Towards large genus asymtotics of intersection numbers on moduli spaces of curves

TL;DR

This work establishes a complete large-genus asymptotic expansion for the Weil-Petersson volumes $V_{g,n}$ of moduli spaces of curves, showing that, up to a universal constant $C$, $V_{g,n}$ scales as $(2g-3+n)!(4\pi^2)^{2g-3+n}/\sqrt{g}$ with coefficients that are polynomials in $n$ and appear in a systematic $1/g$ expansion. The authors derive and exploit recursions for tautological intersection numbers to control asymptotics both for fixed $n$ and for slowly growing $n$, and they provide explicit first-order corrections with detailed polynomial structure in $n$ and $\pi^2$. They also develop an explicit error-term framework, proving expansions up to $O(1/g^s)$ and describing the polynomial dependence of coefficients, illustrating the stability of the large-genus regime. Furthermore, they show that when $n$ grows slowly with $g$, the normalized volumes converge to a universal constant, highlighting the robustness of the asymptotic picture across regimes.

Abstract

We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of $n$-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil-Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in $n$. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.

Theorems & Definitions (23)

• Conjecture 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Corollary 1.7
• Theorem 1.8
• Lemma 2.1
• Remark 3.1