Ballistic random walks in random environment at low disorder

Abstract

We consider random walks in a random environment of the type p₀+γξ_z, where p₀ denotes the transition probabilities of a stationary random walk on ℤ^d, to nearest neighbors, and ξ_z is an i.i.d. random perturbation. We give an explicit expansion, for small γ, of the asymptotic speed of the random walk under the annealed law, up to order 2. As an application, we construct, in dimension d≥2, a walk which goes faster than the stationary walk under the mean environment.