Large genus asymptotics for lengths of separating closed geodesics on random surfaces
Abstract
In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as $g$ goes to infinity, a generic surface $X\in \mathcal{M}_g$ satisfies asymptotically: (1) the separating systole of $X$ is about $2\log g$; (2) there is a half-collar of width about $\frac{\log g}{2}$ around a separating systolic curve of $X$; (3) the length of shortest separating closed multi-geodesics of $X$ is about $2\log g$. As applications, we also discuss the asymptotic behavior of the extremal separating systole, the non-simple systole and the expectation value of lengths of shortest separating closed multi-geodesics as $g$ goes to infinity.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.07538
- arXiv:
- arXiv:2009.07538
- Bibcode:
- 2020arXiv200907538N
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Differential Geometry;
- Mathematics - Probability
- E-Print:
- Journal of Topology, to appear, 64 pages, 11 figures