The chiral Gaussian two-matrix ensemble of real asymmetric matrices

TL;DR

This paper introduces an exactly solvable Gaussian two-matrix ensemble of real asymmetric matrices with a non-Hermiticity parameter, extending the real Ginibre ensemble to a chiral setting relevant for Dirac operators, and provides explicit eigenvalue distributions and correlations.

Contribution

It presents the explicit solution for a chiral Gaussian two-matrix ensemble of real asymmetric matrices, including eigenvalue distributions and correlation functions, generalizing known non-Hermitian matrix kernels.

Findings

Eigenvalue distribution explicitly computed with K-Bessel functions

Correlation functions expressed as Pfaffians involving Laguerre polynomial kernels

Large-N microscopic limits at the origin derived for different non-Hermiticity regimes

Abstract

We solve a family of Gaussian two-matrix models with rectangular Nx(N+v) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter mu. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary, or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalising the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed v at strong and weak non-Hermiticity.