Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s, properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Lowner differential equation partial derivative f/partial derivative t = z zeta(t) + z partial derivative f/zeta(t) - z partial derivative z, with boundary values f(z, 0) = z, in the range z is an element of U = {w is an element of C: \w\ < 1}, t less than or equal to 0. We choose zeta(t) := B(-2t), where B(t) is Brownian motion on all starting at a random-uniform point in partial derivative U. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to partial derivative U has the same law as that of the path f(C(t),t) (where f(z, t) is extended continuously to partial derivative U x (-infinity,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.