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Conformal restriction and Brownian motion. (English) Zbl 1339.60141

Summary: This survey paper is based on the lecture notes for the mini course in the summer school at the Yau Mathematics Science Center, Tsinghua University, 2014. { } We describe and characterize all random subsets \(K\) of a 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 point, 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 sets 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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References:

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