Single-cluster dynamics for the random-cluster model

TL;DR

This paper introduces a single-cluster Monte Carlo algorithm for the random-cluster model, extending the Wolff method to non-integer q, and analyzes its static and dynamic properties compared to existing algorithms.

Contribution

It generalizes the Wolff single-cluster algorithm to non-integer q and investigates its critical dynamics, revealing distinct behavior from the SWCM algorithm for non-integer q.

Findings

The algorithm agrees with SWCM for static quantities.

Autocorrelation functions decay exponentially for integer q.

Non-integer q shows power-law autocorrelation behavior with large dynamic exponents.

Abstract

We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the $q$-state Potts model to non-integer values $q>1$. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer $q$, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents $z_{\rm exp} =0.07 (1), 0.521 (7)$, and $1.007 (9)$ for $q=2, 3$, and 4 respectively. For non-integer $q$, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.