Almost sure multifractal spectrum for the tip of an SLE curve

TL;DR

This paper establishes the almost sure multifractal spectrum at the tip of an SLE curve, providing insights into the conformal map behavior and harmonic measure scaling near the tip.

Contribution

It derives the tip multifractal spectrum for SLE curves and proves its validity with probability one, advancing understanding of conformal maps and harmonic measure in this context.

Findings

Tip multifractal spectrum for SLE curves is explicitly characterized.

Spectrum validity is proven to hold almost surely.

Applications include scaling laws for harmonic measure at the tip.

Abstract

The tip multifractal spectrum of a two-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm-Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability one, and we give applications to the scaling of harmonic measure at the tip.