A comparison of cluster algorithms for the bond-diluted Ising model

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 $τ_w$ and $τ_{\rm sw}$ of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size $L$ 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 $τ_w$ to be bounded from below by $L^{z_w}$ with a dynamical exponent $z_w=γ/ ν\approx 1.75$ for a bond concentration $p < 1$. Furthermore, we numerically show that this lower bound is actually taken for several values of $p$ in the range $0.5 < p < 1$. 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 $p = 0.6$, we find that its dynamical exponent is $z_{\rm sw} = 0.09(4)$.

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