The topology of $SLE_{\kappa}$ is random for $\kappa >4$

TL;DR

This paper proves that for  > 4, the topology of SLE curves is inherently random, showing no homeomorphism can map one SLE curve to another, highlighting the complex and unpredictable nature of these fractal curves.

Contribution

It establishes the non-existence of homeomorphisms between independent SLE curves for  > 4, extending the understanding of their topological randomness.

Findings

No homeomorphism exists between independent SLE curves for  .

The result extends to   when considering curves modulo parametrization.

SLE curves exhibit intrinsic topological randomness for  > 4.

Abstract

We study the topology of $SLE$ curves for $\kappa > 4$. More precisely, we show that, a.s., there is no homeomorphism $\Phi: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to another for $\kappa \in (4,8)$. Furthermore, we extend the result to $\kappa \geq 8$ by showing that there is no homeomorphism taking one $SLE$ curve to another, when viewed as curves modulo parametrization.