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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 709–774 | Cite as

Conformal restriction: the trichordal case

  • Wei Qian
Article
  • 142 Downloads

Abstract

The study of conformal restriction properties in two-dimensions has been initiated by Lawler et al. (J Am Math Soc 16(4):917–955, 2003) who focused on the natural and important chordal case: they characterized and constructed all random subsets of a given simply connected domain that join two marked boundary points and that satisfy the additional restriction property. The radial case (sets joining an inside point to a boundary point) has then been investigated by Wu (Stoch Process Appl 125(2):552–570, 2015). In the present paper, we study the third natural instance of such restriction properties, namely the “trichordal case”, where one looks at random sets that join three marked boundary points. This case involves somewhat more technicalities than the other two, as the construction of this family of random sets relies on special variants of SLE\(_{8/3}\) processes with a drift term in the driving function that involves hypergeometric functions. It turns out that such a random set can not be a simple curve simultaneously in the neighborhood of all three marked points, and that the exponent \(\alpha = 20/27\) shows up in the description of the law of the skinniest possible symmetric random set with this trichordal restriction property.

Mathematics Subject Classification

Primary 60D05 60J67 Secondary 60K35 30C99 

Notes

Acknowledgements

The author is very grateful to Wendelin Werner for suggesting this question, for numerous discussions and suggestions, and for his help throughout the preparation of this paper. The author acknowledges support of the SNF Grant SNF-155922. The author is also part of the NCCR Swissmap.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsETHZZurichSwitzerland

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