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 $\mathit{d}/\mathit{N}$. For ${(loglog\mathit{N})}^4\ll \mathit{d}\lesssim log\mathit{N}$, 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 $\mathit{d}\asymp log\mathit{N}$, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. 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 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.