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 ErdösRenyi graphs gives a good approximation of the spectral distribution for large average degrees. This model and a Wisharttype 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 ErdösRenyi 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 twoband regions.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.02655
 arXiv:
 arXiv:2106.02655
 Bibcode:
 2021arXiv210602655P
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 55 pages, 10 figures