Lengths of saddle connections on random translation surfaces of large genus

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.