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Boundary proximity of SLE
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  • Published: 06 January 2009

Boundary proximity of SLE

  • Oded Schramm1 &
  • Wang Zhou2 

Probability Theory and Related Fields volume 146, pages 435–450 (2010)Cite this article

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  • 12 Citations

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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 ϵ (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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Author information

Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Oded Schramm

  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

    Wang Zhou

Authors
  1. Oded Schramm
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  2. Wang Zhou
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Corresponding author

Correspondence to Wang Zhou.

Additional information

W. Zhou has been supported in part by grants R-155-000-076-112 and R-155-000-083-112 at the National University of Singapore.

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Cite this article

Schramm, O., Zhou, W. Boundary proximity of SLE. Probab. Theory Relat. Fields 146, 435–450 (2010). https://doi.org/10.1007/s00440-008-0195-1

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  • Received: 23 December 2007

  • Published: 06 January 2009

  • Issue Date: March 2010

  • DOI: https://doi.org/10.1007/s00440-008-0195-1

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Keywords

  • SLE
  • Hausdorff dimension

Mathematics Subject Classification (2000)

  • 60D05
  • 28A80
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