Asymptotic direction in random walks in random environment revisited

摘要

Consider a random walk { X _n ?: n ≥0} in an elliptic i.i.d. environment in dimensions d ≥2 and call P ₀ its averaged law starting from 0. Given a direction , A _l ={lim _{n →∞} X _n ? 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; P ₀-a.s. 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: P ₀( A _l ∪ A _{? l} )=1 in the neighborhood of a given direction; there exists an asymptotic direction ν such that P ₀( A _ν ∪ A _{? ν} )=1 and P ₀-a.s we have ; P ₀( A _l ∪ A _{? l} )=1 if and only if l ? ν ≠0. Furthermore, we give a review of some open problems.