A Zero-One Law for Planar Random walks in Random Environment

Abstract

We solve the problem posed by S.A. Kalikow whether the event that the $x$-coordinate of a random walk in a two-dimensional random environment approaches $\infty$ has necessarily probability either zero or one. The answer is yes if we assume the environment to be i.i.d.and in general no if we allow the environment to be just stationary and ergodic.