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

2025/11/16 by Shenxing Zhang, Zhang, Shenxing
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2511.12595

Abstract

Let S_g be a closed surface of genus g and H_g be the moduli space of Abelian differentials on S_g. A stratum of H_g, endowed with the Masur-Veech measure, becomes a probability space. Then the number of closed saddle connections with lengths in [(a)/(√(g)),(b)/(√(g))] on a random translation surface in the stratum is a random variable. We prove that when g→ ∞, the distribution of the random variable converges to a Poisson distributed random variable. This result answers a question of Masur, Rafi and Randecker.

Citations

• Lengths of saddle connections on random translation surfaces of large genus
• Large Genus Asymptotics for Volumes of Strata of Abelian Differentials
• Quasimodularity and large genus limits of Siegel-Veech constants
• Lines of minima and Teichmuller geodesics

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• Benjamini-Schramm limits of high genus translation surfaces: research announcement
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• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves