Concentration of the spectral norm of Erdös-Rényi random graphs

Authors: Gábor Lugosi, Shahar Mendelson and Nikita K. Zhivotovskiy

Bernoulli, Vol. 26, No 3, 2253-2274, August, 2020

We present results on the concentration properties of the spectral norm of the adjacency matrix of an Erdős–Rényi random graph . First, we consider the Erdős–Rényi random graph process and prove that is uniformly concentrated over the range . 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 for all 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 (Combinatorica1 (1981) 233–241) that holds for fixed as .