Rapid mixing of Swendsen-Wang dynamics in two dimensions

TL;DR

This paper establishes comparison results for Swendsen-Wang, heat-bath, and single-bond dynamics, proving rapid mixing in two-dimensional Potts and Ising models at all non-critical temperatures, with implications for various graph classes.

Contribution

It provides the first comparison results linking different dynamics and proves rapid mixing of Swendsen-Wang in 2D at all non-critical temperatures.

Findings

Rapid mixing of heat-bath implies rapid mixing of Swendsen-Wang.

Rapid mixing of Swendsen-Wang and single-bond are equivalent.

Spectral gaps on planar graphs relate to dual graphs with temperature adjustments.

Abstract

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.