Eigenvalue distribution of large weighted bipartite random graphs

Abstract

We study eigenvalue distribution of the adjacency matrix $A^{(N,p,\alpha )}$ of weighted random bipartite graphs $\Gamma ={\Gamma }_{N,p}$. 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. We consider the moments of normalized eigenvalue counting measure ${\sigma }_{N,p,\alpha }$ of $A^{(N,p,\alpha )}$. The weak convergence in probability of normalized eigenvalue counting measures is proved.

References in corpus (1)

• Strong eigenfunction correlations near the Anderson localization transition