SLE Loop Measure and Liouville Quantum Gravity
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| Format: | Preprint |
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2024
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| _version_ | 1866916543058673664 |
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| author | Ang, Morris Cai, Gefei Sun, Xin Wu, Baojun |
| author_facet | Ang, Morris Cai, Gefei Sun, Xin Wu, Baojun |
| contents | As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQG can be used to study the SLE loop measure. Firstly, we show that for $κ\in (8/3,8)$, the loop intensity measure of the conformal loop ensemble agrees with the SLE loop measure as defined by Zhan (2021). The former was initially considered by Kemppainen and Werner (2016) for $κ\in (8/3,4]$, and the latter was constructed for $κ\in (0,8)$. Secondly, we establish a duality for the SLE loop measure between $κ$ and $16/κ$. Thirdly, we obtain the exact formula for the moment of the electrical thickness for the shape (probability) measure of the SLE loop, which in the regime $κ\in (8/3,8)$ was conjectured by Kenyon and Wilson (2004). This relies on the exact formulae for the reflection coefficient and the one-point disk correlation function in Liouville conformal field theory. Finally, we compute several multiplicative constants associated with the SLE loop measure, which are not only of intrinsic interest but also used in our companion paper relating the conformal loop ensemble to the imaginary DOZZ formulae. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_16547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SLE Loop Measure and Liouville Quantum Gravity Ang, Morris Cai, Gefei Sun, Xin Wu, Baojun Probability Mathematical Physics 60J67, 60D05, 81T40 As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQG can be used to study the SLE loop measure. Firstly, we show that for $κ\in (8/3,8)$, the loop intensity measure of the conformal loop ensemble agrees with the SLE loop measure as defined by Zhan (2021). The former was initially considered by Kemppainen and Werner (2016) for $κ\in (8/3,4]$, and the latter was constructed for $κ\in (0,8)$. Secondly, we establish a duality for the SLE loop measure between $κ$ and $16/κ$. Thirdly, we obtain the exact formula for the moment of the electrical thickness for the shape (probability) measure of the SLE loop, which in the regime $κ\in (8/3,8)$ was conjectured by Kenyon and Wilson (2004). This relies on the exact formulae for the reflection coefficient and the one-point disk correlation function in Liouville conformal field theory. Finally, we compute several multiplicative constants associated with the SLE loop measure, which are not only of intrinsic interest but also used in our companion paper relating the conformal loop ensemble to the imaginary DOZZ formulae. |
| title | SLE Loop Measure and Liouville Quantum Gravity |
| topic | Probability Mathematical Physics 60J67, 60D05, 81T40 |
| url | https://arxiv.org/abs/2409.16547 |