Noncrossing partition flow and random matrix models

Abstract

We study a generating function flowing from the one enumerating a set of partitions to the one enumerating the corresponding set of noncrossing partitions; numerical simulations indicate that its limit in the Adjacency random matrix model on bipartite Erd\"os-Renyi graphs gives a good approximation of the spectral distribution for large average degrees. This model and a Wishart-type random matrix model are described using congruence classes on $k$-divisible partitions. We compute, in the $d\to \infty$ limit with $\frac{Z_a}{d}$ fixed, the spectral distribution of an Adjacency and of a Laplacian random block matrix model, on bipartite Erd\"os-Renyi graphs and on bipartite biregular graphs with degrees $Z_1, Z_2$; the former is the approximation previously mentioned; the latter is a mean field approximation of the Hessian of a random bipartite biregular elastic network; it is characterized by an isostatic line and a transition line between the one- and the two-band regions.