TY - JOUR

T1 - Effective polynomial ballisticity conditions for random walk in random environment

AU - Berger, Noam

AU - Drewitz, Alexander

AU - Ramírez, Alejandro F.

N1 - Publisher Copyright:
© 2014 Wiley Periodicals, Inc. © 2014 Wiley Periodicals, Inc.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - The conditions (T)γ, γ{small element of}(0,1), which were introduced by Sznitman in 2002, have had a significant impact on research in random walk in a random environment. Among others, these conditions entail a ballistic behavior as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., d≥2), in order to establish the conditions (T)γ, it is actually enough to check a corresponding condition (P{script}) of polynomial type. In addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than (T)γ, another advantage of the condition (P{script}) is that it is effective in the sense that it can be checked on finite boxes. In particular, this extends the conjectured equivalence of the conditions (T)γ, γ{small element of}(0,1), to all relevant dimensions.

AB - The conditions (T)γ, γ{small element of}(0,1), which were introduced by Sznitman in 2002, have had a significant impact on research in random walk in a random environment. Among others, these conditions entail a ballistic behavior as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., d≥2), in order to establish the conditions (T)γ, it is actually enough to check a corresponding condition (P{script}) of polynomial type. In addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than (T)γ, another advantage of the condition (P{script}) is that it is effective in the sense that it can be checked on finite boxes. In particular, this extends the conjectured equivalence of the conditions (T)γ, γ{small element of}(0,1), to all relevant dimensions.

UR - http://www.scopus.com/inward/record.url?scp=84908877045&partnerID=8YFLogxK

U2 - 10.1002/cpa.21500

DO - 10.1002/cpa.21500

M3 - Article

AN - SCOPUS:84908877045

SN - 0010-3640

VL - 67

SP - 1947

EP - 1973

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 12

ER -