SPECTRAL STATISTICS OF ERDŐS—RÉNYI GRAPHS I: LOCAL SEMICIRCLE LAW

We consider the ensemble of adjacency matrices of Erdős—Rényi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N), the density of eigenvalues of the Erdős—Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N -1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ℓ ∞ -norms of the ℓ 2 -normalized eigenvectors are at most of order N -1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős—Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN >> N 2/3 .