Closed geodesics in homology classes on random hyperbolic surfaces of large genus

Zeev Rudnick

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arXiv · 2607.06263

Closed geodesics in homology classes on random hyperbolic surfaces of large genus

Zeev Rudnick

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Abstract

We study the distribution of closed geodesics in homology classes on random hyperbolic surfaces of large genus. Viewing the surface as a random point in moduli space equipped with the Weil--Petersson probability measure, we investigate the fluctuations of the weighted counting function of closed geodesics in homology classes modulo $q$. We show that, in the large genus limit, the variance is asymptotic to $X\log X$ for every modulus $q>2$, with an exceptional factor of two when $q=2$. We relate our findings to a conjecture of Hooley, and to work of Friedlander and Goldston, on primes in arithmetic progressions. We also introduce a statistical model based on the random allocation of weighted balls which exhibits analogous behavior.

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Zeev Rudnick. 2026-07-07. Closed geodesics in homology classes on random hyperbolic surfaces of large genus. https://arxiv.org/abs/2607.06263

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