Eigenvalue distribution of bipartite large weighted random graphs. Resolvent approach

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 the finite second moment. We consider the resolvents $G^{(N,p,\alpha )}(z)$ of $A^{(N,p,\alpha )}$ and study the functions $f_{1,N}(u,z)=\frac{1}{[\alpha N]}{\sum }_{k=1}^{[\alpha N]}{e}^{-u{a}_k^2{G}_{kk}^{(N,p,\alpha )}(z)}$ and $f_{2,N}(u,z)=\frac{1}{N-[\alpha N]}{\sum }_{k=[\alpha N]+1}^N{e}^{-u{a}_k^2{G}_{kk}^{(N,p,\alpha )}(z)}$ in the limit $N\to \infty$. We derive closed system of equations that uniquely determine the limiting functions $f_1(u,z)$ and $f_2(u,z)$. This system of equations allow us to prove the existence of the limiting measure ${\sigma }_{p,\alpha }$ . The weak convergence in probability of normalized eigenvalue counting measures is proved.

12 pages. arXiv admin note: text overlap with arXiv:0911.5684 by other authors

References in corpus (1)

• Strong eigenfunction correlations near the Anderson localization transition