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

Maury Bramson·Ofer Zeitouni·Martin P. W. Zerner
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Abstract

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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Maury Bramson, Ofer Zeitouni, Martin P. W. Zerner. 2006-06-28. Shortest spanning trees and a counterexample for random walks in random environments. https://doi.org/10.1214/009117905000000783

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