Ballisticity conditions for random walk in random environment
Abstract
Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d 2. In 2002, Sznitman introduced for each γ∈ (0,1) the ballisticity conditions (T)γ and (T'), the latter being defined as the fulfilment of (T)γ for all γ∈ (0,1). He proved that (T') implies ballisticity and that for each γ∈ (0.5,1), (T)γ is equivalent to (T'). It is conjectured that this equivalence holds for all γ∈ (0,1). Here we prove that for γ∈ (γd,1), where γd is a dimension dependent constant taking values in the interval (0.366,0.388), (T)γ is equivalent to (T'). This is achieved by a detour along the effective criterion, the fulfilment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.