On the Christoffel-Darboux kernel for random Hermitian matrices with external source

TL;DR

This paper derives a new representation of the Christoffel-Darboux kernel for eigenvalues of random Hermitian matrices with external sources, connecting it to classical orthogonal polynomials, which simplifies analysis of these ensembles.

Contribution

It provides a novel expression of the Christoffel-Darboux kernel in terms of standard orthogonal polynomials, extending previous results involving multiple orthogonal polynomials.

Findings

New representation of the Christoffel-Darboux kernel

Simplifies analysis of eigenvalue correlations

Links multiple and standard orthogonal polynomials

Abstract

Bleher and Kuijlaars, and Daems and Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the Christoffel-Darboux kernel for multiple orthogonal polynomials. We obtain a representation of this Christoffel-Darboux kernel in terms of the usual orthogonal polynomials.