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

## TL;DR

This paper investigates the boundary collision sets of SLE$_(c)$ curves, determining their Hausdorff dimensions using advanced probabilistic and geometric techniques, thereby deepening understanding of boundary interactions in conformal stochastic processes.

## Contribution

It introduces a method to compute the Hausdorff dimension of boundary collision sets for SLE$_(c)$, combining moment analysis, Girsanov theorem, and imaginary geometry.

## Key findings

- Determined Hausdorff dimensions of boundary collision sets.
- Developed a new approach combining conformal map derivatives and correlation estimates.
- Extended understanding of boundary behavior in SLE processes.

## 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.