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

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 $\mathrm{SLE}_{2}$ 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 Hölder continuous in the capacity parameterization, assuming its driving term is Hölder continuous. We also briefly discuss the case when the curves are a priori known to be Hölder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.