Boundary proximity of SLE

Abstract

This paper examines how close the chordal $\SLE_\kappa$ curve gets to the real line asymptotically far away from its starting point. In particular, when $\kappa\in(0,4)$, it is shown that if $\beta>\beta_\kappa:=1/(8/\kappa-2)$, then the intersection of the $\SLE_\kappa$ curve with the graph of the function $y=x/(\log x)^{\beta}$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $\beta=\beta_\kappa$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^\alpha}$, $x>e^e$, in an unbounded set if $\alpha\le 1$, but not if $\alpha>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_\kappa$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_{\kappa}$ curve and real axis is $2-8/\kappa$ when $4<\kappa<8$.