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 P₀ its averaged law starting from 0. Given a direction $l\in\mathbb{S}^{d-1}$, 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; P₀-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: P₀(A_l∪A_−l)=1 in the neighborhood of a given direction; there exists an asymptotic direction ν such that P₀(A_ν∪A_−ν)=1 and P₀-a.s we have $\lim_{n\to\infty}X_{n}/|X_{n}|=𝟙_{A_{\nu}}\nu-𝟙_{A_{-\nu}}\nu$; P₀(A_l∪A_−l)=1 if and only if l⋅ν≠0. Furthermore, we give a review of some open problems.