2012/06/27 by Noam Berger, Berger, Noam, Alexander Drewitz +4 · 4 citations
Decision Sciences · Mathematics · #60K37 #82D30 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60K37 #msc:82D30
paper · pdf · doi:10.48550/arxiv.1206.6377
21 pages, 2 figures; followed referee's and readers' comments, corrected minor errors; to appear in Comm. Pure Appl. Math
openalex publication_date 2012/06/27 · arxiv created 2013/02/14 · arxiv updated 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The conditions (T)γ, γ∈ (0,1), which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. Among others, these conditions entail a ballistic behaviour 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) 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) 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∈ (0,1), to all relevant dimensions.