Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment

TL;DR

This paper extends the equivalence of certain ballisticity conditions for high-dimensional random walks in random environments, using sharp exit probability estimates and multiscale methods to advance understanding of ballistic behavior.

Contribution

It broadens the range of gamma values for which the conditions $(T)_{ ext{gamma}}$ and $(T')$ are equivalent in dimensions greater than three, employing new exit probability estimates and confirming a conjecture by Sznitman.

Findings

Extended the equivalence range to all gamma in (0,1) for dimensions >3.

Established sharp estimates for atypical quenched exit probabilities.

Confirmed a conjecture by Sznitman regarding exit probabilities.

Abstract

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $\gamma\in(0,1)$ the ballisticity condition $(T)_{\gamma}$ and the condition $(T')$ defined as the fulfillment of $(T)_{\gamma}$ for each $\gamma\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $\gamma\in (0.5,1)$, $(T)_{\gamma}$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $\gamma\in (0,1)$, condition $(T)_{\gamma}$ implies a ballistic law of large numbers. On the other hand, Drewitz and Ram\'{{\i}}rez have shown that in dimensions $d\ge2$ there is a constant $\gamma_d\in(0.366,0.388)$ such that for each $\gamma\in(\gamma_d,1)$, condition $(T)_{\gamma}$ is equivalent to $(T')$. Here, for dimensions larger than three, we extend the previous range of equivalence to all $\gamma\in(0,1)$. For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.