Remarks on the intersection of SLE$_κ(ρ)$ curve with the real line

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

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