The ballisticity conjecture: directional transience implies ballisticity for random walks in iid uniformly elliptic environments

Theorem 1.1: for $d\ge2$, iid uniformly elliptic nearest-neighbour rows (no independence within a row needed) and a fixed unit $\ell$ with $P_0(A_\ell)=1$, there is a deterministic $v$ with $v\cdot\ell>0$ and $X_n/n\to v$ almost surely, so the walk is ballistic in direction $\ell$. For $d\ge3$ positive probability of $A_\ell$ suffices, the velocity is unique, and the directions with positive transience probability are exactly the open hemisphere $\{u:u\cdot v>0\}$, with probability one there and zero elsewhere. Not shown: the strictly (not uniformly) elliptic case, non-iid environments, or central limit theorems and slowdown estimates.

Theorem 1.1: for $d\ge2$, iid uniformly elliptic nearest-neighbour rows (no independence within a row needed) and a fixed unit $\ell$ with $P_0(A_\ell)=1$, there is a deterministic $v$ with $v\cdot\ell>0$ and $X_n/n\to v$ almost surely, so the walk is ballistic in direction $\ell$. For $d\ge3$ positive probability of $A_\ell$ suffices, the velocity is unique, and the directions with positive transience probability are exactly the open hemisphere $\{u:u\cdot v>0\}$, with probability one there and zero elsewhere. Not shown: the strictly (not uniformly) elliptic case, non-iid environments, or central limit theorems and slowdown estimates.