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

TL;DR

This paper estimates the variance of closed geodesics in small regions on the modular surface, proposing a probabilistic model and proving a conjecture about its asymptotic behavior, which differs from previous cases.

Contribution

It introduces a new probabilistic model for closed geodesics and proves the conjectured variance behavior on the modular surface, resolving an open question.

Findings

Variance asymptotics for geodesics in small regions

Proved conjecture for variance behavior on the modular surface

Established difference between variance and expected value in this setting

Abstract

We asymptotically estimate the variance for the distribution of closed geodesics in small random balls or annuli on the modular surface $\Gamma\backslash\mathbb{H}$. 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 $\Gamma\backslash\mathbb{H}$. 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.