On the norm and eigenvalue distribution of large random matrices

Abstract

We study the eigenvalue distribution of N × N symmetric random matrices HN(x, y) = N -1/2h(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 HN converges with probability 1 to a nonrandom function μ(λ) as N → ∞. We derive an equation for the Stieltjes transform of the measure d μ(λ) and show that the latter has a compact support Λμ. We find the upper bound for lim sup N → ∞ ∥ HN ∥ and study asymptotically the case when there are no eigenvalues of HN outside of Λμ when N → ∞.