## Annals of Probability

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

#### Abstract

We construct forests that span ℤd, 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 ℤd, for which the corresponding random walk disobeys a certain zero–one law for directional transience.

#### Article information

Source
Ann. Probab., Volume 34, Number 3 (2006), 821-856.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aop/1151418483

Digital Object Identifier
doi:10.1214/009117905000000783

Mathematical Reviews number (MathSciNet)
MR2243869

Zentralblatt MATH identifier
1102.60091

#### Citation

Bramson, Maury; Zeitouni, Ofer; Zerner, Martin P. W. Shortest spanning trees and a counterexample for random walks in random environments. Ann. Probab. 34 (2006), no. 3, 821--856. doi:10.1214/009117905000000783. https://projecteuclid.org/euclid.aop/1151418483

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