On the Norm and Eigenvalue Distribution of Large Random Matrices

Abstract

We study the eigenvalue distribution of $N \times N$ symmetric random matrices $H_N (x, y) = N^-1/2h(x,y),x,y=1,\ldots,N$, where $h(x, y), x \leq y$ are Gaussian weakly dependent random variables. We prove that the normalized eigenvalue counting function of $H_{N}$ converges with probability 1 to a nonrandom function $\mu(\lambda)$ as $N\rightarrow\infty$. We derive an equation for the Stieltjes transform of the measure $d\mu(\lambda)$ and show that the latter has a compact support $\Lambda_\mu$. We find the upper bound for $\lim\sup_{N\rightarrow\infty}\|H_N\|$ and study asymptotically the case when there are no eigenvalues of $H_N$ outside of $\Lambda_\mu$ when $N \rightarrow \infty$.