Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

Abstract

We consider the adjacency matrix $A$ of the Erdős-Rényi graph on $N$ vertices with edge probability $d/N$. For $(\log \log N{)}^4\ll d\lesssim \log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale $d\asymp \log N$, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. Together with [arXiv:2005.14180], our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of $A$. The proof relies on a three-scale rigidity argument, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.

80 pages, 2 figures. Journal version. To appear in Ann. Probab

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