Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

• László Erdős ^[2] ; Antti Knowles ^[1] ; Horng-Tzer Yau ^[1] ; Jun Yin ^[3]
  1. [1] Harvard University

     Harvard University

     City of Cambridge, Estados Unidos

     
  2. [2] University of Munich
  3. [3] University of Wisconsin

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 2, 2013, págs. 2279-2375
• Idioma: inglés
• DOI: 10.1214/11-AOP734
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• Resumen

  • 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.