On the norm and eigenvalue distribution of large random matrices

We study the eigenvalue distribution of N × N symmetric random matrices H N (x,y) = N -1/2 h(x,y), x,y = 1,...,N, where h(x,y), x ≤ 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 μ(λ) as N → ∞. We derive an equation for the Stieltjes transform of the measure dμ(λ) and show that the latter has a compact support Λ μ . We find the upper bound for lim sup N → ∞ || H N || and study asymptotically the case when there are no eigenvalues of H N outside of Λ μ when N → ∞.