Asymptotic properties of large random matrices with independent entries

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Abstract
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√n]j,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 E{gn(z)} and of the connected correlator E{gn(z1)gn(z2)}−E{gn(z1)}E{gn (z2)}. We also prove that the centralized trace ngn(z)−E{ngn(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.
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