$\hat{I}»$-shaped random matrices, $\hat{I}»$-plane trees, and $\hat{I}»$-Dyck paths

Abstract

We consider random matrices whose shape is the dilation $N\hat{I}»$ of a self-conjugate Young diagram $\hat{I}»$. In the large-$N$ limit, the empirical distribution of the squared singular values converges almost surely to a probability distribution $F^{\hat{I}}»$. The moments of $F^{\hat{I}}»$ enumerate two combinatorial objects: $\hat{I}»$-plane trees and $\hat{I}»$-Dyck paths, which we introduce and show to be in bijection. We also prove that the distribution $F^{\hat{I}}»$ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for $F^{\hat{I}}»$ and we express it as a free convolution of two measures involving a Marchenko-Pastur and a Bernoulli distribution.

24 pages, 4 figures