Local semicircle law for random regular graphs

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.