Block-modified Wishart matrices and free Poisson laws

TL;DR

This paper investigates the spectral distribution of block-modified Wishart matrices, revealing their asymptotic behavior converging to a free Poisson law combined with other distributions under certain conditions.

Contribution

It establishes a formula for the eigenvalue distribution of block-modified Wishart matrices involving free probability concepts, extending understanding of their asymptotic spectral properties.

Findings

Eigenvalue distribution converges to a free Poisson law convolved with another measure.

The asymptotic spectral distribution is characterized explicitly in terms of the law of the linear map.

Provides conditions under which the eigenvalue distribution formula holds.

Abstract

We study the random matrices of type $\tilde{W}=(id\otimes\varphi)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is a self-adjoint linear map. We prove that, under suitable assumptions, we have the $d\to\infty$ eigenvalue distribution formula $\delta m\tilde{W}\sim\pi_{mn\rho}\boxtimes\nu$, where $\rho$ is the law of $\varphi$, viewed as a square matrix, $\pi$ is the free Poisson law, $\nu$ is the law of $D=\varphi(1)$, and $\delta=tr(D)$.