Shortest Spanning Trees and a Counterexample for Random Walks in Random Environments

We construct forests that span${\Bbb Z}^{d}$, d ≥ 2, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For d ≥ 3, two independent copies of such forests, pointing in opposite directions, can be pruned so as to...

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Bibliographic Details
Published in:The Annals of probability 2006-05, Vol.34 (3), p.821-856
Main Authors: Bramson, Maury, Zeitouni, Ofer, Zerner, Martin P. W.
Format: Article
Language:English
Subjects:
05C80
60K37
82D30
Decision trees
Ellipticity
Insulation
Mathematical functions
Mathematical models
Polynomials
Probabilities
random environment
Random variables
Random walk
spanning tree
Studies
Trees
Umbrellas
Vertices
zero–one law
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Summary:We construct forests that span${\Bbb Z}^{d}$, d ≥ 2, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For d ≥ 3, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in d ≥ 3 a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on${\Bbb Z}^{d}$, for which the corresponding random walk disobeys a certain zero-one law for directional transience.
ISSN:0091-1798
2168-894X
DOI:10.1214/009117905000000783