article
Spectral statistics of Erdős-Rényi graphs I: Local semicircle law
published
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Antti
Knowles
author
Horng
Yau
author
Jun
Yin
author
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 Statistics2013
Annals of Probability10.1214/11-AOP734
413 B2279 - 2375
yes
L. Erdös, A. Knowles, H. Yau, and J. Yin, “Spectral statistics of Erdős-Rényi graphs I: Local semicircle law,” <i>Annals of Probability</i>, vol. 41, no. 3 B. Institute of Mathematical Statistics, pp. 2279–2375, 2013.
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>
Erdös, László, et al. “Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law.” <i>Annals of Probability</i>, vol. 41, no. 3 B, Institute of Mathematical Statistics, 2013, pp. 2279–375, doi:<a href="https://doi.org/10.1214/11-AOP734">10.1214/11-AOP734</a>.
Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2013. <a href="https://doi.org/10.1214/11-AOP734">https://doi.org/10.1214/11-AOP734</a>.
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.
L. Erdös, A. Knowles, H. Yau, J. Yin, Annals of Probability 41 (2013) 2279–2375.
Erdös, L., Knowles, A., Yau, H., & Yin, J. (2013). Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/11-AOP734">https://doi.org/10.1214/11-AOP734</a>
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