  We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random
walks in an i.i.d. random environment that is a small perturbation of the
environment corresponding to simple random walk. Under a centering condition on
the measure governing the environment, we prove that the exit laws are close to
those of a symmetric random walk, which we identify as a perturbed simple
random walk. We obtain bounds on total variation distances as well as local
results comparing exit probabilities on boundary segments. As an application,
we prove transience of the random walks in random environment. Our work
includes the results on isotropic random walks in random environment of
Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581-645].
Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a
somewhat different approach was given in the first author's thesis [Long-time
behavior of random walks in random environment (2013) Z\"{u}rich Univ.]. Here,
we extend this approach to certain anisotropic walks and provide a further step
towards a fully perturbative theory of random walks in random environment.
