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Random Walks in Degenerate Random Environments

Published online by Cambridge University Press:  20 November 2018

Mark Holmes  and
Thomas S. Salisbury
Mark Holmes
Affiliation:
Department of Statistics, University of Auckland, Auckland, New Zealand. e-mail: [email protected]
Thomas S. Salisbury
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON. e-mail: [email protected]
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Abstract

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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

60K37Random walknon-elliptic random environmentzero-one lawcoupling
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 5 , 01 October 2014 , pp. 1050 - 1077
Copyright
Copyright © Canadian Mathematical Society 2014

References

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