Local law and complete eigenvector delocalization for supercritical Erd\H{o}s-R\'enyi graphs

TL;DR

This paper establishes a local law and complete eigenvector delocalization for supercritical Erdős-Rényi graphs, extending previous results down to the critical scale and introducing new multilinear large deviation estimates for sparse vectors.

Contribution

It proves local laws and eigenvector delocalization for Erdős-Rényi graphs at the critical scale, using novel multilinear large deviation estimates for sparse vectors.

Findings

Local law for adjacency matrix in supercritical regime

Complete eigenvector delocalization in the same regime

Extension of results down to the critical scale

Abstract

We prove a local law for the adjacency matrix of the Erd\H{o}s-R\'enyi graph $G(N, p)$ in the supercritical regime $ pN \geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from [11] by extending them all the way down to the critical scale $pN = O(\log N)$. A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^2$ and $\ell^\infty$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN = O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.