Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment

Abstract

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each γ ∈ (0, 1) the ballisticity condition (T)_γ and the condition (T') defined as the fulfillment of (T)_γ for each γ ∈ (0, 1). Sznitman proved that (T') implies a ballistic law of large numbers. Furthermore, he showed that for all γ ∈ (0.5, 1), (T)_γ is equivalent to (T'). Recently, Berger has proved that in dimensions larger than three, for each γ ∈ (0, 1), condition (T)_γ implies a ballistic law of large numbers. On the other hand, Drewitz and Ramírez have shown that in dimensions d ≥ 2 there is a constant γ_d ∈ (0.366, 0.388) such that for each γ ∈ (γ_d, 1), condition (T)_γ is equivalent to (T'). Here, for dimensions larger than three, we extend the previous range of equivalence to all γ ∈ (0, 1). For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.