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arXiv · 2511.12595

Distribution of lengths of closed saddle connections on moduli space of large genus translation surface

Shenxing Zhang
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Abstract

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.

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Shenxing Zhang. 2025-11-16. Distribution of lengths of closed saddle connections on moduli space of large genus translation surface. https://arxiv.org/abs/2511.12595

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