Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917006308016128 |
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| author | Liu, Haoyu Zhuang, Zijie |
| author_facet | Liu, Haoyu Zhuang, Zijie |
| contents | We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $δ_0>0$ such that for $κ\in (8 - δ_0,8)$, the range of an SLE$_κ$ curve almost surely contains a topological Sierpiński carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLE$_κ$ is almost surely conformally non-removable, and the conformal welding problem for SLE$_κ$ does not have a unique solution. Our result also implies that for $κ\in (8 - δ_0,8)$, the adjacency graph of the complementary connected components of the SLE$_κ$ curve is disconnected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8 Liu, Haoyu Zhuang, Zijie Probability Mathematical Physics Complex Variables We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $δ_0>0$ such that for $κ\in (8 - δ_0,8)$, the range of an SLE$_κ$ curve almost surely contains a topological Sierpiński carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLE$_κ$ is almost surely conformally non-removable, and the conformal welding problem for SLE$_κ$ does not have a unique solution. Our result also implies that for $κ\in (8 - δ_0,8)$, the adjacency graph of the complementary connected components of the SLE$_κ$ curve is disconnected. |
| title | Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8 |
| topic | Probability Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2506.09609 |