A NATURAL PARAMETRIZATION FOR THE SCHRAMM-LOEWNER EVOLUTION

• 成果类型: Article
• 署名作者: Lawler, Gregory F.; Sheffield, Scott
• 署名单位: University of Chicago; University of Chicago; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP560
• 发表日期: 2011
• 页码: 1896-1937
• 关键词: erased random-walks; conformal-invariance

摘要

The Schramm-Loewner evolution (SLE kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When kappa < 8, an instance of SLE kappa is a random planar curve with almost sure Hausdorff dimension d = 1 + kappa/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For kappa < 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE kappa that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is nontrivial (i.e., the curve is not entirely traversed in zero time) for kappa < 4(7 - root 33) = 5.021 ....