## The Annals of Probability

### Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

#### Abstract

We prove a local law for the adjacency matrix of the Erdős–Rényi 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 (Ann. Probab. 41 (2013) 2279–2375) 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3278-3302.

Dates
Revised: January 2019
First available in Project Euclid: 22 October 2019

https://projecteuclid.org/euclid.aop/1571731451

Digital Object Identifier
doi:10.1214/19-AOP1339

Mathematical Reviews number (MathSciNet)
MR4021251

#### Citation

He, Yukun; Knowles, Antti; Marcozzi, Matteo. Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs. Ann. Probab. 47 (2019), no. 5, 3278--3302. doi:10.1214/19-AOP1339. https://projecteuclid.org/euclid.aop/1571731451

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