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

TL;DR

This work analyzes the asymptotics of Weil-Petersson volumes $V_{g,n}$ as genus grows and leverages these results to understand the geometry of random hyperbolic surfaces under the WP measure. It derives precise leading-order and $1/g$-level corrections for $V_{g,n}$ and the volume polynomials $V_{g,n}(L)$, showing, for example, $V_{g,n}=(4\pi^2)^{2g+n-3}(2g-3+n)!\,\frac{1}{\sqrt{g\pi}}\left(1+\frac{c_n}{g}+O(\frac{1}{g^2})\right)$ and the ratios $\frac{V_{g,n+1}}{2gV_{g,n}}=4\pi^{2}+O(\frac{1}{g})$, $\frac{V_{g,n}}{V_{g-1,n+2}}=1+O(\frac{1}{g})$, with $V_{g,n}(L)$ tied to psi-class intersections. These analytic results feed geometric consequences: short non-separating geodesics occur with probability $\asymp \varepsilon^{2}$, separating systoles scale like $\log g$, and the Cheeger constant remains bounded away from zero with high probability, while diameter and embedded-ball radii are poly-logarithmic in $g$. The work thus connects volume growth, intersection theory, and large-genus geometry of moduli spaces, informing the typical shape of high-genus hyperbolic surfaces.

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.

Theorems & Definitions (22)

• Conjecture 1.1: Zograf
• Theorem 1.2
• Theorem 2.1: Wolpert
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Corollary 3.4
• Theorem 3.5