The length of closed geodesics on random Riemann Surfaces
Abstract
Description
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 the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $γ_1\le γ_2\le ... \le γ_i ,...$, then for fixed $k$, if one allows the genus to go to infinity, the length of $γ_{k}$ is independent of the genus.
6 figures
6 figures