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

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.