The uniform spanning tree (UST) and the loop-erased random walk (LERW) are
strongly related probabilistic processes. We consider the limits of these m
odels on a fine grid in the plane, as the mesh goes to zero. Although the e
xistence 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 t
runk 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 inv
ariant in dimension 2. We make a precise statement of the conformal invaria
nce conjecture for the LERW, and show that this conjecture implies an expli
cit construction of the scaling limit, as follows. Consider the Lowner diff
erential equation
partial derivative f/partial derivative t = z zeta(t) + z partial derivativ
e 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 LER
W 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) (w
here 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.