Intersection probabilities for a chordal SLE path and a semicircle

Abstract

We derive a number of estimates for the probability that a chordal SLE$(\kappa )$ path in the upper half plane $\mathbb{H}$ intersects a semicircle centred on the real line. We prove that if $0<\kappa <8$ and $\gamma :[0,\mathrm{\infty })\to \mathbb{H}$ is a chordal SLE$(\kappa )$ in $\mathbb{H}$ from $0$ to $\mathrm{\infty }$ then $\mathbb{P}(\gamma [0,\mathrm{\infty })\cap \mathcal{C}(x,rx)\ne \mathrm{\varnothing }\asymp r^{4a-1}$, where $a=2/\kappa$ and $\mathcal{C}(x,rx)$ denotes the semicircle centered at $x>0$ of radius $rx$. For $4<\kappa <8$ we estimate the probability that an entire semicircle on the real line is swalled at once by a chordal SLE$(\kappa )$ path.