Non-Hermitian extensions of Wishart random matrix ensembles

TL;DR

This paper explores non-Hermitian extensions of Wishart random matrix ensembles, providing explicit solutions for eigenvalue correlations across different symmetry classes and non-Hermiticity levels, with applications in complex systems.

Contribution

It introduces a unified framework for non-Hermitian Wishart ensembles with explicit finite-size solutions and interpolations between Hermitian and maximally non-Hermitian cases.

Findings

Explicit finite-size eigenvalue correlation functions for all three symmetry classes.

Derived Bessel kernels in the microscopic large-N limit at the origin.

Interpolating non-Hermiticity parameter connects Hermitian and non-Hermitian regimes.

Abstract

We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart-Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian two-matrix models, where the complex eigenvalues of the product of the two independent rectangular matrices are sought, with the matrix elements of both matrices being either real, complex or quaternion real. We also present the more general case depending on a non-Hermiticity parameter, that allows us to interpolate between the corresponding three Hermitian Wishart ensembles with real eigenvalues and the maximally non-Hermitian case. All three symmetry classes are explicitly solved for finite matrix size NxM for all complex eigenvalue correlations functions (and real or mixed correlations for real matrix elements). These are given in terms of the corresponding kernels built from orthogonal or skew-orthogonal Laguerre polynomials in the complex plane. We then present the corresponding three Bessel kernels in the complex plane in the microscopic large-N scaling limit at the origin, both at weak and strong non-Hermiticity with M-N greater or equal to 0 fixed.