Hausdorff dimension of the SLE curve intersected with the real line

Abstract

We establish an upper bound on the asymptotic probability of an SLE$(\kappa)$ curve hitting two small intervals on the real line as the interval width goes to zero, for the range $4 < \kappa < 8$. As a consequence we are able to prove that the random set of points in $\mathbb{R}$ hit by the curve has Hausdorff dimension $2 - 8/\kappa$, almost surely.