The Loewner difference equation and convergence of loop-erased random walk

Abstract

We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.

44 pages, no figures. This article can be viewed as a companion paper to "Convergence of loop-erased walk in the natural parametrization" (arXiv:1603.05203) and a part of this paper formed part of an earlier version of arXiv:1603.05203. Minor changes in v2