Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

Erdős, László University of Munich Knowles, Antti Harvard University Yau, Horng-Tzer Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28016

Annals of Probability, 41, (3), 587-640, 2013
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 $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (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 $\ell^{\infty}$-norms of the $\ell^{2}$-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 $pN\gg N^{2/3}$.
Erd˝os-R´enyi graphs, Local semicircle law, Density of states
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@inproceedings{erdős,2013spectral,
  title={Spectral statistics of Erdős–Rényi graphs I: Local semicircle law},
  author={Erdős, László, Knowles, Antti, Yau, Horng-Tzer, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823163800223257413},
  booktitle={Annals of Probability},
  volume={41},
  number={3},
  pages={587-640},
  year={2013},
}
Erdős, László, Knowles, Antti, Yau, Horng-Tzer, and Jun Yin. Spectral statistics of Erdős–Rényi graphs I: Local semicircle law. 2013. Vol. 41. In Annals of Probability. pp.587-640. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823163800223257413.
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