[{"author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"last_name":"Knowles","full_name":"Knowles, Antti","first_name":"Antti"},{"last_name":"Yau","full_name":"Yau, Horng-Tzer","first_name":"Horng"},{"last_name":"Yin","full_name":"Yin, Jun","first_name":"Jun"}],"date_created":"2018-12-11T11:59:34Z","year":"2013","abstract":[{"text":"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.","lang":"eng"}],"page":"2279 - 2375","status":"public","volume":41,"doi":"10.1214/11-AOP734","date_published":"2013-05-01T00:00:00Z","quality_controlled":0,"intvolume":" 41","title":"Spectral statistics of Erdős-Rényi graphs I: Local semicircle law","citation":{"apa":"Erdös, L., Knowles, A., Yau, H., & Yin, J. (2013). Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. *Annals of Probability*. Institute of Mathematical Statistics. https://doi.org/10.1214/11-AOP734","ista":"Erdös L, Knowles A, Yau H, Yin J. 2013. Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. Annals of Probability. 41(3 B), 2279–2375.","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law.” *Annals of Probability*. Institute of Mathematical Statistics, 2013. https://doi.org/10.1214/11-AOP734.","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Annals of Probability 41 (2013) 2279–2375.","mla":"Erdös, László, et al. “Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law.” *Annals of Probability*, vol. 41, no. 3 B, Institute of Mathematical Statistics, 2013, pp. 2279–375, doi:10.1214/11-AOP734.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “Spectral statistics of Erdős-Rényi graphs I: Local semicircle law,” *Annals of Probability*, vol. 41, no. 3 B. Institute of Mathematical Statistics, pp. 2279–2375, 2013.","ama":"Erdös L, Knowles A, Yau H, Yin J. Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. *Annals of Probability*. 2013;41(3 B):2279-2375. doi:10.1214/11-AOP734"},"oa":1,"publisher":"Institute of Mathematical Statistics","type":"journal_article","issue":"3 B","_id":"2781","date_updated":"2021-01-12T06:59:41Z","extern":1,"publication":"Annals of Probability","publist_id":"4109","publication_status":"published","month":"05","main_file_link":[{"url":"http://arxiv.org/abs/1103.1919","open_access":"1"}],"day":"01"}]