The directional zero-one law for random walks in iid random environments in dimension at least three

Theorem 1.1: for $d\ge3$, iid nearest-neighbour rows that are almost surely strictly positive (no uniform lower bound, no moment condition) and any fixed real $\ell\ne0$, $P_0(A_\ell)\in\{0,1\}$. The October companion extends the law to stationary ergodic finite-range-dependent environments under uniform ellipticity, and the family's ballisticity paper gives, for uniformly elliptic iid environments in $d\ge3$, the hemisphere description of the transient directions. Not shown: ballisticity or a velocity under strict ellipticity alone, jumps beyond nearest neighbours, or general mixing environments (where the law fails by Bramson-Zeitouni-Zerner).

Theorem 1.1: for $d\ge3$, iid nearest-neighbour rows that are almost surely strictly positive (no uniform lower bound, no moment condition) and any fixed real $\ell\ne0$, $P_0(A_\ell)\in\{0,1\}$. The October companion extends the law to stationary ergodic finite-range-dependent environments under uniform ellipticity, and the family's ballisticity paper gives, for uniformly elliptic iid environments in $d\ge3$, the hemisphere description of the transient directions. Not shown: ballisticity or a velocity under strict ellipticity alone, jumps beyond nearest neighbours, or general mixing environments (where the law fails by Bramson-Zeitouni-Zerner).