We show that random walk in uniformly elliptic i.i.d. environment in dimension ≥5 has at most one non zero limiting velocity. In particular this proves a law of large numbers in the distributionally symmetric case and establishes connections between different conjectures.
"Limiting velocity of high-dimensional random walk in random environment." Ann. Probab. 36 (2) 728 - 738, March 2008. https://doi.org/10.1214/07-AOP338