Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm

TL;DR

This paper investigates the dynamic critical behavior of the Chayes-Machta algorithm for the Fortuin-Kasteleyn model in 2D and 3D, revealing bounds and disproving a conjecture about the dynamic critical exponent.

Contribution

It provides the first detailed Monte Carlo analysis of the Chayes-Machta dynamics for noninteger q in higher dimensions, challenging existing bounds and conjectures.

Findings

Li-Sokal bound is close but not sharp in 2D

Li-Sokal bound is far from sharp in 3D

The conjecture z ≥ β/ν is false for some q in both dimensions

Abstract

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge \alpha/\nu is close to but probably not sharp in d=2, and is far from sharp in d=3, for all q. The conjecture z \ge \beta/\nu is false (for some values of q) in both d=2 and d=3.