Concentration of the spectral norm of Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper investigates the concentration behavior of the spectral norm of Erdős-Rényi random graph adjacency matrices, providing uniform concentration results and sharp moment inequalities across a broad range of edge probabilities.

Contribution

It establishes uniform concentration of the spectral norm over a wide probability range and derives improved sub-Gaussian moment bounds for the spectral norm.

Findings

Spectral norm concentrates uniformly for p in [C log n / n, 1]

Sharp sub-Gaussian moment inequalities for spectral norm

Results align with classical asymptotic behavior as n grows large

Abstract

We present results on the concentration properties of the spectral norm $\|A_p\|$ of the adjacency matrix $A_p$ of an Erd\H{o}s-R\'enyi random graph $G(n,p)$. First we consider the Erd\H{o}s-R\'enyi random graph process and prove that $\|A_p\|$ is uniformly concentrated over the range $p\in [C\log n/n,1]$. The analysis is based on delocalization arguments, uniform laws of large numbers, together with the entropy method to prove concentration inequalities. As an application of our techniques we prove sharp sub-Gaussian moment inequalities for $\|A_p\|$ for all $p\in [c\log^3n/n,1]$ that improve the general bounds of Alon, Krivelevich, and Vu (2001) and some of the more recent results of Erd\H{o}s et al. (2013). Both results are consistent with the asymptotic result of F\"uredi and Koml\'os (1981) that holds for fixed $p$ as $n\to \infty$.