Schramm-Loewner Evolution (SLEκ) is a one-parameter family of random fractal curves that describes the conjectural scaling limits of interfaces in two-dimensional statistical mechanics models. In this talk, I will present a result with Haoyu Liu (PKU) showing that there exists δ > 0 such that for κ ∈ (8 −δ, 8), the range of an SLEκ curve almost surely contains a topological Sierpi´nski carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLEκ is almost surely conformally non-removable, and the conformal welding problem for SLEκ does not have a unique solution. Our result also implies that for κ ∈ (8 − δ, 8), the adjacency graph of the complementary connected components of the SLEκ curve is disconnected. During this talk, I will explain the main ideas inspired by Mandelbrot’s fractal percolation model and discuss some open problems.
