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

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.