Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations

TL;DR

This paper studies skew orthogonal polynomials associated with certain non-Hermitian random matrix ensembles, providing explicit formulas and evaluations for their kernels using advanced algebraic and integral techniques.

Contribution

It introduces new explicit expressions for skew orthogonal polynomials in real and quaternion real Ginibre ensembles using Schur polynomial theory and integral evaluations.

Findings

Explicit formulas for kernels in these ensembles

Connections between skew orthogonal polynomials and characteristic polynomial averages

Application of Schur polynomial theory and Selberg integral in evaluations

Abstract

There are some distinguished ensembles of non-Hermitian random matrices for which the joint PDF can be written down explicitly, is unchanged by rotations, and furthermore which have the property that the eigenvalues form a Pfaffian point process. For these ensembles, in which the elements of the matrices are either real, or real quaternion, the kernel of the Pfaffian is completely determined by certain skew orthogonal polynomials, which permit an expression in terms of averages over the characteristic polynomial, and the characteristic polynomial multiplied by the trace. We use Schur polynomial theory, knowledge of the value of a Schur polynomial averaged against real, and real quaternion Gaussian matrices, and the Selberg integral to evaluate these averages.