Ballisticity Criteria for Random Walk in Random Environments

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 $(T^{\prime })$.