Block-modified Wishart matrices: the easy case

TL;DR

This paper investigates the asymptotic $*$-distribution of block-modified Wishart matrices when the modification map is 'easy' or super-easy, revealing a compound free Poisson law in the large dimension limit.

Contribution

It provides a detailed analysis of the asymptotic distribution of block-modified Wishart matrices with easy or super-easy modifications, extending previous work with Nechita.

Findings

Asymptotic $*$-distribution converges to a compound free Poisson law

Results apply to 'easy' and super-easy linear maps in quantum algebra

Generalizes previous results on Wishart matrices and their modifications

Abstract

Associated to any complex Wishart matrix $W$ of parameters $(dn,dm)$ and any linear map $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is the "block-modified" matrix $\tilde{W}=(id\otimes\varphi)W$. Following some previous work with Nechita, we study here the asymptotic $*$-distribution of $\tilde{W}$, in the $d\to\infty$ limit, in the case where the modification map $\varphi$ is "easy", or more generally super-easy, in the quantum algebra/representation theory sense. Under suitable assumptions on $\varphi$ we obtain in this way a compound free Poisson law.