Numerical solution of Riemann--Hilbert problems: random matrix theory and orthogonal polynomials
Mathematical Physics
2012-10-09 v1 Complex Variables
math.MP
Numerical Analysis
Probability
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.
Keywords
Cite
@article{arxiv.1210.2199,
title = {Numerical solution of Riemann--Hilbert problems: random matrix theory and orthogonal polynomials},
author = {Sheehan Olver and Thomas Trogdon},
journal= {arXiv preprint arXiv:1210.2199},
year = {2012}
}