CONVERGENCE RATES FOR LOOP-ERASED RANDOM WALK AND OTHER LOEWNER CURVES

• 成果类型: Article
• 署名作者: Viklund, Fredrik Johansson
• 署名单位: Uppsala University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP872
• 发表日期: 2015
• 页码: 119-165
• 关键词: uniform spanning-trees; differential-equation; conformal-mappings; holder continuity; sle; exponent

摘要

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.