Hausdorff dimension of the SLE curve intersected with the real line

TL;DR

This paper proves that the Hausdorff dimension of the intersection of an SLE(kappa) curve with the real line is almost surely 2-8/kappa for 4<kappa<8, using probability bounds on hitting small intervals.

Contribution

It establishes the Hausdorff dimension of SLE curves intersecting the real line for the first time within the specified kappa range.

Findings

Hausdorff dimension of SLE intersection is 2-8/kappa

Upper bounds on hitting probabilities for small intervals

Almost sure dimension result

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 SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely.