Shortest Spanning Trees and a Counterexample for Random Walks in Random Environments

We construct forests that span${\Bbb Z}^{d}$, 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${\Bbb Z}^{d}$, for which the corresponding random walk disobeys a certain zero-one law for directional transience.