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

TL;DR

This paper establishes a new coupling method between loop-erased random walk (LERW) and SLE(2) using the Loewner difference equation, advancing understanding of their convergence when curves are parametrized by capacity.

Contribution

It introduces a novel coupling approach based on the Green's function and Loewner difference equation, differing from previous methods, to analyze LERW convergence to SLE(2).

Findings

Constructed a new coupling of LERW and SLE(2)

Used Green's function as a martingale observable

Connected LERW length parametrization to Minkowski content

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.