Multiple chordal SLE(0) and classical Calogero-Moser system
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915313472241664 |
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| author | Zhang, Jiaxin |
| author_facet | Zhang, Jiaxin |
| contents | We develop a general theory of multiple chordal $\mathrm{SLE}(0)$ systems of type $(n, m)$ for positive integers $n$ and $m$ with $m \leq \lfloor n/2 \rfloor$, extending the construction of~\cite{ABKM20} beyond the previously studied case $n = 2m$.
By applying integrals of motion associated with the Loewner evolution, we show that, in the $\mathbb{H}$-uniformization with the marked point $q = \infty$, the traces of type $(n, m)$ multiple chordal $\mathrm{SLE}(0)$ systems correspond to the real locus of real rational functions with $n$ real simple critical points, $m$ simple poles, and a pole of order $n - 2m + 1$ at infinity.
Furthermore, we demonstrate that, under a common capacity parametrization, the Loewner dynamics evolve according to the classical Calogero-Moser Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple chordal SLE(0) and classical Calogero-Moser system Zhang, Jiaxin Probability Mathematical Physics Complex Variables We develop a general theory of multiple chordal $\mathrm{SLE}(0)$ systems of type $(n, m)$ for positive integers $n$ and $m$ with $m \leq \lfloor n/2 \rfloor$, extending the construction of~\cite{ABKM20} beyond the previously studied case $n = 2m$. By applying integrals of motion associated with the Loewner evolution, we show that, in the $\mathbb{H}$-uniformization with the marked point $q = \infty$, the traces of type $(n, m)$ multiple chordal $\mathrm{SLE}(0)$ systems correspond to the real locus of real rational functions with $n$ real simple critical points, $m$ simple poles, and a pole of order $n - 2m + 1$ at infinity. Furthermore, we demonstrate that, under a common capacity parametrization, the Loewner dynamics evolve according to the classical Calogero-Moser Hamiltonian. |
| title | Multiple chordal SLE(0) and classical Calogero-Moser system |
| topic | Probability Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2505.17129 |