Lengths of saddle connections on random translation surfaces of large genus

TL;DR

This paper studies the distribution of saddle connections on large genus random translation surfaces, showing they follow a Poisson distribution and are independent across disjoint length intervals as genus grows.

Contribution

It establishes the asymptotic Poisson distribution and independence of saddle connection counts on random translation surfaces of large genus.

Findings

Number of saddle connections in scaled intervals converges to Poisson distribution.

Counts for disjoint length intervals are asymptotically independent.

Distribution becomes predictable as genus tends to infinity.

Abstract

We determine the distribution of the number of saddle connections on a random translation surface of large genus. More specifically, for genus $g$ tending to infinity, the number of saddle connections with lengths in a given interval $[\frac{a}{g}, \frac{b}{g}]$ converges in distribution to a Poisson distributed random variable. Furthermore, the numbers of saddle connections associated to disjoint intervals of lengths are independent.