---
_id: '2781'
abstract:
- lang: eng
text: '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.'
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Antti
full_name: Knowles, Antti
last_name: Knowles
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: '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'
apa: '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'
chicago: '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.'
ieee: '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.'
ista: '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.'
mla: '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.'
short: L. Erdös, A. Knowles, H. Yau, J. Yin, Annals of Probability 41 (2013) 2279–2375.
date_created: 2018-12-11T11:59:34Z
date_published: 2013-05-01T00:00:00Z
date_updated: 2021-01-12T06:59:41Z
day: '01'
doi: 10.1214/11-AOP734
extern: 1
intvolume: ' 41'
issue: 3 B
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1103.1919
month: '05'
oa: 1
page: 2279 - 2375
publication: Annals of Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4109'
quality_controlled: 0
status: public
title: 'Spectral statistics of Erdős-Rényi graphs I: Local semicircle law'
type: journal_article
volume: 41
year: '2013'
...