Asymptotic coefficients of Weil-Petersson volumes in the large genus

TL;DR

This paper proves a conjecture about the polynomial nature of asymptotic coefficients of Weil-Petersson volumes in the large genus limit, confirming they are polynomials in rational coefficients and powers of pi.

Contribution

It establishes that the asymptotic coefficients are polynomials in [^{-2}] as conjectured by Mirzakhani-Zograf.

Findings

Confirmed the polynomial nature of asymptotic coefficients in [^{-2}].

Proved Mirzakhani-Zograf's conjecture.

Enhanced understanding of Weil-Petersson volume asymptotics.

Abstract

Mirzakhani-Zograf proved the large genus asymptotic expansions of Weil-Petersson volumes and showed that the asymptotic coefficients are polynomials in $\mathbb Q[\pi^{-2},\pi^2]$. They also conjectured that these are actually polynomials in $\mathbb Q[\pi^{-2}]$. In this paper, we prove Mirzakhani-Zograf's conjecture.