STRONG PATH CONVERGENCE FROM LOEWNER DRIVING FUNCTION CONVERGENCE

• 成果类型: Article
• 署名作者: Sheffield, Scott; Sun, Nike
• 署名单位: Massachusetts Institute of Technology (MIT); Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP627
• 发表日期: 2012
• 页码: 578-610
• 关键词: erased random-walks; conformal-invariance; sle

摘要

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal SLE kappa as a scaling limit, with kappa < 8 (nonspace-filling). Existing SLE kappa convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from infinity) converge to Brownian motion. Unfortunately, this is not sufficient, and additional arguments are required to complete the proofs. We show that driving function convergence is sufficient if it can be established for both parametrization directions and a generic observation point.