Dynamic Critical Behavior of the Chayes-Machta Algorithm for the Random-Cluster Model. I. Two Dimensions

TL;DR

This study uses Monte Carlo simulations to analyze the dynamic critical behavior of the Chayes-Machta algorithm for the random-cluster model in two dimensions, revealing insights into critical exponents and bounds across various q values.

Contribution

It provides the first detailed numerical estimates of the dynamic critical exponent z_{CM} for the Chayes-Machta algorithm across a range of q in two dimensions, testing conjectures and bounds.

Findings

Violates Ossola-Sokal conjecture for 1 b7 q b7 1.95

Li-Sokal bound is nearly sharp but likely non-sharp by a power

Evidence on corrections to scaling in static observables

Abstract

We study, via Monte Carlo simulation, 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 ferromagnet to non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95 the Ossola-Sokal conjecture z_{CM} \ge \beta/\nu is violated, though we also present plausible fits compatible with this conjecture. We show that the Li-Sokal bound z_{CM} \ge \alpha/\nu is close to being sharp over the entire range 1 \le q \le 4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.