Random Walks in Degenerate Random Environments. (1st October 2014)

• Record Type: Journal Article
• Authors: Holmes, Mark; Salisbury, Thomas S.
• Is Part Of: Canadian journal of mathematics. Volume 66:Number 5(2014)
• Page Start: 1050
• Page End: 1077
• Publication Date: 2014-10-01
• Subjects: 60K37; Random walk -- non-elliptic random environment -- zero-one law -- coupling; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2013-017-3
• Languages: English
• ISSNs: 0008-414X

Abstract

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.