We present results on the concentration properties of the spectral norm ||Ap|| of the adjacency matrix Ap of an Erdös-Rényi random graph G(n, p). First, we consider the Erdös-Rényi random graph process and prove that ||Ap|| is uniformly concentrated over the range p E [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 ||Ap|| for all p E [c log3 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 Erdös 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 -> infinite.
