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.
Keywords
Cite
@article{arxiv.1611.01406,
title = {The Loewner difference equation and convergence of loop-erased random walk},
author = {Gregory F. Lawler and Fredrik Viklund},
journal= {arXiv preprint arXiv:1611.01406},
year = {2017}
}
Comments
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