Intersection probabilities for a chordal SLE path and a semicircle

TL;DR

This paper provides estimates for the probability of a chordal SLE path intersecting semicircles in the upper half-plane, including asymptotic behaviors and implications for the path's diameter and swallowing probabilities.

Contribution

It derives new asymptotic estimates for intersection probabilities of SLE paths with semicircles, extending understanding of SLE geometry and boundary interactions.

Findings

Probability asymptotically behaves as r^(4a-1) for intersection with semicircles.

Estimated the diameter of SLE paths between boundary points.

Bounded the probability of a semicircle being swallowed by SLE for 4<kappa<8.

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<kappa<8 and gamma:[0,infinity) to closure(H) is a chordal SLE in H from 0 to infinity, then P(gamma[0,infinity) cap C(x;rx) neq emptyset) asymp 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 infinity.