On Asymptotic Properties of Large Random Matrices with Independent Entries

Alexei M. Khorunzhy; Boris A. Khoruzhenko; Leonid A. Pastur

doi:10.1063/1.531589

paper

On Asymptotic Properties of Large Random Matrices with Independent Entries

arXiv:cond-mat/9606174 · doi:10.1063/1.531589

Abstract

We study the normalized trace $g_n(z)=n^{-1} \mbox{tr} \, (H-zI)^{-1}$ of the resolvent of $n \times n$ real symmetric matrices $H = [(1 + δ_{j k}) W_{j k} / n]_{j, k = 1}^{n}$ assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of $g_{n} (z)$ for $∣ℑ z ∣ \geq η_{0}$ where $η_{0}$ is determined by the second moment of $W_{j k}$ . By using this method we find the asymptotic form of the expectation $E {g_{n} (z)}$ and of the connected correlator $E {g_{n} (z_{1}) g_{n} (z_{2})} - E {g_{n} (z_{1})} E {g_{n} (z_{2})}$ . We also prove that the centralized trace $n g_{n} (z) - E {n g_{n} (z)}$ has the Gaussian distribution in the limit $n = \infty$ . Basing on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles.

29 pages, LaTeX (revtex style files required) submitted to Journ Math Phys

On Asymptotic Properties of Large Random Matrices with Independent Entries

Abstract

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