We derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of Kalikow. The approach is based on the existence of a renewal structure under an assumption of “transience in the direction $l$ .” This extends, to a multidimensional context, previous work of Kesten. Our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution.
"A Law of Large Numbers for Random Walks in Random Environment." Ann. Probab. 27 (4) 1851 - 1869, October 1999. https://doi.org/10.1214/aop/1022874818