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.
"Shortest spanning trees and a counterexample for random walks in random environments." Ann. Probab. 34 (3) 821 - 856, May 2006. https://doi.org/10.1214/009117905000000783