Spectral statistics for the difference of two Wishart matrices

TL;DR

This paper derives the joint eigenvalue distribution and spectral properties of the difference between two independent Wishart matrices, providing exact finite-dimensional formulas and asymptotic results relevant for random matrix theory.

Contribution

It introduces two novel methods to derive the eigenvalue distribution of the Wishart difference, including closed-form expressions and asymptotic spectral densities.

Findings

Exact joint eigenvalue density formulas derived

Spectral densities validated with Monte Carlo simulations

Asymptotic spectral density obtained using Stieltjes transform

Abstract

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.