We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d. environments.
"A Stationary, mixing and perturbative counterexample to the 0-1-law for random walk in random environment in two dimensions." Electron. J. Probab. 18 1 - 33, 2013. https://doi.org/10.1214/EJP.v18-1880