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

Abstract

We construct forests that span ${\mathbb{Z}}^d$, $d\ge 2$, 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 $d\ge 3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\ge 3$ 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.

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

References in corpus (1)

• Random walks in random environments

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