing at all non-critical temperatures.

Abstract

We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing of the Swendsen-Wang process for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the two-dimensional Potts models at all non-critical temperatures.