Comparison of cluster algorithms for the bond-diluted Ising model

Abstract

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times Formula Presented and Formula Presented of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size Formula Presented when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time Formula Presented to be bounded from below by Formula Presented with a dynamical exponent Formula Presented for a bond concentration Formula Presented. Furthermore, we numerically show that this lower bound is actually taken for several values of Formula Presented in the range Formula Presented. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at Formula Presented, we find that its dynamical exponent is Formula Presented.