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.

Journal of Topology, to appear, 64 pages, 11 figures

References in corpus (4)

• Large Genus Asymptotics for Intersection Numbers and Principal Strata Volumes of Quadratic Differentials
• A high-genus asymptotic expansion of Weil-Petersson volume polynomials
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• The simple separating systole for hyperbolic surfaces of large genus

Cited by in corpus (6)

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• Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus
• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps
• Averages of determinants of Laplacians over moduli spaces for large genus
• Shape of filling-systole subspace in surface moduli space and critical points of systole function