Boundary proximity of SLE

Abstract

This paper examines how close the chordal $\SLE_κ$ 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 $\SLE_κ$ 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 $\SLE_κ$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_κ$ curve and real axis is $2-8/\kappa$ when $4<\kappa <8$.

18 pages, new results are added, typos are corrected

References in corpus (1)

• The Loewner equation: maps and shapes