Random Walks in Degenerate Random Environments

Holmes, Mark ; Salisbury, Thomas S.

Canadian journal of mathematics, Tome 66 (2014) no. 5, pp. 1050-1077

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-017-3

Mots-clés : 60K37, Random walk, non-elliptic random environment, zero-one law, coupling

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