A Dimension Spectrum for SLE Boundary Collisions

TL;DR

This paper analyzes the fractal dimensions of boundary collision points of SLE(kappa) curves for kappa > 4, providing a detailed spectrum of the size of these collision sets using advanced probabilistic techniques.

Contribution

It introduces a dimension spectrum for boundary collision points of SLE curves, extending understanding of their fractal geometry for kappa > 4.

Findings

Computed the Hausdorff dimension spectrum of boundary collision sets.

Established the use of Girsanov's theorem for analyzing moments and events.

Provided two-point correlation estimates for the collision points.

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.