Ballisticity conditions for random walk in random environment
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2011
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Abstract
Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d a parts per thousand yen 2. In 2002, Sznitman introduced for each gamma epsilon (0,1) the ballisticity conditions (T) (gamma) and (T'), the latter being defined as the fulfillment of (T) (gamma) for all gamma epsilon (0,1). He proved that (T') implies ballisticity and that for each gamma epsilon (0.5,1) , (T) (gamma) is equivalent to (T'). It is conjectured that this equivalence holds for al;l gamma epsilon (0,1), where gamma (d) is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) (gamma) is equivalent to (T'). This is achieved by a detour along the effective criterion, the fulfillment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.
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Random walk in random environment, Slowdowns, Ballisticity conditions, Asymptotic direction