QUENCHED EXIT ESTIMATES AND BALLISTICITY CONDITIONS FOR HIGHER-DIMENSIONAL RANDOM WALK IN RANDOM ENVIRONMENT

• 成果类型: Article
• 署名作者: Drewitz, Alexander; Ramirez, Alejandro F.
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Pontificia Universidad Catolica de Chile
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP637
• 发表日期: 2012
• 页码: 459-534

摘要

Consider a random walk in an uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each gamma is an element of (0, 1) the ballisticity condition (T)(gamma) and the condition (T') defined as the fulfillment of (T)gamma for each gamma is an element of (0, 1). Sznitman proved that (T') implies a ballistic law of large numbers. Furthermore, he showed that for all gamma is an element of (0.5, 1), (T)gamma is equivalent to (T'). Recently, Berger has proved that in dimensions larger than three, for each gamma is an element of (0, 1), condition (T)gamma implies a ballistic law of large numbers. On the other hand. Drewitz and Ramirez have shown that in dimensions d >= 2 there is a constant gamma(d) is an element of (0.366, 0.388) such that for each gamma is an element of (gamma d, I), condition (T)gamma is equivalent to (T'). Here, for dimensions larger than three, we extend the previous range of equivalence to all gamma is an element of (0, 1). For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.