Dynamic and static properties of the invaded cluster algorithm

TL;DR

This study investigates the dynamic and static properties of the invaded cluster algorithm in simulating critical two-dimensional Ising and 3-state Potts models, revealing a crossover behavior and measuring the dynamic exponent.

Contribution

It introduces a scaling form for the crossover phenomenon and provides new measurements of the dynamic exponent for the invaded cluster algorithm.

Findings

The autocorrelation time peaks in the crossover region.

The dynamic exponent z for the 3-state Potts model is approximately 0.346.

For the Ising model, the exponent z is less than 0.21, possibly zero.

Abstract

Simulations of the two-dimensional Ising and 3-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sub-lattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time tau(l,L) for an l*l sub-lattice attains a maximum in the crossover region, and a dynamic exponent z for the IC algorithm is defined according to tau_max ~ L^z. Simulation results for the 3-state model yield z=.346(.002) which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that z<.21 and possibly that tau_max ~ log L so that z=0 -- similar to previous results for the SW and Wolff algorithms.