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Intersection probabilities for a chordal SLE path and a semicircle. (English) Zbl 1187.82034

Summary: We derive a number of estimates for the probability that a chordal SLE path in the upper half plane \(H\) intersects a semicircle centred on the real line. We prove that if \(0<\kappa<8\) and \(\gamma:[0,\infty)\to \overline{H}\) is a chordal SLE in \(H\) from 0 to \(\infty\), then there exist constants \(K_1\) and \(K_2\) such that
\[ K_1 r^{(4a-1)}< P(\gamma [0,\infty)\cap C(x;rx)\neq \emptyset)< K_2 r(4a-1), \]
where \(a=2/\kappa \) and \(C(x;rx)\) denotes the semicircle centred at \(x>0\) of radius \(rx\), \(0<r<1/3\), in the upper half plane. As an application of our results, for \(0<\kappa<8\), we derive an estimate for the diameter of a chordal SLE path in \(H\) between two real boundary points 0 and \(x>0\). 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 path in \(H\) from 0 to \(\infty\).

MSC:

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G99 Stochastic processes
60J65 Brownian motion
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