Asymptotic properties of large random matrices with independent entries

We study the normalized trace gn (z) =n−1 tr (H−zI) −1 of the resolvent of n×n real symmetric matrices H= (1+δjk) Wjk√nj, k=1n assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of gn (z) for | Iz|≥η0 where η0 is determined by the second moment of Wjk. By using this method we find the asymptotic form of the expectation Egn (z) and of the connected correlator Egn (z1) gn (z2) −Egn (z1) Egn (z2). We also prove that the centralized trace ngn (z) −Engn (z) has the Gaussian distribution in the limit n=∞. Based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles.