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 SLE2 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 Holder continuous in the capacity parameterization, assuming its driving term is Holder continuous. We also briefly discuss the case when the curves are a priori known to be Holder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.
To implement these ideas in a particular setting we need to show that the assumptions we used are satisfied uniformly in @math , with high probability in terms of @math . If a convergence rate for the driving terms (or martingale observable in rough domains) is known, then we believe it is possible to derive the remaining required information from existing results in the literature on discrete models without too much effort, and we derive the needed SLE derivative estimates, from estimates in @cite_12 , in this paper. Indeed, as already mentioned, the event that the geometric condition fails implies annuli crossing events that are reasonably well-understood for the models known to converge to SLE.
@cite_12
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The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ − 6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(κ; ρ1 ,...,ρ n) with respect to SLE(κ).