In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants.
Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the
corresponding closed geodesic on the pair of pants.
Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem.
I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and S. Taylor, and its generalizations
to other hyperbolic groups.