Concentration Of The Spectral Norm Of Erdos-Renyi Random Graphs

We present results on the concentration properties of the spectral norm parallel to A(p)parallel to of the adjacency matrix A(p) of an Erdos-Renyi random graph G(n, p). First, we consider the Erdos-Renyi random graph process and prove that parallel to A(p)parallel to is uniformly concentrated over the range p is an element of [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 parallel to A(p)parallel to for all p is an element of [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 Erdos et al. (Ann. Probab. 41 (2013) 2279-2375). Both results are consistent with the asymptotic result of Furedi and Komlos (Combinatorica 1 (1981) 233-241) that holds for fixed p as n -> infinity.