Schramm-Loewner evolution contains a topological Sierpiński 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 Sierpiński 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.

38 pages, 10 figures; minor revisions