The transience–ballisticity conjecture for uniformly elliptic RWRE

Let $d>1$, let $\mathbb{P}$ be an i.i.d. uniformly elliptic environment on $\mathbb{Z}^d$, and let $l$ be a direction. The RWRE is transient in direction $l$ when

$$P_{0}\left(\lim_{n\to\infty}X_n\cdot l=\infty\right)=1,$$

and ballistic in direction $l$ when

$$P_0\text{-a.s.},\qquad \liminf_{n\to\infty}\frac{X_n\cdot l}{n}>0.$$

Transience–ballisticity conjecture. Every uniformly elliptic RWRE in an i.i.d. environment that is transient in a given direction is ballistic in the same direction. This conjecture concerns whether uniform ellipticity rules out the separation between directional transience and directional ballisticity known for some elliptic or one-dimensional random walks in random environments. Its status is open in the source.