The length of closed geodesics on random Riemann surfaces

Abstract

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers’ theorem gives a global bound on the length of the first 3g − 3 geodesics. We use Brooks and Makover’s construction of random Riemann surfaces to investigate the distribution of short (< log(g)) geodesics on random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders geodesics by length \({\gamma_1\le \gamma_2\le \cdots \le \gamma_i ,\ldots}\), then for fixed k, if one allows the genus to go to infinity, the length of γ k is independent of the genus.