Sweeny dynamics for the random-cluster model with small $Q$

TL;DR

This paper investigates the dynamical behavior of the Sweeny algorithm for the 2D random-cluster model with small Q, revealing Q-dependent critical slowing-down and proposing an improved hybrid method to achieve constant autocorrelation times.

Contribution

It uncovers the Q-dependent critical slowing-down in the Sweeny algorithm and introduces a combined Sweeny-Kawasaki approach to eliminate this issue for all quantities.

Findings

Critical speeding-up becomes more pronounced as Q decreases.

Autocorrelation time diverges as Q approaches zero for certain local quantities.

The improved Sweeny-Kawasaki method achieves autocorrelation times of order O(1).

Abstract

The Sweeny algorithm for the $Q$-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As $Q$ decreases, the so-called critical speeding-up for non-local quantities becomes more and more pronounced. However, for some quantity of specific local pattern -- e.g., the number of half faces on the square lattice, we observe that, as $Q \to 0$, the integrated autocorrelation time $\tau$ diverges as $Q^{-\zeta}$, with $\zeta \simeq 1/2$, leading to the non-ergodicity of the Sweeny method for $Q \to 0$. Such $Q$-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight $v=\sqrt{Q}$, can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order $O(1)$.