Distribution of lengths of closed saddle connections on moduli space of large genus translation surface
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918413844086784 |
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| author | Zhang, Shenxing |
| author_facet | Zhang, Shenxing |
| contents | Let $S_g$ be a closed surface of genus $g$ and $\mathcal{H}_g$ be the moduli space of Abelian differentials on $S_g$. A stratum of $\mathcal{H}_g$, endowed with the Masur-Veech measure, becomes a probability space. Then the number of closed saddle connections with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ on a random translation surface in the stratum is a random variable. We prove that when $g\to \infty$, the distribution of the random variable converges to a Poisson distributed random variable. This result answers a question of Masur, Rafi and Randecker. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of lengths of closed saddle connections on moduli space of large genus translation surface Zhang, Shenxing Complex Variables 30F30, 30F60 Let $S_g$ be a closed surface of genus $g$ and $\mathcal{H}_g$ be the moduli space of Abelian differentials on $S_g$. A stratum of $\mathcal{H}_g$, endowed with the Masur-Veech measure, becomes a probability space. Then the number of closed saddle connections with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ on a random translation surface in the stratum is a random variable. We prove that when $g\to \infty$, the distribution of the random variable converges to a Poisson distributed random variable. This result answers a question of Masur, Rafi and Randecker. |
| title | Distribution of lengths of closed saddle connections on moduli space of large genus translation surface |
| topic | Complex Variables 30F30, 30F60 |
| url | https://arxiv.org/abs/2511.12595 |