We consider random matrices whose shape is the dilation Nλ of a self-conjugate Young diagram λ. In the large-N limit, the empirical distribution of the squared singular values converges almost surely to a probability distribution F^λ. The moments of F^λ enumerate two combinatorial objects: λ-plane trees and λ-Dyck paths, which we introduce and show to be in bijection. We also prove that the distribution F^λ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for F^λ and we express it as a free convolution of two measures involving a Marchenko–Pastur and a Bernoulli distribution.
λ-shaped random matrices, λ-plane trees, and λ-Dyck paths / Bisi, Elia; Cunden, Fabio Deelan. - In: ELECTRONIC JOURNAL OF PROBABILITY. - ISSN 1083-6489. - ELETTRONICO. - 30:(2025), pp. 11.1-11.24. [10.1214/25-ejp1268]
λ-shaped random matrices, λ-plane trees, and λ-Dyck paths
Bisi, Elia
;
2025
Abstract
We consider random matrices whose shape is the dilation Nλ of a self-conjugate Young diagram λ. In the large-N limit, the empirical distribution of the squared singular values converges almost surely to a probability distribution F^λ. The moments of F^λ enumerate two combinatorial objects: λ-plane trees and λ-Dyck paths, which we introduce and show to be in bijection. We also prove that the distribution F^λ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for F^λ and we express it as a free convolution of two measures involving a Marchenko–Pastur and a Bernoulli distribution.File | Dimensione | Formato | |
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