Conformal restriction: the chordal case

Details Conformal restriction: the chordal case Conformal restriction: the chordal case

2002-09-25

arxiv.org/abs/math/0209343v2

J.Am.Math.Soc.16:917-955,2003

Mathematics

Probability

To appear in JAMS

Scientific paper

10.1090/S0894-0347-03-00430-2

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).

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