The variance of closed geodesics in balls and annuli on the modular surface

We asymptotically estimate the variance for the distribution of closed geodesics in small random balls or annuli on the modular surface $\mathrm{\Gamma }\mathrm{\backslash }\mathbb{H}$. A probabilistic model in which closed geodesics are modeled using random geodesic segments is proposed, and we rigorously analyze this model using mixing of the geodesic flow in $\mathrm{\Gamma }\mathrm{\backslash }\mathbb{H}$. This leads to a conjecture for the asymptotic behavior of the variance, which unlike in previously explored cases is not equal to the expected value. We prove this conjecture for small balls and annuli, resolving a question left open by Humphries and Radziwill.