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

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≥1. We consider spatial dimension d=2 and 1.25≤q≤4 in steps of 0.25, on lattices up to 10242, and obtain estimates for the dynamic critical exponent zCM. We present evidence that when 1≤q≲1.95 the Ossola–Sokal conjecture zCM≥β/ν is violated, though we also present plausible fits compatible with this conjecture. We show that the Li–Sokal bound zCM≥α/ν is close to being sharp over the entire range 1≤q≤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.

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