Spectral statistics of Erd\H{o}s-R\'{e}nyi graphs I: Local semicircle law

TL;DR

This paper proves that the eigenvalue distribution of Erdős-Rényi graphs follows the semicircle law under certain conditions, and shows that eigenvectors are delocalized, with results applicable to spectral universality in a companion study.

Contribution

It establishes the local semicircle law for Erdős-Rényi graphs with sparse regimes and eigenvector delocalization, advancing understanding of spectral properties in random graph models.

Findings

Eigenvalue density follows the semicircle law for pN→∞.

Eigenvectors are completely delocalized with high probability.

Results support spectral universality in a subsequent paper.

Abstract

We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi 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 Erd\H{o}s-R\'{e}nyi 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 Erd\H{o}s-R\'{e}nyi 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 $pN\gg N^{2/3}$.