Open Access
2008 Intersection probabilities for a chordal SLE path and a semicircle
Tom Alberts, Michael Kozdron
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Electron. Commun. Probab. 13: 448-460 (2008). DOI: 10.1214/ECP.v13-1399


We derive a number of estimates for the probability that a chordal SLE$_\kappa$ path in the upper half plane $\mathbb{H}$ intersects a semicircle centred on the real line. We prove that if $0<\kappa <8$ and $\gamma:[0,\infty) \to \overline{\mathbb{H}}$ is a chordal SLE$_\kappa$ in $\mathbb{H}$ from $0$ to $\infty$, then $P\{\gamma[0,\infty) \cap \mathcal{C}(x;rx) \neq \emptyset\} \asymp r^{4a-1}$ where $a=2/\kappa$ and $\mathcal{C}(x;rx)$ denotes the semicircle centred at $x<0$ of radius $rx$, $00$. For $4<\kappa<8$, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE$_\kappa$ path in $\mathbb{H}$ from $0$ to $\infty$.


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Tom Alberts. Michael Kozdron. "Intersection probabilities for a chordal SLE path and a semicircle." Electron. Commun. Probab. 13 448 - 460, 2008.


Accepted: 14 August 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1187.82034
MathSciNet: MR2430712
Digital Object Identifier: 10.1214/ECP.v13-1399

Primary: 82B21
Secondary: 60G99 , 60J65 , 60K35

Keywords: Hausdorff dimension , intersection probability , restriction property , Schramm-Loewner evolution , Schwarz-Christoffel transformation , swallowing time

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