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

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 $\H_g$, together 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 flat 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.