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Title: Boundary proximity of SLE
Authors: Schramm, O.
Zhou, W. 
Keywords: Hausdorff dimension
Issue Date: Dec-2009
Citation: Schramm, O., Zhou, W. (2009-12). Boundary proximity of SLE. Probability Theory and Related Fields 146 (3) : 435-450. ScholarBank@NUS Repository.
Abstract: 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 ε (0, 4), it is shown that if β > β K:= 1/(8/K - 2), then the intersection of the SLEK curve with the graph of the function y = x/(log x)β, x > e, is a.s. bounded, while it is a.s. unbounded if β = βK. The critical SLE4 curve a.s. intersects the graph of x > ee, in an unbounded set if α ≤ 1, but not if α > 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. © Springer-Verlag 2008.
Source Title: Probability Theory and Related Fields
ISSN: 01788051
DOI: 10.1007/s00440-008-0195-1
Appears in Collections:Staff Publications

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