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 ZdZd, that is, the probability that the random walk jumps from a site xx to a nearest neighboring site x+ex+e is given by p(e)+eξ(x,e)p(e)+eξ(x,e), where p(e)p(e) is deterministic, {{ξ(x,e):|e|1=1}:x∈Zd}{{ξ(x,e):|e|1=1}:x∈Zd} are i.i.d. and e>0e>0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in ee 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 ee for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d=2d=2.