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 Rd. 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 δZ2 (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 one 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 H\"older continuity, iii) the branches are also rough,
in the sense that their Hausdorff dimension exceeds one, iv) there is a random
dense subset of R2, of dimension strictly between one and two, 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 (LERW) in two
dimensions. The proofs proceed through the derivation of scale-invariant power
bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex