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

Abstract

The object under consideration in this article is the total volume Vg,n(x1, …, xn) of the moduli space of hyperbolic surfaces of genus g with n boundary components of lengths x1, …, xn, for the Weil–Petersson volume form. We prove the existence of an asymptotic expansion of the quantity Vg,n(x1, …, xn) in terms of negative powers of the genus g, true for fixed n and any x1, …, xn ≥ 0. The first term of this expansion appears in the work of Mirzakhani and Petri [Comment. Math. Helvetici 94, 869–889 (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.