Uniform asymptotics and the Weil–Petersson constant

Abstract

In their celebrated work on the large-genus asymptotic expansion of Weil–Petersson volumes, Mirzakhani and Zograf left open the question of whether the Weil–Petersson constant satisfies $C_{\mathrm{WP}}=π^{-1/2}$. We resolve the conjecture affirmatively. Our strategy is to consider a family of spectral curves interpolating between $y=z-4z^3$ and $y=\sin(2πz)/(2π)$. For each fixed number $n\geq2$ of marked points, we normalize the cusp coefficients using the action integral and the derivative at its first positive zero. We prove that the logarithmic derivatives of these normalized coefficients with respect to the family parameter are $O_n(g^{-1})$, uniformly along the family. The proof combines the descendant recursion, positivity, the dilaton equation, KdV, and coefficient asymptotics for the Painlevé I equation.