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 ${\mathbb H}$, 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$_6$.