We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of $d$-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased Markovian random walk in energy space. For the $d=1$, 2, 3 Ising model, the mean first-passage time $\tau$ scales with the number of spins $N=L^d$ as $\tau \propto N^2{L}^z$. The exponent $z$ is found to decrease as the dimensionality $d$ is increased. In the mean-field limit of infinite dimensions we find that $z$ vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by $z>0$ for finite $d$ can be overcome by two complementary approaches—cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that $\tau \propto N^2$ up to logarithmic corrections for the $d=1$, 2 Ising model.

• Received 10 December 2004

DOI:https://doi.org/10.1103/PhysRevE.72.046704