  We prove a local law for the adjacency matrix of the Erd\H{o}s-R\'enyi graph
$G(N, p)$ in the supercritical regime $ pN \geq C\log N$ where $G(N,p)$ has
with high probability no isolated vertices. In the same regime, we also prove
the complete delocalization of the eigenvectors. Both results are false in the
complementary subcritical regime. Our result improves the corresponding results
from [11] by extending them all the way down to the critical scale $pN = O(\log
N)$.
  A key ingredient of our proof is a new family of multilinear large deviation
estimates for sparse random vectors, which carefully balance mixed $\ell^2$ and
$\ell^\infty$ norms of the coefficients with combinatorial factors, allowing us
to prove strong enough concentration down to the critical scale $pN = O(\log
N)$. These estimates are of independent interest and we expect them to be more
generally useful in the analysis of very sparse random matrices.
