Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus

TL;DR

This paper investigates the asymptotic growth of Weil-Petersson volumes for moduli spaces of hyperbolic surfaces as genus increases and explores implications for the geometry of random hyperbolic surfaces, including Cheeger constants and geodesic lengths.

Contribution

It provides new asymptotic estimates for Weil-Petersson volumes and applies these to analyze geometric properties of large genus random hyperbolic surfaces.

Findings

Weil-Petersson volumes grow asymptotically with genus.

Random hyperbolic surfaces have bounded Cheeger constants.

Shortest geodesic lengths exhibit specific asymptotic behavior.

Abstract

In this paper we study the asymptotic behavior of Weil-Petersson volumes of moduli spaces of hyperbolic surfaces of genus $g$ as $g \rightarrow \infty.$ We apply these asymptotic estimates to study the geometric properties of random hyperbolic surfaces, such as the Cheeger constant and the length of the shortest simple closed geodesic of a given combinatorial type.