  We consider random $d$-regular graphs on $N$ vertices, with degree $d$ at
least $(\log N)^4$. We prove that the Green's function of the adjacency matrix
and the Stieltjes transform of its empirical spectral measure are well
approximated by Wigner's semicircle law, down to the optimal scale given by the
typical eigenvalue spacing (up to a logarithmic correction). Aside from
well-known consequences for the local eigenvalue distribution, this result
implies the complete (isotropic) delocalization of all eigenvectors and a
probabilistic version of quantum unique ergodicity.
