Asymptotics of correlators of sparse bipartite random graphs

Abstract

We study asymptotic behaviour of the correlation functions of bipartite sparse random $N\times N$ matrices. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha }{1-\alpha }$ and the average number of edges attached to one vertex is $\alpha \cdot p$ or $(1-\alpha )\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with all moments finite. It is shown that the main term of the correlation function of $k$-th and $m$-th moments of the integrated density of states is $N^{-1}{n}_{k,m}$. The closed system of recurrent relations for coefficients $\{n_{k,m}{\}}_{k,m=1}^{\infty }$ was obtained.

Better notations, slightly better explanation