Shortest spanning trees and a counterexample for random walks in random environments

Details Shortest spanning trees and a counterexample for random walks in random environments Shortest spanning trees and a counterexample for random walks in random environments

2005-01-29

arxiv.org/abs/math/0501533v2

Annals of Probability 2006, Vol. 34, No. 3, 821-856

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000783 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000783

We construct forests that span $\mathbb{Z}^d$, $d\geq2$, 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 $d\geq3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain zero--one law for directional transience.

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