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# Random Walks in Degenerate Random Environments

Published:2013-07-16
Printed: Oct 2014
• Mark Holmes,
Department of Statistics, University of Auckland, Auckland, New Zealand
• Thomas S. Salisbury,
Department of Mathematics and Statistics, York University, Toronto, ON
 Format: LaTeX MathJax PDF

## Abstract

We study the asymptotic behaviour of random walks in i.i.d. random environments on $\mathbb{Z}^d$. 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.
 Keywords: random walk, non-elliptic random environment, zero-one law, coupling
 MSC Classifications: 60K37 - Processes in random environments