Convergence of loop-erased random walk in the natural parameterization

Abstract

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to ${\mathrm{SLE}}_2$ parameterized 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. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.