Eigenvalue distribution of bipartite large weighted random graphs. Resolvent approach

We study an eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of the weighted random bipartite graphs $\Gamma= \Gamma_{N,p,\alpha}$. 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 $\alpha p$ for the first part and $(1-\alpha) p$ for the second part of vertices. To each edge of the graph $e_{ij}$, we assign the 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
\begin{gather*}f_{1,N}(u,z)=\frac{1}{[\alpha N]}\sum_{k=1}^{[\alpha N]}e^{-ua_k^2G_{kk}^{(N,p,\alpha)}(z)} \end{gather*}
and
\begin{gather*}f_{2,N}(u,z)=\frac{1}{N-[\alpha N]}\sum_{k=[\alpha N]+1}^Ne^{-ua_k^2G_{kk}^{(N,p,\alpha)}(z)}\end{gather*}
in the limit $N\to \infty$. We derive a closed system of equations that uniquely determine the limiting functions $f_{1}(u,z)$ and $f_{2}(u,z)$. This system of equations allows us to prove the existence of the limiting measure $\sigma_{p, \alpha}$. The weak convergence in probability of the normalized eigenvalue counting measures is proved.