Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure

TL;DR

This paper introduces four new random matrix ensembles involving Gaussian Wigner and Wishart matrices, deriving their eigenvalue densities and correlation functions with a biorthogonal structure, validated through simulations.

Contribution

It develops a biorthogonal framework for eigenvalue densities of these ensembles and provides a generalized determinantal formula for correlation functions.

Findings

Eigenvalue densities exhibit biorthogonal structure.

Derived a generalized Andre9ief's formula for correlation functions.

Validated results with Monte Carlo simulations.

Abstract

We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andr\'{e}ief's integration formula, is used to compactly express the $r$-point correlation function of eigenvalues. This representation circumvents the complications encountered in the usual approaches, and the answer is obtained immediately by examining the joint density of eigenvalues. We validate our analytical results using Monte Carlo simulations.