Multicanonical sampling of rare events in random matrices

TL;DR

This paper introduces a multicanonical Monte Carlo method to efficiently estimate extremely rare large deviations in the largest eigenvalue of various random matrix ensembles, surpassing the limitations of naive sampling.

Contribution

The paper presents a versatile multicanonical sampling approach for rare event estimation in random matrices, applicable across different ensembles and statistics.

Findings

Successfully estimated probabilities down to 10^{-200}

Effective in Gaussian orthogonal, sparse, and uniform density matrices

Outperforms naive sampling methods in rare event detection

Abstract

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble (GOE), sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of $\sim 10^{-200}$, a region where naive random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.