Spectral density of generalized Wishart matrices and free multiplicative convolution.
We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas 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.
Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, PL 50-284, Wrocław, Poland.
Wojciech
W
Młotkowski
Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, ul. S. Łojasiewicza 11, PL 30-348 Kraków, Poland.
Maciej A
MA
Nowak
Sorbonne Universités, Université Paris VI, Laboratoire de Physique de la Matière Condensée (LPTMC), CNRS UMR 7600, t.13, 5ème ét. BC.121, 4 pl. Jussieu, F 75252 Paris Cedex 05, France.
Karol A
KA
Penson
Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, ul. S. Łojasiewicza 11, PL 30-348 Kraków, Poland.|Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46 PL 02-668 Warszawa, Poland.
Karol
K
Życzkowski