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 $\gamma:[0,\infty) \to \overline{\mathbb{H}}$ is a chordal SLE$_\kappa$ in $\mathbb{H}$ from $0$ to $\infty$, then $P\{\gamma[0,\infty) \cap \mathcal{C}(x;rx) \neq \emptyset\} \asymp r^{4a-1}$ where $a=2/\kappa$ and $\mathcal{C}(x;rx)$ denotes the semicircle centred at $x>0$ of radius $rx$, $00$. 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$_\kappa$ path in $\mathbb{H}$ from $0$ to $\infty$.