Random matrices associated to Young diagrams

Abstract

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.