@article{2781,
abstract = {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 ℓ2-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 »N2/3.},
author = {László Erdös and Knowles, Antti and Yau, Horng-Tzer and Yin, Jun},
journal = {Annals of Probability},
number = {3 B},
pages = {2279 -- 2375},
publisher = {Institute of Mathematical Statistics},
title = {{Spectral statistics of Erdős-Rényi graphs I: Local semicircle law}},
doi = {10.1214/11-AOP734},
volume = {41},
year = {2013},
}