Asymptotic coefficients of Weil-Petersson volumes in the large genus

TL;DR

This work resolves a conjecture on the large-genus asymptotics of Weil-Petersson volumes by proving that all expansion coefficients are polynomials in $\mathbb{Q}[\pi^{-2}]$. Building on Mirzakhani-Zograf's algorithm and a suite of recursion formulas for intersection numbers and WP volumes, the authors derive explicit first-order corrections $e^1_{n,|\mathbf{d}|}$ and show their dependence on $n$ and $|\mathbf{d}|$ is polynomial with controlled $\pi^{-2}$-terms. They provide two explicit forms for $e^1_{n,|\mathbf{d}|}$ depending on the number of zero entries in $\mathbf{d}$, and prove that related coefficients $h_n^i$, $b_n^i$, and $c_n^i$ are polynomials with degrees bounded in $i$. The results have implications for random hyperbolic geometry and related quantum gravity models, confirming the anticipated algebraic structure of large-genus asymptotics and enabling precise asymptotic estimates of WP volumes with potential applications to JT gravity and beyond.

Abstract

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

Theorems & Definitions (25)

• Conjecture 1.1: Zograf
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Conjecture 1.6: Kimura
• Theorem 1.7
• Theorem 1.8
• Remark 2.1
• Lemma 2.2