Boundary proximity of SLE |
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Authors: | Oded Schramm Wang Zhou |
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Institution: | 1. Microsoft Research, Redmond, WA, USA 2. Department of Statistics and Applied Probability, Faculty of Science, National University of Singapore, Block S16, Level 6, 6 Science Drive 2, Singapore, 117546, Singapore
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Abstract: | This paper examines how close the chordal SLE κ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ? (0, 4), it is shown that if β > β κ := 1/(8/κ ? 2), then the intersection of the SLE κ 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 = β κ . The critical SLE4 curve a.s. intersects the graph of $y=x^{{-({\rm log\,log\,x})}^{\alpha}}, x > e^e$ , x > e e , 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 SLE κ 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 SLE κ path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLE κ curve and the real axis is 2 ? 8/κ when 4 < κ < 8. |
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