Ergodicity of the Wang--Swendsen--Koteck\'y algorithm on several classes of lattices on the torus

TL;DR

This paper proves the ergodicity of the Wang--Swendsen--Kotecký algorithm for certain classes of lattices on the torus, expanding understanding of its behavior in statistical mechanics models.

Contribution

It establishes ergodicity conditions for the WSK algorithm on various toroidal lattices, including quadrangulations and Eulerian triangulations, for specific q-values.

Findings

WSK is ergodic for q≥4 on quadrangulations of girth ≥4

Ergodicity holds for q≥5 on certain Eulerian triangulations

Includes many lattices relevant in statistical mechanics

Abstract

We prove the ergodicity of the Wang--Swendsen--Koteck\'y (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of interest in statistical mechanics.