# A dimension spectrum for SLE boundary collisions

Tom Alberts,
Ilia Binder,
Fredrik Viklund

February 2016
SLE

### Abstract

We consider chordal SLE$(\kappa)$ curves for $\kappa > 4$, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension $\min(2−8/\kappa,1)$. We study the random sets of points at which the curve collides with the real line at a specified “angle” and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov’s theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

Publication

*Comm. Math. Phys.*