  We consider a general class of random matrices whose entries are centred
random variables, independent up to a symmetry constraint. We establish precise
high-probability bounds on the averages of arbitrary monomials in the resolvent
matrix entries. Our results generalize the previous results of [5,16,17] which
constituted a key step in the proof of the local semicircle law with optimal
error bound in mean-field random matrix models. Our bounds apply to random band
matrices, and improve previous estimates from order 2 to order 4 in the cases
relevant for applications. In particular, they lead to a proof of the diffusion
approximation for the magnitude of the resolvent of random band matrices. This,
in turn, implies new delocalization bounds on the eigenvectors. The
applications are presented in a separate paper [3].
