On Asymptotic Properties of Large Random Matrices with Independent Entries

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 $H=\big[(1+δ_{jk})W_{jk}/\sqrt n\big]_{j,k=1}^n$ 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 $|\Im z| \ge η_0$ where $η_0$ is determined by the second moment of $W_{jk}$. By using this method we find the asymptotic form of the expectation ${\bf E}\{g_n(z)\}$ and of the connected correlator ${\bf E}\{g_n(z_1)g_n(z_2)\}- {\bf E}\{g_n(z_1)\} {\bf E}\{g_n(z_2)\}$. We also prove that the centralized trace $ng_n(z)- {\bf E}\{ng_n(z)\}$ 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.