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

Abstract

Abstract. 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 Löwner differential equationwith boundary values f (z, 0) = z, in the range z ∈ U = {w ∈ C : |w| < 1}, t 0. We choose ζ(t) := B(−2t), where B(t) is Brownian motion on ∂U starting at a random-uniform point in ∂U. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂U has the same law as that of the path f (ζ(t), t) (where f (z, t) is extended continuously to ∂U × (−∞, 0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.1991 Mathematics Subject Classification. Primary 60D05. Secondary 60J15.

SLE and the free field: Partition functions and couplings

“…In [24], Lawler, Schramm and Werner prove convergence of the Peano path of a UST to SLE 8 for appropriate boundary conditions. The approach here will be closer to the one in [32].…”

“…The notion of scaling limit of a uniform spanning tree is analyzed in [32]. In [24], Lawler, Schramm and Werner prove convergence of the Peano path of a UST to SLE 8 for appropriate boundary conditions.…”

“…domains with two marked boundary points; the conformal invariance requirement reads * SLE = SLE ( ) for a conformal equivalence : → ( ) (in other words, SLE is a covariant functor on the groupoid of configurations). Under the conformal invariance requirement and an additional domain Markov property, the collection of measures is classified by a positive parameter > 0 ( [32]). Another type of conformally invariant scaling limits involves distributions.…”

Schramm-Loewner evolution

“…While one often can only prove a result such as (11), there are many cases where one can give a stronger estimate:…”

“…The Schramm-Loewner evolution (SLE) was introduced by Oded Schramm [11] as a candidate for scaling limits of models in statistical physics. It has led to a much greater rigorous understanding of scaling limits of critical models in twodimensional statistical physics.…”