POISSON STATISTICS AND LOCALIZATION AT THE SPECTRAL EDGE OF SPARSE ERDOS-RENYI GRAPHS

We consider the adjacency matrix A of the Erdos-Renyi graph on N ver-tices with edge probability d/N. For (log log N)4 << d < log N, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson point process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale d kappa log N, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known dis-tribution. Together with (Comm. Math. Phys. 388 (2021) 507-579), 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 argu-ment, 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.