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.

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