Average characteristic polynomials for multiple orthogonal polynomial ensembles

TL;DR

This paper links the determinants of blocks of a Riemann-Hilbert matrix for multiple orthogonal polynomials to average characteristic polynomials over associated ensembles, extending classical results to more complex polynomial types.

Contribution

It establishes a novel connection between Riemann-Hilbert matrix blocks and average characteristic polynomials for MOP, generalizing classical orthogonal polynomial results.

Findings

Det Y_{1,1} equals average characteristic polynomial.

Det Y_{2,2} equals average inverse characteristic polynomial.

Results extended to products and ratios of characteristic polynomials.

Abstract

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann-Hilbert matrix Y consisting of four blocks Y_{1,1}, Y_{1,2}, Y_{2,1} and Y_{2,2}. In this paper, we show that det Y_{1,1} (det Y_{2,2}) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel-Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.