Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces

TL;DR

This work analyzes Weil-Petersson random hyperbolic surfaces in the large-$n$ regime with fixed genus, establishing that many short-scale geometric and spectral features are governed by sharp large-$n$ asymptotics. The authors develop a novel large-$n$ expansion for Weil-Petersson volumes $V_{g,n}(\ell_1,\dots,\ell_k)$, deriving an explicit leading term expressed as a product of modified Bessel functions $I_0$, and obtain precise constants via asymptotics of intersection numbers. These volume asymptotics underpin new results on the Laplacian spectrum, showing a linear-in-$n$ number of small eigenvalues with high probability, and on geodesic structure, proving that most closed geodesics of length up to $L(n)=O(\log n)$ are non-simple. The methods blend Mirzakhani–Zograf recursion with generating-function techniques and a careful analysis of Li–Xu type relations, yielding a unified framework that contrasts with the large-genus sinh-approximation and connects to broader themes in quantum gravity and random geometry.

Abstract

We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small Laplacian eigenvalues is linear in $n$ with high probability. By work of Otal and Rosas [41], this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to $\log(n)$ scales are non-simple. Our main technical contribution is a novel large-$n$ asymptotic formula for the Weil-Petersson volume $V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of the moduli space $\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of genus-$g$ hyperbolic surfaces with $k$ geodesic boundary components and $n-k$ cusps with $k$ fixed, building on work of Manin and Zograf [30].

Theorems & Definitions (43)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 2.1: Mi07a
• Lemma 2.2: Li.Xu2009