A note on biorthogonal ensembles

TL;DR

This paper explores biorthogonal ensembles of random matrices, revealing their correlation kernels as averages of characteristic polynomial ratios and introducing a new chiral unitary ensemble with a source term.

Contribution

It demonstrates that the correlation kernel is an average of a ratio of characteristic polynomials and introduces a novel biorthogonal matrix ensemble with a source term.

Findings

Kernel expressed as average of a ratio of characteristic polynomials

Type I multiple polynomials as inverse characteristic polynomial averages

Introduction of a new chiral unitary ensemble with a source term

Abstract

We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.