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September 2019 Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs
Yukun He, Antti Knowles, Matteo Marcozzi
Ann. Probab. 47(5): 3278-3302 (September 2019). DOI: 10.1214/19-AOP1339


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.


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Yukun He. Antti Knowles. Matteo Marcozzi. "Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs." Ann. Probab. 47 (5) 3278 - 3302, September 2019.


Received: 1 September 2018; Revised: 1 January 2019; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145317
MathSciNet: MR4021251
Digital Object Identifier: 10.1214/19-AOP1339

Primary: 05C80, 15B52, 60B20

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 5 • September 2019
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