Optimality of Wang-Landau sampling and performance limitations of flat histogram methods

Abstract

We determine the optimal scaling of local-update flat-histogram methods (such as multicanonical, broad histograms, tempering, or Wang-Landau sampling) with system size by using a perfect flat-histogram scheme based on the exact density of states of 2D Ising models. We find that the Wang-Landau algorithm shows the same scaling as the perfect scheme and is thus optimal. However, even for the perfect scheme the scaling with the number of spins N is slower than the minimal N2 of an unbiased random walk in energy space for both, local and N-fold way updates. While it still follows a power law N2Lz for the ferromagnetic and fully frustrated Ising model we find exponential scaling of the tunneling times in the case of the +/-J Ising spin glass. The tunneling times of the +/-J Ising spin glass are found to follow a fat-tailed Fréchet extremal value distribution.