Spectral density of generalized Wishart matrices and free multiplicative convolution
Abstract
We investigate the level density for several ensembles of positive random matrices of a Wishartlike structure, W =X X^{†} , where X stands for a nonHermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the MarchenkoPastur (MP) distribution, MP^{⊠s}, which for an integer s yield FussCatalan 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.
 Publication:

Physical Review E
 Pub Date:
 July 2015
 DOI:
 10.1103/PhysRevE.92.012121
 arXiv:
 arXiv:1407.1282
 Bibcode:
 2015PhRvE..92a2121M
 Keywords:

 05.30.Ch;
 05.40.Ca;
 02.50.Fz;
 Quantum ensemble theory;
 Noise;
 Stochastic analysis;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 10 latex pages including 4 figures, Ver 4, minor improvements and references update