Vengerovsky V. Asymptotic properties of spectra of sparse random matrices

Ensembles of weighted adjacency matrices of the random graph G(N,p/N), weighted Laplace of the random graph G(N,p/N), and regularized matrices of transition probabilities of the random graph G(N,p/N). Analysis of asymptotic spectral properties of the above-mentioned ensembles. Method of Stieltjes transforms and Wigner’s idea of combinatorial representation of moments. Weak convergence in probability of normalized counting measures and limiting Stieltjes transform of above-mentioned ensembles are obtained; we obtain a system of recurrent relations is found that let us possible to determine the main asymptotic coefficients of correlators of moments for the case of ensembles of the weighted adjacency matrices of G(N,p/N); it is proved that the spectra of first and second ensembles are unbounded; an explicit expression of the limiting measure of the adjacency matrix of G(N,p/N) with parameter p<1 is found; the limiting moments for the ensembles are found. The results are important for random matrix theoryand for further study of asymptotic properties of sparse random matrices.