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

Zeev Rudnick

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

Geometric Topology 2026-07-07 v1 Number Theory

Authors: Zeev Rudnick

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 $q=2$ . This contrasts with Hooley's conjecture for primes in arithmetic progressions, where the variance is expected to be $X\log q$ . We suggest an explanation for this discrepancy, by comparing our result with the corresponding theory for function fields over a finite field.

Keywords

closed geodesics homology classes random hyperbolic surfaces large genus Weil-Petersson measure variance asymptotics

Cite

@article{arxiv.2607.06263,
  title  = {Closed geodesics in homology classes on random hyperbolic surfaces of large genus},
  author = {Zeev Rudnick},
  journal= {arXiv preprint arXiv:2607.06263},
  year   = {2026}
}