Spectral statistics of Erdős-Rényi graphs I: Local semicircle law László Erdös ; https://orcid.org/0000-0001-5366-9603 Knowles, Antti Yau, Horng-Tzer Yin, Jun 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. Institute of Mathematical Statistics 2013 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/2781 Erdös L, Knowles A, Yau H, Yin J. Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. <i>Annals of Probability</i>. 2013;41(3 B):2279-2375. doi:<a href="https://doi.org/10.1214/11-AOP734">10.1214/11-AOP734</a> info:eu-repo/semantics/altIdentifier/doi/10.1214/11-AOP734 info:eu-repo/semantics/openAccess