Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in Z^d, d\geq 3, of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupianien. The analysis is based on propagating estimates on the variational distance between the exit measure and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities.