A multifractal boundary spectrum for SLE${}_{\kappa }(\rho )$

Abstract

We study SLE${}_{\kappa }(\rho )$ curves, with $\kappa$ and $\rho$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure Hausdorff dimension of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $g_t$, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

50 pages and 13 figures. Final version -- to appear in Probability Theory and Related Fields