Ballisticity conditions for random walk in random environment

TL;DR

This paper proves that for a certain range of parameters, the ballisticity conditions $(T)_ ext{gamma}$ and $(T')$ are equivalent in random walks in random environments, advancing understanding of their relationship.

Contribution

It establishes the equivalence of $(T)_ ext{gamma}$ and $(T')$ for $ ext{gamma}$ above a dimension-dependent threshold, extending previous results.

Findings

Proves $(T)_ ext{gamma}$ is equivalent to $(T')$ for $ ext{gamma} > ext{dimension-dependent constant}$.

Develops techniques controlling atypical quenched exit distributions.

Advances theoretical understanding of ballisticity conditions in random environments.

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 $(T'),$ 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.