Remarks on the intersection of SLE$_{\kappa}(\rho)$ curve with the real line

Abstract

SLE$_{\kappa}(\rho)$ is a variant of SLE$_{\kappa}$ where $\rho$ characterizes the repulsion (if $\rho>0$) or attraction $(\rho<0)$ from the boundary. This paper examines the probabilities of SLE$_{\kappa}(\rho)$ to get close to the boundary. We show how close the chordal SLE$_{\kappa}(\rho)$ curves get to the boundary asymptotically, and provide an estimate for the probability that the SLE$_{\kappa}(\rho)$ curve hits graph of functions. These generalize the similar result derived by Schramm and Zhou for standard SLE$_{\kappa}$ curves.