Conformal restriction and Brownian motion

Abstract

This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014. We describe and characterize all random subsets $K$ of simply connected domain which satisfy the “conformal restriction” property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $\infty $. In the radial case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $i$. It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.