Random matrices associated to Young diagrams

TL;DR

This paper studies the spectral properties of Young diagram-shaped random matrices, revealing new limiting distributions related to Beta and classical random matrix laws, with explicit formulas and generalizations.

Contribution

It introduces a new class of random matrices shaped by Young diagrams and derives their spectral distribution, generalizing known random matrix laws like Marchenko-Pastur.

Findings

Empirical spectral distribution converges to a distribution generalizing Catalan numbers.

Limiting distribution is a product of rescaled Beta random variables with a hypergeometric transform.

Special cases recover Marchenko-Pastur and Dykema-Haagerup measures.

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 generalisation 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 factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko-Pastur law as product of independent uniform and arcsine random variables.