Short geodesics and small eigenvalues on random hyperbolic punctured spheres
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909063200112640 |
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| author | Hide, Will Thomas, Joe |
| author_facet | Hide, Will Thomas, Joe |
| contents | We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with $n$ cusps in the regime $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.Pe19}, we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with $n$ cusps has at least $k=o(n)$ arbitrarily small eigenvalues tends to $1$ as $n\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_15568 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Short geodesics and small eigenvalues on random hyperbolic punctured spheres Hide, Will Thomas, Joe Geometric Topology Differential Geometry Probability Spectral Theory 58J50, 32G15 We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with $n$ cusps in the regime $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.Pe19}, we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with $n$ cusps has at least $k=o(n)$ arbitrarily small eigenvalues tends to $1$ as $n\to\infty$. |
| title | Short geodesics and small eigenvalues on random hyperbolic punctured spheres |
| topic | Geometric Topology Differential Geometry Probability Spectral Theory 58J50, 32G15 |
| url | https://arxiv.org/abs/2209.15568 |