Dynamics of the Wang-Landau algorithm and complexity of rare events for the three-dimensional bimodal Ising spin glass

TL;DR

This paper analyzes the performance and limitations of the Wang-Landau algorithm in simulating three-dimensional bimodal Ising spin glasses, revealing extreme variability and the impact of rare, slow-equilibrating configurations on computational complexity.

Contribution

It provides a detailed study of the dynamics and rare event complexity of the Wang-Landau algorithm applied to 3D spin glasses, highlighting challenges in sampling and ground state determination.

Findings

Round-trip times follow a fat-tailed Frechet distribution.

Average round-trip time becomes ill-defined for large systems.

Rare events significantly hinder the algorithm's efficiency.

Abstract

We investigate the performance of flat-histogram methods based on a multicanonical ensemble and the Wang-Landau algorithm for the three-dimensional +/- J spin glass by measuring round-trip times in the energy range between the zero-temperature ground state and the state of highest energy. Strong sample-to-sample variations are found for fixed system size and the distribution of round-trip times follows a fat-tailed Frechet extremal value distribution. Rare events in the fat tails of these distributions corresponding to extremely slowly equilibrating spin glass realizations dominate the calculations of statistical averages. While the typical round-trip time scales exponential as expected for this NP-hard problem, we find that the average round-trip time is no longer well-defined for systems with N >= 8^3 spins. We relate the round-trip times for multicanonical sampling to intrinsic properties of the energy landscape and compare with the numerical effort needed by the genetic Cluster-Exact Approximation to calculate the exact ground state energies. For systems with N >= 8^3 spins the simulation of these rare events becomes increasingly hard. For N >= 14^3 there are samples where the Wang-Landau algorithm fails to find the true ground state within reasonable simulation times. We expect similar behavior for other algorithms based on multicanonical sampling.