λ-shaped random matrices, λ-plane trees, and λ-Dyck paths

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 Fλ. The moments of Fλ 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 Fλ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for Fλ and we express it as a free convolution of two measures involving a Marchenko–Pastur and a Bernoulli distribution.