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

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