The Annals of Probability

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

Maury Bramson, Ofer Zeitouni, and Martin P. W. Zerner

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1151418483

Digital Object Identifier
doi:10.1214/009117905000000783

Mathematical Reviews number (MathSciNet)
MR2243869

Zentralblatt MATH identifier
1102.60091

Subjects
Primary: 60K37: Processes in random environments
Secondary: 05C80: Random graphs [See also 60B20] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk random environment spanning tree zero–one law

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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