Schramm-Loewner evolution contains a topological Sierpi\'nski carpet when $\kappa$ is close to 8

TL;DR

This paper proves that for values of  close to 8, SLE curves almost surely contain a topological Sierpiski carpet, revealing complex topological and conformal properties in this regime.

Contribution

It demonstrates the presence of a topological Sierpiski carpet within SLE curves near =8 and explores implications for conformal non-removability and the structure of the curve's complement.

Findings

SLE contains a topological Sierpiski carpet near =8

SLE is conformally non-removable in this regime

The adjacency graph of the curve's complement is disconnected

Abstract

We consider the Schramm-Loewner evolution (SLE$_\kappa$) for $\kappa \in (4,8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $\delta_0>0$ such that for $\kappa \in (8 - \delta_0,8)$, the range of an SLE$_\kappa$ curve almost surely contains a topological Sierpi\'nski carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLE$_\kappa$ is almost surely conformally non-removable, and the conformal welding problem for SLE$_\kappa$ does not have a unique solution. Our result also implies that for $\kappa \in (8 - \delta_0,8)$, the adjacency graph of the complementary connected components of the SLE$_\kappa$ curve is disconnected.