Sparse random matrices: Spectral edge and statistics of rooted trees

Following Furedi and Komlos, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply th is method to sparse N x N random matrices A(N. p) that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm parallel toA(N,p)parallel to in the limit 1 much less than p much less than N. We show that the value p(c) = log N is t he critical one for lim parallel to A(N,p) /rootp parallel to to be bounded or not. We discuss relations of this result with the Erdos-Renyi limit the orem and properties of large random graphs. In the proof, the principal iss ue is that the averaged vertex degree of plane rooted trees of k edges rema ins bounded even when k --> infinity. This observation implies fairly preci se estimates for the moments of A(N.p) They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectra l statistics at the border of the limiting spectra of large random matrices .