Convergence of loop-erased random walk in the natural parametrization

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".

82 pages, 6 figures. Reorganized and shortened. Part of the previous version has been moved to form part of a new separate paper, "The Loewner difference equation and convergence of loop-erased random walk". Several typos have been corrected and figures have been added