On Asymptotic Properties of Large Random Matrices with Independent Entries

TL;DR

This paper analyzes the asymptotic behavior of the resolvent trace of large symmetric random matrices with independent entries, establishing Gaussian fluctuations and supporting universality of local eigenvalue statistics.

Contribution

It develops a rigorous asymptotic analysis method for moments of the resolvent trace and proves Gaussian fluctuations, advancing understanding of eigenvalue universality in non-i.i.d. matrix ensembles.

Findings

Asymptotic form of the expected resolvent trace derived

Centralized trace converges to Gaussian distribution

Results support universality of local eigenvalue statistics

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 $H=\big[(1+\delta_{jk})W_{jk}/\sqrt n\big]_{j,k=1}^n$ 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 $|\Im z| \ge \eta_0$ where $\eta_0$ 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.