Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

TL;DR

This paper develops skew-orthogonal polynomial methods in the complex plane for asymmetric real random matrices, deriving explicit integral representations and applying them to various ensembles relevant in physics.

Contribution

It introduces new integral representations for skew-orthogonal polynomials in the complex plane for asymmetric real matrices, including explicit forms for Laguerre and Hermite cases.

Findings

Derived integral representations for SOP and their Cauchy transforms.

Applied SOP to chiral and elliptic ensembles, connecting to physical models.

Reproduced known SOPs for the elliptic real Ginibre ensemble.

Abstract

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.