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

TL;DR

This paper investigates how SLE$_{}()$ curves interact with the boundary, providing asymptotic estimates for boundary proximity and hitting probabilities, extending known results from standard SLE to the () variant.

Contribution

It generalizes previous boundary interaction results for SLE to the () case, analyzing boundary proximity and hitting probabilities.

Findings

Asymptotic behavior of SLE$_{}()$ near the boundary

Probability estimates for SLE$_{}()$ hitting boundary graphs

Extension of Schramm and Zhou's results to SLE$_{}()$

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.