# Spectral statistics of Erdős-Rényi graphs I: Local semicircle law

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.

Journal Article | Published
Author
Erdös, LászlóISTA ; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
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 = 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.
Publishing Year
Date Published
2013-05-01
Journal Title
Annals of Probability
Volume
41
Issue
3 B
Page
2279 - 2375
IST-REx-ID

### Cite this

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
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
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.
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.
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.
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.
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