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

Abstract

The object under consideration in this article is the total volume V g, n( x1, …, x n) of the moduli space of hyperbolic surfaces of genus g with n boundary components of lengths x1, …, x n, for the Weil–Petersson volume form. We prove the existence of an asymptotic expansion of the quantity V g, n( x1, …, x n) in terms of negative powers of the genus g, true for fixed n and any x1, …, x n ≥ 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.