Boundary proximity of SLE

• 成果类型: Article
• 署名作者: Schramm, Oded; Zhou, Wang
• 署名单位: National University of Singapore; Microsoft
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0195-1
• 发表日期: 2010
• 页码: 435-450
• 关键词: erased random-walks; conformal-invariance

摘要

This paper examines how close the chordal SLEk curve gets to the real line asymptotically far away from its starting point. In particular, when k is an element of (0, 4), it is shown that if beta > beta(k) := 1/(8/k - 2), then the intersection of the SLEk curve with the graph of the function y = x/(log x)(beta), x > e, is a.s. bounded, while it is a.s. unbounded if beta = beta(k). The critical SLE4 curve a.s. intersects the graph of y = x(-(log log x)alpha), x > e(e), in an unbounded set if alpha <= 1, but not if alpha > 1. Under a very mild regularity assumption on the function y(x), we give a necessary and sufficient integrability condition for the intersection of the SLEk path with the graph of y to be unbounded. When the intersection is bounded a.s., we provide an estimate for the probability that the SLEk path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLEk curve and the real axis is 2 - 8/k when 4 < k < 8.