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Conformal restriction: The chordal case

Authors: Gregory Lawler, Oded Schramm and Wendelin Werner
Journal: J. Amer. Math. Soc. 16 (2003), 917-955
MSC (2000): Primary 60K35, 82B27, 60J69, 30C99
Published electronically: June 2, 2003
MathSciNet review: 1992830
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Abstract: We characterize and describe all random subsets $K$of a given simply connected planar domain (the upper half-plane ${\mathbb H}$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points ($0$ and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $H$ of ${\mathbb H}$is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from ${\mathbb H}$ onto $H$ with $\Phi(0)=0$ and $\Phi(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution processes with parameter $\kappa \le 8/3$ and on their distortion under conformal maps. We show in particular that SLE$_{8/3}$ is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE$_6$.

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Additional Information

Gregory Lawler
Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201

Oded Schramm
Affiliation: Microsoft Corporation, One Microsoft Way, Redmond, Washington 98052

Wendelin Werner
Affiliation: Département de Mathématiques, Bât. 425, Université Paris-Sud, 91405 ORSAY cedex, France

Keywords: Conformal invariance, restriction property, random fractals, SLE
Received by editor(s): October 9, 2002
Published electronically: June 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society