Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus

Abstract

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume $V^{g,n}$ of the moduli spaces of hyperbolic surfaces of genus $g$ with $n$ punctures as $g \to \infty$. Then we discuss basic geometric properties of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure as $g \to \infty$.