Closed geodesics in short intervals for random hyperbolic surfaces

Abstract

We study the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, and compare it with the classical problem of primes in short intervals. Viewing the surface $M$ as a random point in moduli space equipped with the Weil--Petersson measure, we investigate the random variable $Ψ_M(x;H)$ counting closed geodesics with norms in the interval $[X, X+H]$, weighted by primitive length, where $H=o(X)$. This is analogous to the Chebyshev function in prime number theory. Our main result establishes that in the large genus limit, \[ \lim_{g\to \infty}\mathrm{Var}(Ψ_M(X;H)) \sim 2\,H \log X, \] when $X\to \infty$, $H=o(X)$. Goldston and Montgomery related the variance for primes in short intervals to the form factor associated with zeros of the Riemann zeta function, and conjectured that it is asymptotic to \[ H\log(X/H). \] We show that for automorphic L-functions of degree $d>1$, the early-time GUE form factor already follows from the Riemann Hypothesis, thereby recovering the variance $H\log X$ in the very short interval regime predicted by Bui, Keating and Smith. In the geometric setting, the appearance of $\log X$ reflects the much higher spectral density of Laplace eigenvalues relative to zeros of finite-degree $L$-functions, while the additional factor of $2$ is explained by the expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.