Spectral Statistics of Erd{H o}s-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

Laszlo Erdos Institute of Mathematics, University of Munich Antti Knowles Department of Mathematics, Harvard University Horng-Tzer Yau Department of Mathematics, Harvard University Jun Yin Department of Mathematics, Harvard University

Probability mathscidoc:1608.28019

Communications in Mathematical Physics, 314, (3), 587-640, 2012
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.
Erd˝os-R´enyi graphs, universality, Dyson Brownian motion
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@inproceedings{laszlo2012spectral,
  title={Spectral Statistics of Erd{H o}s-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues},
  author={Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824094901659009427},
  booktitle={Communications in Mathematical Physics},
  volume={314},
  number={3},
  pages={587-640},
  year={2012},
}
Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin. Spectral Statistics of Erd{H o}s-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues. 2012. Vol. 314. In Communications in Mathematical Physics. pp.587-640. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824094901659009427.
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