SPECTRAL STATISTICS OF ERDOS-RENYI GRAPHS I: LOCAL SEMICIRCLE LAW

• 成果类型: Article
• 署名作者: Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
• 署名单位: University of Munich; Harvard University; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP734
• 发表日期: 2013
• 页码: 2279-2375
• 关键词: random matrices universality; bulk universality; wigner matrices; Orthogonal polynomials; EIGENVALUE; asymptotics; edge; delocalization; deformation; Respect

摘要

We consider the ensemble of adjacency matrices of Erdos-Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p equivalent to p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN -> infinity (with a speed at least logarithmic in N), the density of eigenvalues of the Erdos-Renyi 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(infinity)-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 Erdos-Renyi 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.