Random Walk in a Random Environment, a process where a randomly chosen transition probability is assigned to each point, has been exhaustively studied in the one dimensional nearest neighbor case but is much more difficult in any more general case. This paper develops crude techniques for handling transience results and zero-one laws in a generalized set up.
"Generalized Random Walk in a Random Environment." Ann. Probab. 9 (5) 753 - 768, October, 1981. https://doi.org/10.1214/aop/1176994306