Commutation relations for two-sided radial SLE
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arXiv
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| Format: | Preprint |
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2024
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| author | Krusell, Ellen Wang, Yilin Wu, Hao |
| author_facet | Krusell, Ellen Wang, Yilin Wu, Hao |
| contents | We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dubédat. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_κ$ for $κ\in (0,8)$: it is either a two-sided radial SLE$_κ$ with spiral of constant spiraling rate or a chordal SLE$_κ$ weighted by a power of the conformal radius of its complement. Namely, for fixed $κ$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. We also discuss the semiclassical limit of the commutation relation as $κ\to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commutation relations for two-sided radial SLE Krusell, Ellen Wang, Yilin Wu, Hao Probability Complex Variables 60J67 We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dubédat. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_κ$ for $κ\in (0,8)$: it is either a two-sided radial SLE$_κ$ with spiral of constant spiraling rate or a chordal SLE$_κ$ weighted by a power of the conformal radius of its complement. Namely, for fixed $κ$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. We also discuss the semiclassical limit of the commutation relation as $κ\to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal. |
| title | Commutation relations for two-sided radial SLE |
| topic | Probability Complex Variables 60J67 |
| url | https://arxiv.org/abs/2405.07082 |