Random Walks in Degenerate Random Environments. (1st October 2014) - Holmes, Mark; Salisbury, Thomas S.
Abstract: We study the asymptotic behaviour of random walks in i.i.d. random environments on Z^d . The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience and, in 2-dimensions, the existence of a deterministic limiting velocity.