Ballisticity conditions for random walk in random environment

Abstract

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for each $\gamma \in (0,1)$ the ballisticity conditions $(T{)}_{\gamma }$ and $(T^{\prime }),$ the latter being defined as the fulfilment of $(T{)}_{\gamma }$ for all $\gamma \in (0,1).$ He proved that $(T^{\prime })$ implies ballisticity and that for each $\gamma \in (0.5,1),$ $(T{)}_{\gamma }$ is equivalent to $(T^{\prime })$. It is conjectured that this equivalence holds for all $\gamma \in (0,1).$ Here we prove that for $\gamma \in ({\gamma }_d,1),$ where ${\gamma }_d$ is a dimension dependent constant taking values in the interval $(0.366,0.388),$ $(T{)}_{\gamma }$ is equivalent to $(T^{\prime }).$ 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.