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 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.