Spectral density of generalized Wishart matrices and free multiplicative convolution

TL;DR

This paper analyzes the spectral density of generalized Wishart matrices using free probability, deriving new formulas for specific cases and exploring convolutions related to random matrix products and distributions.

Contribution

It introduces new formulas for spectral densities of free multiplicative powers of Marchenko-Pastur distributions and explores convolutions involving the Bures distribution.

Findings

Derived level densities for free multiplicative powers s=3 and s=1/3.

Obtained the spectral density for the generalized Bures distribution.

Explained the reason behind the convolution of arcsine and MP distributions.

Abstract

We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{\dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\rm MP}^{\boxtimes s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$ independent square random matrices, $X=X_1\cdots X_s$. New formulae for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.