Wang-Landau Algorithm: a Theoretical Analysis of the Saturation of the Error

TL;DR

This paper provides a theoretical analysis of the Wang-Landau algorithm's convergence, identifying the cause of error saturation and proposing a power-law scaling of the refinement parameter to improve accuracy.

Contribution

It offers a new analytical approach showing how scaling the refinement parameter as a power law can overcome error saturation in the Wang-Landau algorithm.

Findings

Error saturation is due to decreasing variations of the refinement parameter.

Power-law scaling of the refinement parameter mitigates error saturation.

Analysis extends to the N-fold way variation of the method.

Abstract

In this work we present a theoretical analysis of the convergence of the Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced years ago to calculate the density of states in statistical models. We study the dynamical behavior of the error in the calculation of the density of states.We conclude that the source of the saturation of the error is due to the decreasing variations of the refinement parameter. To overcome this limitation, we present an analytical treatment in which the refinement parameter is scaled down as a power law instead of exponentially. An extension of the analysis to the N-fold way variation of the method is also discussed.