A high-genus asymptotic expansion of Weil-Petersson volume polynomials

TL;DR

This paper establishes an asymptotic expansion for Weil-Petersson volumes of hyperbolic surfaces as genus grows large, providing explicit second-order terms and utilizing Mirzakhani's topological recursion.

Contribution

It proves the existence of a genus-based asymptotic expansion for Weil-Petersson volumes and explicitly computes the second term, extending previous work.

Findings

Asymptotic expansion exists for fixed boundary components and large genus.

Explicit second-term formula for the expansion.

Utilizes Mirzakhani's topological recursion as a key tool.

Abstract

The object under consideration in this article is the total volume $V_{g,n}(x_1, \ldots, x_n)$ of the moduli space of hyperbolic surfaces of genus $g$ with $n$ boundary components of lengths $x_1, \ldots, x_n$, for the Weil-Petersson volume form. We prove the existence of an asymptotic expansion of the quantity $V_{g,n}(x_1, \ldots, x_n)$ in terms of negative powers of the genus $g$, true for fixed $n$ and any $x_1, \ldots, x_n \geq 0$. The first term of this expansion appears in work of Mirzakhani and Petri (2019), and we compute the second term explicitly. The main tool used in the proof is Mirzakhani's topological recursion formula, for which we provide a comprehensive introduction.