Skip to main content
Article thumbnail
Location of Repository

Spectral Statistics of Erd{\H o}s-R\'enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

By Laszlo Erdos, Antti Knowles, Horng-Tzer Yau and Jun Yin

Abstract

We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ 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. Under the assumption $p N \gg N^{2/3}$, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least $4 + \epsilon$ moments

Topics: Mathematics - Probability, Mathematical Physics, 15B52, 60B20, 05C80
Year: 2012
DOI identifier: 10.1007/s00220-012-1527-7
OAI identifier: oai:arXiv.org:1103.3869
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • http://arxiv.org/abs/1103.3869 (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.