On the mixing time in the Wang-Landau algorithm

TL;DR

This paper investigates the mixing time of the Wang-Landau algorithm by analyzing the spectral gap of the transition matrix in energy space for Ising models, estimating how mixing time scales with lattice size.

Contribution

It introduces a method to analyze the mixing time via the spectral gap of the transition matrix built from the exact density of states.

Findings

Spectral gap is inversely proportional to mixing time.

Mixing time depends on lattice size, with an estimated mixing exponent.

Preliminary results suggest a relationship between energy space properties and convergence speed.

Abstract

We present preliminary results of the investigation of the properties of the Markov random walk in the energy space generated by the Wang-Landau probability. We build transition matrix in the energy space (TMES) using the exact density of states for one-dimensional and two-dimensional Ising models. The spectral gap of TMES is inversely proportional to the mixing time of the Markov chain. We estimate numerically the dependence of the mixing time on the lattice size, and extract the mixing exponent.