Critical speeding-up in a local dynamics for the random-cluster model

TL;DR

This paper investigates the dynamic critical behavior of the Sweeny local bond-update dynamics in the random-cluster model, revealing a phenomenon called critical speeding-up where certain observables decorrelate rapidly near criticality.

Contribution

It demonstrates the occurrence of critical speeding-up in the random-cluster model and provides numerical evidence that the dynamic critical exponent is close to the theoretical lower bound.

Findings

Global observable S_2 decorrelates rapidly near criticality.

Dynamic critical exponent z_{exp} is close to the lower bound /.

Possible faster decay than cluster dynamics.

Abstract

We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2,3, by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S_2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep, so that the integrated autocorrelation time tends to zero as the critical point is approached. We also show that the dynamic critical exponent z_{exp} is very close (possibly equal) to the rigorous lower bound \alpha/\nu, and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.