Characteristic polynomials in real Ginibre ensembles

TL;DR

This paper derives new formulas for characteristic polynomials in real Ginibre ensembles and their chiral variants, enabling interpolation between different symmetric and asymmetric matrix ensembles.

Contribution

It introduces a simple method to compute characteristic polynomials for real Ginibre ensembles and their chiral counterparts without eigenvalue decomposition, including new results for the chiral case.

Findings

Derived average characteristic polynomials for real Ginibre ensembles

Established kernels in terms of Hermite and Laguerre polynomials

Interpolated between asymmetric Ginibre and orthogonal ensembles

Abstract

We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian Orthogonal Ensemble, as well as their chiral counterparts.