Scaling limits of looperased random walks and uniform spanning trees
Abstract
The uniform spanning tree (UST) and the looperased random walk (LERW) are 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 the 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 2 dimensions. 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 Loewner differential equation ${\partial f\over\partial t} = z {\zeta(t)+z \over \zeta(t)z} {\partial f\over\partial z}$ with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st w<1\}$, $t\le 0$. We choose $\zeta(t):= \B(2t)$, where $\B(t)$ is Brownian motion on $\partial \U$ starting at a randomuniform point in $\partial \U$. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 1999
 arXiv:
 arXiv:math/9904022
 Bibcode:
 1999math......4022S
 Keywords:

 Probability;
 Mathematical Physics;
 60D05;
 60J15
 EPrint:
 (for V2) inserted another figure and two more references