We prove a weak version of the law of large numbers for multi-dimensional finite range random walks in certain mixing elliptic random environments. This already improves previously existing results, where a law of large numbers was known only under strong enough transience. We also prove that for such walks the zero-one law implies a law of large numbers.
"On the Zero-One Law and the Law of Large Numbers for Random Walk in Mixing Random Environment." Electron. Commun. Probab. 10 36 - 44, 2005. https://doi.org/10.1214/ECP.v10-1130