Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime

Abstract

We consider a random walk in random environment in the low disorder regime on Zd. That is, the probability that the random walk jumps from a site x to a nearest neighboring site x+e is given by p(e)+ǫξ(x,e), where p(e) is deterministic, {{ξ(x,e) : |e|1 = 1} : x ∈ Zd} are i.i.d. and ǫ > 0 is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in ǫ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in ǫ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d = 2