Random matrices with equispaced external source

TL;DR

This paper analyzes Hermitian random matrix models with equispaced external sources, deriving strong asymptotics for associated multiple orthogonal polynomials and their average characteristic polynomials.

Contribution

It introduces a novel analysis of multiple orthogonal polynomials with growing weights via Riemann-Hilbert problems in the context of equispaced external sources.

Findings

Asymptotic formulas for multiple orthogonal polynomials

Characterization of polynomials through Riemann-Hilbert problems

Insights into eigenvalue distributions in large matrices

Abstract

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 x 1 vector-valued Riemann-Hilbert problems, and to perform an asymptotic analysis of the Riemann-Hilbert problems.