Scaling limit of the loop-erased random walk Green's function

TL;DR

This paper proves that the scaled probability of a loop-erased random walk passing through a specific edge converges to a conformally covariant quantity related to SLE(2), using combinatorial methods rather than SLE techniques.

Contribution

It establishes the scaling limit of the LERW Green's function in a planar domain without relying on SLE techniques, using combinatorial identities and asymptotic analysis.

Findings

The scaled LERW Green's function converges to an explicit conformally covariant limit.

The proof avoids SLE techniques, relying on combinatorial identities.

Sharp asymptotics for loop measures and spinor observables are obtained.

Abstract

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the SLE(2) Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a "spinor" observable for random walk started from one of the vertices of the marked edge.