## Electronic Journal of Probability

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

#### Abstract

It is conjectured that in dimensions d ≥ 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 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 &gt; 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 almost (in a sense to be made precise) exponential decay.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 24, 17 pp.

Dates
Accepted: 5 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067130

Digital Object Identifier
doi:10.1214/EJP.v20-3655

Mathematical Reviews number (MathSciNet)
MR3325093

Zentralblatt MATH identifier
1328.60225

Rights

#### Citation

Guerra, Enrique; Ramirez, Alejandro. Almost exponential decay for the exit probability from slabs of ballistic RWRE. Electron. J. Probab. 20 (2015), paper no. 24, 17 pp. doi:10.1214/EJP.v20-3655. https://projecteuclid.org/euclid.ejp/1465067130

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