Random walks in degenerate random environments

Details Random walks in degenerate random environments Random walks in degenerate random environments

2011-05-25

arxiv.org/abs/1105.5105v1

Mathematics

Probability

Scientific paper

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. 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. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment. 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. Particular emphasis is placed on the 2-dimensional setting in 2-valued environments where the random walk evolves by choosing uniformly among a random subset of nearest neighbours of their current location. We give explicit velocity calculations in some such cases where renewals occur in a straightforward way.

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