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 Wishart--like structure, $W=X{X}^{†}$, 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, ${\mathrm{MP}}^{⊠s}$, 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.

arXiv comment10 latex pages including 4 figures, Ver 4, minor improvements and references update

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Cited by in corpus (10)

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• Products of Random Matrices from Polynomial Ensembles
• Distinguishing Random and Black Hole Microstates
• Reflected entropy in random tensor networks II: a topological index from the canonical purification
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Complete diagrammatics of the single ring theorem
• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
• Free self-decomposability and unimodality of the Fuss-Catalan distributions