Convergence rates for loop-erased random walk and other Loewner curves

TL;DR

This paper establishes explicit convergence rates for loop-erased random walk (LERW) curves approaching SLE$_2$ curves, using a new geometric measure called the tip structure modulus, and discusses regularity conditions for Loewner curves.

Contribution

It introduces the tip structure modulus as a geometric tool to analyze Loewner curves and provides explicit power-law convergence rates for LERW to SLE$_2$ under boundary regularity assumptions.

Findings

Power-law convergence rate for LERW to SLE$_2$ in supremum norm.

Tip structure modulus as a geometric condition for Hölder continuity of Loewner curves.

Explicit relation between curve regularity and convergence rates.

Abstract

We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial SLE$_2$ path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be H\"{o}lder continuous in the capacity parameterization, assuming its driving term is H\"{o}lder continuous. We also briefly discuss the case when the curves are a priori known to be H\"{o}lder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.