Almost exponential decay for the exit probability from slabs of ballistic RWRE

TL;DR

This paper investigates the decay rate of exit probabilities for random walks in random environments, showing that a weaker ballisticity condition implies nearly exponential decay, advancing understanding of ballistic behavior in such systems.

Contribution

It proves that the condition $(T')$, weaker than exponential decay, implies almost exponential decay of exit probabilities in ballistic RWRE.

Findings

$(T')$ implies almost exponential decay of exit probabilities

The decay rate is faster than any polynomial but not fully exponential

Advances understanding of ballisticity conditions in RWRE

Abstract

It is conjectured that in dimensions $d\ge 2$ any random walk in an i.i.d. uniformly elliptic random environment (RWRE) which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which would need to be proven in order to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman \cite{Sz02} in 2001 and 2002 are the so called conditions $(T')$ and $(T)$: given a slab of width $L$ orthogonal to $l$, condition $(T')$ in direction $l$ is the requirement that the annealed exit probability of the walk through the side of the slab in the half-space $\{x:x\cdot l<0\}$, decays faster than $e^{-CL^\gamma}$ for all $\gamma\in (0,1)$ and some constant $C>0$, while condition $(T)$ in direction $l$ is the requirement that the decay is exponential $e^{-CL}$. It is believed that $(T')$ implies $(T)$. In this article we show that $(T')$ implies at least an {\it almost} (in a sense to be made precise) exponential decay.