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 $(T'),$ the latter being defined as the fulfilment of $(T)_\gamma$ for all $\gamma\in (0,1)$. He proved that $(T')$ implies ballisticity and that for each $\gamma\in (0.5,1)$, $(T)_\gamma$ is equivalent to $(T')$. 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 $(T')$.