Concentration of the spectral norm of Erdös-Rényi random graphs

We present results on the concentration properties of the spectral norm $\parallel A_p\parallel$ 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 $\parallel A_p\parallel$ 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 $\parallel A_p\parallel$ 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 (Combinatorica1 (1981) 233–241) that holds for fixed $p$ as $n\to \infty$.