Experiments suggesting that the distribution of the hyperbolic length of closed geodesics sampling by word length is Gaussian

TL;DR

This paper uses computer experiments to investigate the distribution of hyperbolic lengths of closed geodesics on a pair of pants surface, suggesting that the lengths follow a Gaussian distribution as word length increases.

Contribution

It provides empirical evidence that the distribution of hyperbolic lengths of closed geodesics, sampled by word length, is approximately Gaussian.

Findings

Distribution of lengths appears normal based on experiments

Length distribution becomes more Gaussian as word length increases

Supports conjecture of Gaussian behavior in hyperbolic length distribution

Abstract

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length in the pair of pants surface. Our experiments strongly suggest that the distribution is normal.