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

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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.