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August 2020 Concentration of the spectral norm of Erdős–Rényi random graphs
Gábor Lugosi, Shahar Mendelson, Nikita Zhivotovskiy
Bernoulli 26(3): 2253-2274 (August 2020). DOI: 10.3150/19-BEJ1192

Abstract

We present results on the concentration properties of the spectral norm $\|A_{p}\|$ of the adjacency matrix $A_{p}$ of an Erdős–Rényi random graph $G(n,p)$. First, we consider the Erdős–Rényi 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 $p\in[c\log^{3}n/n,1]$ that improve the general bounds of Alon, Krivelevich, and Vu (Israel J. Math. 131 (2002) 259–267) and some of the more recent results of Erdős et al. (Ann. Probab. 41 (2013) 2279–2375). Both results are consistent with the asymptotic result of Füredi and Komlós (Combinatorica 1 (1981) 233–241) that holds for fixed $p$ as $n\to\infty$.

Citation

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Gábor Lugosi. Shahar Mendelson. Nikita Zhivotovskiy. "Concentration of the spectral norm of Erdős–Rényi random graphs." Bernoulli 26 (3) 2253 - 2274, August 2020. https://doi.org/10.3150/19-BEJ1192

Information

Received: 1 March 2019; Revised: 1 October 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193959
MathSciNet: MR4091108
Digital Object Identifier: 10.3150/19-BEJ1192

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 3 • August 2020
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