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Asymptotic direction in random walks in random environment revisited

Alexander Drewitz and Alejandro F. Ramírez
Brazilian Journal of Probability and Statistics
Vol. 24, No. 2, Contributions to the XII Brazilian School of Probability (July 2010), pp. 212-225
Stable URL: http://www.jstor.org/stable/43601387
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Asymptotic direction in random walks in random environment revisited
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Abstract

Consider a random walk {Xn : n ≥ 0} in an elliptic i.i.d. environment in dimensions d ≥ 2 and call P₀ its averaged law starting from 0. Given a direction l ∊ sd-1, Al = {limn→∞Xn · l = ∞} is called the event that the random walk is transient in the direction l. Recently Simenhaus proved that the following are equivalent: the random walk is transient in the neighborhood of a given direction; P₀-as · there exists a deterministic asymptotic direction; the random walk is transient in any direction contained in the open half space defined by this asymptotic direction. Here we prove that the following are equivalent: P₀(Al ⋃ A-l) = 1 in the neighborhood of a given direction; there exists an asymptotic direction ν such that P₀(Aν ⋃ A-ν) = 1 and P₀-a.s we have limn→∞ ${X_n}/|{X_n}| = {1_{{A_\upsilon }}}\upsilon - {1_{A - \upsilon }}\upsilon $ P₀(Al ⋃ A-l) = 1 if and only if l · ν ≠ 0. Furthermore, we give a review of some open problems.

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