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Intersection probabilities for a chordal SLE path and a semicircle

   
@article{ECP1399,
	author = {Tom Alberts and Michael Kozdron},
	title = {Intersection probabilities for a chordal SLE path and a semicircle},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {13},
	year = {2008},
	keywords = {Schramm-Loewner evolution; restriction property; Hausdorff dimension; swallowing time; intersection probability; Schwarz-Christoffel transformation},
	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 $\infty$.},
	pages = {no. 43, 448-460},
	issn = {1083-589X},
	doi = {10.1214/ECP.v13-1399},    
        url = {http://ecp.ejpecp.org/article/view/1399}}