A Proof of Sznitman's Conjecture about Ballistic RWRE

We consider random walk in a uniformly elliptic i.i.d. random environment in DOUBLE-STRUCK CAPITAL Z(d) for d >= 2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, Sznitman defined the so-called conditions (T) and (T '). The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L. The second one is the requirement that for all gamma is an element of (0, 1) condition (T)(gamma) is satisfied, which in turn is defined as the requirement that the decay is like e-CL gamma for some C > 0. In this article we prove a conjecture of Sznitman from 2002, stating that (T) and (T ') are equivalent. Hence, this closes the circle proving the equivalence of conditions (T), (T '), and (T)(gamma) for some gamma is an element of (0, 1) as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition (P)(M) for M >= 15d + 5 introduced by Berger, Drewitz, and Rami rez in 2014. (c) 2019 Wiley Periodicals, Inc.