Ballisticity conditions for random walk in random environment

Alexander Drewitz, Alejandro F. Ramírez

Published 2009-03-25Version 1

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 $(T').$ 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.