Intersection probabilities for a chordal SLE path and a semicircle

Abstract

We derive a number of estimates for the probability that a chordal SLE κ path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0 < κ < 8 andwhere a = 2/κ 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 < κ < 8, we derive an estimate for the diameter of a chordal SLE κ path in H between two real boundary points 0 and x > 0. For 4 < κ < 8, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE κ path in H from 0 to ∞.2000 Mathematics Subject Classification. 82B21, 60K35, 60G99, 60J65

A Dimension Spectrum for SLE Boundary Collisions

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The Green function for the radial Schramm–Loewner evolution

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