Performance and Improvements of Flat-histogram Monte Carlo Simulations

Abstract

We study the performance of Monte Carlo simulations that samples a broad histogram in energy by determining the mean first passage time to span the entire energy space of d-dimensional Ising-Potts models. For the d = 1, 2,3 Ising model, the mean first passage time τ of flat-histogram Monte Carlo methods with single-spin flip updates, such as the Wang-Landau algorithm or the multicanonical method, scales with the number of spins N = L ^d as τ ∼ N ² L ^z. The exponent z is found to be decrease as the dimensionality d is increased. In the mean field limit of infinite dimensions we find that z vanishes up to a logarithmic correction. We then demonstrate how the flat-histogram algorithms can be improved by two complementary approaches — cluster dynamics and ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that τ ∼ N ² up to logarithmic corrections for the d = 1, 2 Ising model.