Abstract
We continue our study of the asymptotics of moduli space volumes in large genus, following Mirzakhani and Mirzakhani-Zograf. We deduce some asymptotic properties of random hyperbolic surfaces in large genus: for example, we compute the expectation of the number of simple closed geodesics (separating or non-separating) of given length. We then tackle the statistical study of the length spectrum, including even non-simple geodesics. Following the work of Anantharaman-Monk, we define the notion of "local topological type" for a periodic geodesic, and express the expectation of the number of closed geodesics of given length and given local topological type.
References
M. Mirzakhani, Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus, J. Differential Geom. 2013.
N. Anantharaman, L. Monk, A high-genus asymptotic expansion of Weil-Petersson volume polynomials, J. Math. Phys. 2022
N. Anantharaman, L. Monk, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps, ArXiv 2023, § 4 and 5
