Almut Burchard - Minimal and random spanning trees in two dimension

This talk will describe joint work with M. Aizenman, C. Newman, and D. Wilson, on the subject of (continuum) scaling limits for stochastic tree processes. Three examples will be discussed in some detail:

(1)  The uniformly random spanning tree on a planar square lattice,

(2)  the minimal spanning tree on a planar square lattice with random edge weights,

and

(3)  the Euclidean (minimal) spanning tree on a Poisson point process in the plane.

The infinite-volume limit is known to exist in each case. Here, we consider scaling limits where the typical distance between neighboring sites is taken to zero.

As a first step, we construct a common configuration space, which is analogous to a space of curves. The formulation of the tree processes as probability measures on this space remains meaningful in the scaling limit. Scaling limits exist--at least along suitable subsequences--by a compactness argument. Furthermore, we describe some basic properties of the limiting measures, such as bounds on the dimension of tree branches, and bounds on the number and degree of branching points. The main step in the proof of these statements is to establish a family of scale-invariant bounds on the probability of repeated crossings of annuli by the random tree. The bounds are verified separately for each of the three models.