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