Asymptotics of correlators of sparse bipartite random graphs

TL;DR

This paper analyzes the asymptotic behavior of correlation functions in bipartite sparse random matrices, deriving a main term and a recursive system for coefficients that describe their moments.

Contribution

It introduces a new asymptotic analysis framework for correlation functions in bipartite sparse random graphs, including a recursive system for key coefficients.

Findings

Main term of correlation functions scales as N^{-1}

Derived a closed recursive system for coefficients n_{k,m}

Established asymptotic relations for moments of the integrated density of states

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 $\displaystyle\frac{\alpha}{1-\alpha}$ 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.