## Hearing the shape of a random matrix

I will consider the singular values of Young diagram shaped random matrices. In the 'large shape' limit, the empirical distribution of the squares of the singular values converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers that count (coloured) plane trees. For block-shaped random matrices, the limiting measure 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. I will also discuss some connections to operator-valued free probability. Based on joint works with Elia Bisi, Marilena Ligabò and Tommaso Monni.

Area: CS8 - Combinatorial structures in probability and statistics (Elia Bisi)

Keywords: Random matrices,Young diagrams, r -plane trees, generalized Catalan numbers, operator-valued free probability

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