Conformal restriction: The chordal case |
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| Authors: | Gregory Lawler Oded Schramm Wendelin Werner |
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| Affiliation: | Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201 ; Microsoft Corporation, One Microsoft Way, Redmond, Washington 98052 ; Département de Mathématiques, Bât. 425, Université Paris-Sud, 91405 ORSAY cedex, France |
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| Abstract: | We characterize and describe all random subsets  of a given simply connected planar domain (the upper half-plane  , say) which satisfy the ``conformal restriction' property, i.e.,  connects two fixed boundary points ( and  , say) and the law of  conditioned to remain in a simply connected open subset  of  is identical to that of  , where  is a conformal map from  onto  with  and  . The construction of this family relies on the stochastic Loewner evolution processes with parameter  and on their distortion under conformal maps. We show in particular that SLE is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE . |
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| Keywords: | Conformal invariance restriction property random fractals SLE |
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