Multiple orthogonal polynomials in random matrix theory

TL;DR

This paper explores multiple orthogonal polynomials and their role in random matrix theory, focusing on determinantal point processes, Riemann-Hilbert problems, and asymptotic analysis of various models.

Contribution

It provides recent results on models like non-intersecting Brownian motions and two matrix models, highlighting the use of vector equilibrium problems in asymptotic analysis.

Findings

Correlation kernels expressed via Riemann-Hilbert problems.

Vector equilibrium problems describe limiting eigenvalue densities.

Asymptotic analysis involves external fields and constraints.

Abstract

Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point processes that are called multiple orthogonal polynomial (MOP) ensembles. The correlation kernel in such an ensemble is expressed in terms of the solution of a Riemann-Hilbert problem, that is of size (r+1) x (r+1) in the case of r weights. A number of models give rise to a MOP ensemble, and we discuss recent results on models of non-intersecting Brownian motions, Hermitian random matrices with external source, and the two matrix model. A novel feature in the asymptotic analysis of the latter two models is a vector equilibrium problem for two or more measures, that describes the limiting mean eigenvalue density. The vector equilibrium problems involve both an external field and an upper constraint.