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May 2013 Spectral statistics of Erdős–Rényi graphs I: Local semicircle law
László Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin
Ann. Probab. 41(3B): 2279-2375 (May 2013). DOI: 10.1214/11-AOP734

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 $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (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 $\ell^{\infty}$-norms of the $\ell^{2}$-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 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 $pN\gg N^{2/3}$.

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László Erdős. Antti Knowles. Horng-Tzer Yau. Jun Yin. "Spectral statistics of Erdős–Rényi graphs I: Local semicircle law." Ann. Probab. 41 (3B) 2279 - 2375, May 2013. https://doi.org/10.1214/11-AOP734

Information

Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1272.05111
MathSciNet: MR3098073
Digital Object Identifier: 10.1214/11-AOP734

Subjects:
Primary: 15B52, 82B44

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3B • May 2013
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