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

TL;DR

This paper proves that as the genus of a translation surface grows large, the distribution of the number of closed saddle connections within a scaled length interval converges to a Poisson distribution, answering a question by Masur, Rafi, and Randecker.

Contribution

It establishes the asymptotic Poisson distribution of saddle connection counts on high-genus translation surfaces, a novel result in the field.

Findings

Number of saddle connections follows a Poisson distribution as genus increases.

Distribution converges when lengths are scaled by a0g.

Answers a previously open question by Masur, Rafi, and Randecker.

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.