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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 43 open journal systems 


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 ∞.


Full text: PDF | PostScript

Pages: 448-460

Published on: August 14, 2008


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Electronic Communications in Probability. ISSN: 1083-589X