Scaling limits for minimal and random spanning trees in two dimensions

A general formulation is presented for continuum scaling limits of stochast ic spanning trees. A spanning tree is expressed in this limit through a con sistent collection of subtrees, which includes a tree for every finite set of endpoints in R-d. Tightness of the distribution, as delta --> 0, is esta blished for the following two-dimensional examples: the uniformly random sp anning tree on delta Z(2), the minimal spanning tree on delta Z(2) (with ra ndom edge lengths), and the Euclidean minimal spanning tree on a Poisson pr ocess of points in R-2 with density delta(-2). In each case, sample trees a re 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 delta > 0); (ii) the tree branches;are given by curves whi ch are regular in the sense of Holder continuity; (iii) the branches are al so rough, in the sense that their Hausdorff dimension exceeds 1; (iv) there is a random dense subset of R-2, 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 counta bly many points in R-2; and (vi) the branching numbers are uniformly bounde d. The results include tightness for the loop-erased random walk in two dim ensions. The proofs proceed through the derivation of scale-invariant power hounds on the probabilities of repeated crossings of annuli. (C) 1999 John Wiley & Sons, Inc.