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Asymptotic properties of large random matrices with independent entries. (English) Zbl 0866.15014
The authors develop a rigorous method of asymptotic analysis of moments of the normalized \(\text{trace\;}g_n(z)=n^{-1}\text{Tr}(H-z)^{-1}\) of the resolvent of \(n\times n\) real symmetric matrices \(H=[(1+\delta_{jk})W_{jk}\sqrt n]^n_{j,k=1}\), assuming that their entries are independent but not necessarily identically distributed random variables. The paper starts with the presentation of basic analytical tools. Next, it finds the asymptotic form of the expectation \({\mathbf E}\{g_n(z)\}\) and of the connected correlator \({\mathbf E}\{g_n(z_1)g_n(z_2)\}-{\mathbf E}\{g_n(z_1)\}{\mathbf E}\{g_n(z_2)\}\). It gives a simple proof to improve the asymptotic order from \(O(n^{-2})\) to \(O(n^{-5/2})\), provided the fifth absolute moment of \(W_{jk}\) is uniformly bounded. Further, it is shown that the centralized \(\text{trace\;}ng_n(z)-{\mathbf E}\{ng_n(z)\}\) has the Gaussian distribution in the limit as \(n\to\infty\). Finally, the authors present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles.

MSC:
15B52 Random matrices (algebraic aspects)
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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