Scaling limits for minimal and random spanning trees in two dimensions

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in ℝ^d. Tightness of the distribution, as δ → 0, is established for the following two-dimensional examples: the uniformly random spanning tree on δℤ², the minimal spanning tree on δSℤ² (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in ℝ² with density δ^-2. In each case, sample trees are proven to have the following properties, with probability 1 with respect to any of the limiting measures: (i) there is a single route to infinity (as was known for δ > 0); (ii) the tree branches are given by curves which are regular in the sense of Hölder continuity; (iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds 1; (iv) there is a random dense subset of ℝ², of dimension strictly between 1 and 2, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints; (v) branching occurs at countably many points in ℝ²; and (vi) the branching numbers are uniformly bounded. The results include tightness for the loop-erased random walk in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.