Abstract
We have just defined the notion of "local topological type" for a closed curve traced on a surface. We define the associated "volume function" (for simple curves, we fall back on Mirzakhani volume polynomials). We give first bounds on these volume functions, which can be used to study the statistics of the length spectrum of random hyperbolic surfaces. We prove, for example, the Mirzakhani-Petri theorem, according to which the length spectrum converges in distribution, and when the genus tends to infinity, to a Poisson point process. Finally (and independently of these techniques), we demonstrate that the Cheeger constant of a random surface of large genus remains far from zero, which also shows the existence of a spectral hole.
References
N. Anantharaman, L. Monk, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps https://arxiv.org/abs/2304.02678, Sections 4 and 5
M. Mirzakhani, B. Petri, Lengths of closed geodesics on random surfaces of large genus, Comment. Math. Helv. 94 (2019)
M. Mirzakhani, Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus, J. Differential Geom. 2013, Section 4.5
