Convergence of loop-erased random walk in the natural parametrization

TL;DR

This paper proves that loop-erased random walk (LERW) in the plane converges to SLE(2) when parametrized by length, providing a new method applicable to other models converging to SLE.

Contribution

It establishes the convergence of LERW to SLE(2) with a specific parametrization and introduces a general method for proving similar convergence results.

Findings

LERW converges to SLE(2) in the scaling limit

Develops a method for proving convergence of models to SLE

Provides new estimates and lemmas related to LERW

Abstract

Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree. This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content. In doing this we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest: for example, two-point estimates, estimates on maximal content, and a "separation lemma".