Mathematics – Probability
Scientific paper
2011-03-09
Mathematics
Probability
Scientific paper
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. We prove that, as long as $p N \to \infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erd\H{o}s-R\'enyi 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 [12] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $p N \gg N^{2/3}$.
Erdos Laszlo
Knowles Antti
Yau Horng-Tzer
Yin Jun
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