The square summability of the CLE complementary component diameters
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909802586701824 |
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| author | Doherty, Cillian Miller, Jason |
| author_facet | Doherty, Cillian Miller, Jason |
| contents | We show that the sum of the squares of the diameters of the complementary connected components of the CLE$_κ$ carpet/gasket is almost surely finite for $κ\in (8/3, 4) \cup (4, 8)$. This is a prerequisite for the application of a result of Ntalampekos which allows the CLE$_κ$ carpet/gasket to be uniformized to a round Sierpiński packing, in analogy with the classical Koebe uniformization theorem for finitely connected domains. Our result is new in the case that $κ\in (4,8)$ and we provide a new proof for $κ\in (8/3, 4)$. In both cases we use the link between CLE and space-filling SLE. The square-summability of diameters has been proved for $κ\in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method. Our work completes the proof that this property holds for all $κ$ for which CLE$_κ$ is defined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The square summability of the CLE complementary component diameters Doherty, Cillian Miller, Jason Probability We show that the sum of the squares of the diameters of the complementary connected components of the CLE$_κ$ carpet/gasket is almost surely finite for $κ\in (8/3, 4) \cup (4, 8)$. This is a prerequisite for the application of a result of Ntalampekos which allows the CLE$_κ$ carpet/gasket to be uniformized to a round Sierpiński packing, in analogy with the classical Koebe uniformization theorem for finitely connected domains. Our result is new in the case that $κ\in (4,8)$ and we provide a new proof for $κ\in (8/3, 4)$. In both cases we use the link between CLE and space-filling SLE. The square-summability of diameters has been proved for $κ\in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method. Our work completes the proof that this property holds for all $κ$ for which CLE$_κ$ is defined. |
| title | The square summability of the CLE complementary component diameters |
| topic | Probability |
| url | https://arxiv.org/abs/2509.19204 |