Conformal restriction: The chordal case

Lawler, Gregory¹ ; Schramm, Oded² ; Werner, Wendelin³

¹ Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201
² Microsoft Corporation, One Microsoft Way, Redmond, Washington 98052
³ Département de Mathématiques, Bât. 425, Université Paris-Sud, 91405 ORSAY cedex, France

Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 917-955

Résumé

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$_6$.

Détail

DOI : 10.1090/S0894-0347-03-00430-2

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