Asymptotic direction in random walks in random environment revisited

Abstract

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 I is an element of S(d-1), A(l) = {lim(n ->infinity) Xn . l = infinity} is called the event that the random walk is transient in the direction I. 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) boolean OR A(-l)) = 1 in the neighborhood of a given direction; there exists an asymptotic direction v such that P(0) (A(upsilon) boolean OR A(-upsilon)) = 1 and P(0)-a.s we have lim(n ->infinity) X(n)/vertical bar X(n)vertical bar = 1(A upsilon)upsilon - 1(A-upsilon)upsilon; P(0) (A(l) boolean OR A(-l)) = 1 if and only if l . upsilon not equal 0. Furthermore, we give a review of some open problems.