$λ$-shaped random matrices, $λ$-plane trees, and $λ$-Dyck paths
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909998225817600 |
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| author | Bisi, Elia Cunden, Fabio Deelan |
| author_facet | Bisi, Elia Cunden, Fabio Deelan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $λ$-shaped random matrices, $λ$-plane trees, and $λ$-Dyck paths Bisi, Elia Cunden, Fabio Deelan Probability Mathematical Physics Combinatorics Primary: 60B20, 60F15, 62E15, 44A60, 05C05, 05A15. Secondary: 33C20, 46L54 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. |
| title | $λ$-shaped random matrices, $λ$-plane trees, and $λ$-Dyck paths |
| topic | Probability Mathematical Physics Combinatorics Primary: 60B20, 60F15, 62E15, 44A60, 05C05, 05A15. Secondary: 33C20, 46L54 |
| url | https://arxiv.org/abs/2403.07418 |