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

Shenxing Zhang

TL;DR

The paper proves that the number of closed saddle connections of length in $[a/\sqrt{g}, b/\sqrt{g}]$ on a random translation surface from a fixed stratum converges to a Poisson law in the large genus limit. Central to the method is a new surgery that collapses a closed saddle connection, together with a factorial-moment approach that translates geometric counts into Poisson parameters. The main result shows that for the stratum $\mathcal{H}(2^{g-1})$, the counts in disjoint intervals converge to independent Poisson variables with means $\lambda_{[a,b]}=3\pi(b^2-a^2)$, and this extends to general strata with higher-order zeros, yielding analogous Poisson limits with modified means. The work hinges on Siegel-Veech constant asymptotics, precise control of exceptional loci, and measure-preserving transformations in period coordinates, highlighting a robust Poisson-limit phenomenon for closed-geodesic data in large-genus translation-surface moduli spaces.

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.

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

TL;DR

The paper proves that the number of closed saddle connections of length in on a random translation surface from a fixed stratum converges to a Poisson law in the large genus limit. Central to the method is a new surgery that collapses a closed saddle connection, together with a factorial-moment approach that translates geometric counts into Poisson parameters. The main result shows that for the stratum , the counts in disjoint intervals converge to independent Poisson variables with means , and this extends to general strata with higher-order zeros, yielding analogous Poisson limits with modified means. The work hinges on Siegel-Veech constant asymptotics, precise control of exceptional loci, and measure-preserving transformations in period coordinates, highlighting a robust Poisson-limit phenomenon for closed-geodesic data in large-genus translation-surface moduli spaces.

Abstract

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

Paper Structure

This paper contains 21 sections, 12 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Moduli space of Abeilian differientials
  4. Flat metric
  5. Stratum and Masur-Veech measure
  6. Siegel-Veech constant
  7. Configuration
  8. Siegel-Veech formula and Siegel-Veech constant
  9. Factorial moment
  10. Some claims
  11. Surgery
  12. Open up a higher-order zero
  13. Move zero along closed curve and pinch
  14. Move the zero locally by period mapping.
  15. $F_2$ is well-defined almost everywhere
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $[a_1,b_1],\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\to \infty$, the vector of random variables converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where for $i=1,\cdots,k$. That is,

Figures (5)

  • Figure 1: The collapsing surgery
  • Figure 2: The opening surgery
  • Figure 3: The resulting surface
  • Figure 4: Moving and cut
  • Figure 5: Pinching

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3: Siegel-Veech formula veech1998siegel
  • Theorem 2.4: vallejos2024random Appendix Poposition A.1
  • Theorem 2.5: The method of moment from bollobas2001random, Theorem 1.23
  • Claim 1
  • ...and 16 more