Numerical study of higher order analogues of the Tracy-Widom distribution

TL;DR

This paper numerically investigates a family of distributions from critical unitary random matrix ensembles, including Tracy-Widom and its higher order analogues, by solving Riemann-Hilbert problems.

Contribution

It introduces a numerical approach to compute and analyze higher order Tracy-Widom analogues via Riemann-Hilbert problem solutions.

Findings

Distributions are expressed as Fredholm determinants.

Numerical solutions reveal properties of these distributions.

Plots illustrate the behavior of higher order analogues.

Abstract

We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy-Widom distribution and higher order analogues of it. We compute the distributions numerically by solving a Riemann-Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit.