%0 Journal Article
%T Spectral density of generalized Wishart matrices and free multiplicative convolution
%A Wojciech Mlotkowski
%A Maciej A. Nowak
%A Karol A. Penson
%A Karol Zyczkowski
%J Physics
%D 2014
%I arXiv
%R 10.1103/PhysRevE.92.012121
%X 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.
%U http://arxiv.org/abs/1407.1282v4