Random matrices with external source and multiple orthogonal polynomials

TL;DR

This paper links the average characteristic polynomial of certain Hermitian random matrices to multiple orthogonal polynomials, providing new formulas and a Riemann-Hilbert characterization for cases with two eigenvalues of the external source.

Contribution

It introduces a novel connection between random matrix characteristic polynomials and multiple orthogonal polynomials, including a Christoffel-Darboux formula and Riemann-Hilbert problem formulation.

Findings

Characteristic polynomial characterized by multiple orthogonality conditions.

Derived a Christoffel-Darboux formula for two eigenvalues.

Expressed the kernel via a Riemann-Hilbert problem.

Abstract

We show that the average characteristic polynomial P_n(z) = E [\det(zI-M)] of the random Hermitian matrix ensemble Z_n^{-1} \exp(-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a_j of A, there is a weight and P_n has n_j orthogonality conditions with respect to this weight, if n_j is the multiplicity of a_j. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.