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We study the performance of Monte Carlo simulations that sample a broadhistogram in energy by determining the mean first-passage time to span theentire energy space of d-dimensional ferromagnetic Ising/Potts models. We firstshow that flat-histogram Monte Carlo methods with single-spin flip updates suchas the Wang-Landau algorithm or the multicanonical method perform sub-optimallyin comparison to an unbiased Markovian random walk in energy space. For thed=1,2,3 Ising model, the mean first-passage time \tau scales with the number ofspins N=L^d as \tau \propto N^2L^z. The critical exponent z is found todecrease as the dimensionality d is increased. In the mean-field limit ofinfinite dimensions we find that z vanishes up to logarithmic corrections. Wethen demonstrate how the slowdown characterized by z>0 for finite d can beovercome by two complementary approaches - cluster dynamics in connection withWang-Landau sampling and the recently developed ensemble optimizationtechnique. Both approaches are found to improve the random walk in energy spaceso that \tau \propto N^2 up to logarithmic corrections for the d=1 and d=2Ising model

Overcoming the slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles