Numerical solution of Riemann--Hilbert problems: random matrix theory and orthogonal polynomials

TL;DR

This paper reviews a numerical framework for solving Riemann--Hilbert problems and applies it to compute orthogonal polynomials, Fredholm determinants, and level statistics in random matrix theory.

Contribution

It introduces a numerical method for Riemann--Hilbert problems and demonstrates its application to orthogonal polynomials and random matrix statistics.

Findings

Computed level densities and gap statistics for unitary ensembles

Successfully calculated Hastings--McLeod solution of Painlevé II

Validated the numerical approach against known results

Abstract

In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with an approach to compute Fredholm determinants, we are able to calculate level densities and gap statistics for general finite-dimensional unitary ensembles. We also include a description of how to compute the Hastings--McLeod solution of the homogeneous Painlev\'e II equation.