Conformal restriction: The chordal case

Abstract

We characterize and describe all random subsets $K$ of a given simply connected planar domain (the upper half-plane $\H$, 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 $D$ of $\H$ is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from $\H$ onto $D$ with $\Phi(0)=0$ and $\Phi(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution (SLE) 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 relate it to the outer boundaries of planar Brownian motion and SLE(6).