On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review

TL;DR

This review compares numerical methods for evaluating probability distributions in random matrix theory, highlighting Fredholm determinants as more efficient, and presents new formulas and a Matlab toolbox for further research.

Contribution

It introduces a comparative analysis of numerical approaches, advocates for Fredholm determinants, and reports new determinantal formulas for eigenvalue distributions.

Findings

Fredholm determinants are simpler and more efficient for numerical evaluation.

New determinantal formulas for eigenvalue distributions at the spectral edges.

A Matlab toolbox is provided for further computational experiments.

Abstract

In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlev\'e transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) beta-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the k-th largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.