Probability Seminar——Schramm-Loewner Evolution contains a topological Sierpiński carpet when $\kappa$ is close to 8
报告人：庄子杰（宾夕法尼亚大学）
时间：2025-06-09 15:10-16:10
地点：智华楼-四元厅
Abstract:
Schramm-Loewner Evolution (SLE$_\kappa$) 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 $\delta>0$ such that for $\kappa \in (8 - \delta, 8)$, 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 $\kappa \in (8 - \delta, 8)$, the adjacency graph of the complementary connected components of the SLE$_\kappa$ curve is disconnected.
During this talk, I will explain the main ideas inspired by Mandelbrot’s fractal percolation model and discuss some open problems.
Bio:
Zijie Zhuang is a PhD student at University of Pennsylvania, working under the supervision of Jian Ding and Xin Sun. He obtained his bachelor’s degree from Peking University in 2021. His research focuses on statistical mechanics and random geometry, in particular percolation theory and Schramm-Loewner Evolution.
