We prove a law of large numbers for a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the nonnestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman and Zerner.
"A law of large numbers for random walks in random mixing environments." Ann. Probab. 32 (1B) 880 - 914, January 2004. https://doi.org/10.1214/aop/1079021467