Intersection probabilities for a chordal SLE path and a semicircle
Tom Alberts, New York University
Michael J Kozdron, University of Regina
Abstract
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 < κ < 8 and γ:[0,∞) → H is a chordal SLE in H from 0 to ∞, then
there exist constants K1 and K2 such that
K1 r(4a-1) <
P ( γ[0,∞) ∩ C(x;rx) ≠ ∅ )
< K2 r(4a-1)
where 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 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 path in H from 0 to ∞.
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