Uniform large-genus asymptotics of Weil-Petersson volumes
Abstract
Let Vg,n be the Weil--Petersson volume of the moduli space of hyperbolic surfaces of genus g with n cusps. The fixed-n large-genus expansion of Mirzakhani and Zograf has correction terms polynomial in n of degree 2i at order g−i, and ceases to be uniform when n is of order √g. We prove an asymptotic expansion in powers of 1/g that holds uniformly in all n≥0, with coefficients analytic in a density parameter that ranges over a compact interval, including the limits n/g→0 and n/g→∞. Its leading term is explicit, contains no undetermined constant and has relative error O(1/g); truncation at an order proportional to √g gives relative error O(exp(−a√g)) for some a>0. For fixed n this confirms Zograf's conjecture that the constant in the Mirzakhani--Zograf asymptotics is 1/√π. We also determine, at every cusp density, the limits of the volume ratios Vg,n/Vg−1,n+2 and Vg,n+1/(2gVg,n) and, in all degrees, the asymptotics of the ratios of adjacent coefficients of the one-boundary volume polynomial; this answers two questions of Mirzakhani. The proof deforms mixed κ1--ψ intersection numbers into pure ψ intersection numbers along a family whose recursion has nonnegative coefficients, fixes the overall constant through Painlevé I without assuming resurgence, and recovers individual volumes by a local limit theorem in the number of cusps. As an application, the spectral gap of a Weil--Petersson random surface with n≫√g cusps tends to zero in probability; this removes the restriction n/g→0 from a theorem of Shen and Wu.
Related
- Uniform asymptotics and the Weil--Petersson constant
- The Weil-Petersson diameter of moduli space for large genus
- Eynard-Orantin Topological Recursions for Higher Weil-Petersson Volumes on Moduli Spaces of Super Riemann Surfaces
- A discrete analogue of 𝒩=2 super Weil-Petersson volumes
- q-deformed Weil-Petersson Volumes from the Spectral Edge of DSSYK and Supersymmetric Extensions
BibTeX
@misc{li2026uniform,
title = {{Uniform large-genus asymptotics of Weil-Petersson volumes}},
author = {Zhiyi Li and Qihang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35492},
eprint = {2610.35492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35492},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform large-genus asymptotics of Weil-Petersson volumes
AU - Zhiyi Li
AU - Qihang Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.35492
UR - https://arxiv.org/abs/2610.35492
AB - Let V_g,n be the Weil--Petersson volume of the moduli space of hyperbolic surfaces of genus g with n cusps. The fixed-n large-genus expansion of Mirzakhani and Zograf has correction terms polynomial in n of degree 2i at order g^−i, and ceases to be uniform when n is of order √g. We prove an asymptotic expansion in powers of 1/g that holds uniformly in all n≥0, with coefficients analytic in a density parameter that ranges over a compact interval, including the limits n/g→0 and n/g→∞. Its leading term is explicit, contains no undetermined constant and has relative error O(1/g); truncation at an order proportional to √g gives relative error O(exp(−a√g)) for some a>0. For fixed n this confirms Zograf's conjecture that the constant in the Mirzakhani--Zograf asymptotics is 1/√π. We also determine, at every cusp density, the limits of the volume ratios V_g,n/V_g−1,n+2 and V_g,n+1/(2gV_g,n) and, in all degrees, the asymptotics of the ratios of adjacent coefficients of the one-boundary volume polynomial; this answers two questions of Mirzakhani. The proof deforms mixed κ_1--ψ intersection numbers into pure ψ intersection numbers along a family whose recursion has nonnegative coefficients, fixes the overall constant through Painlevé I without assuming resurgence, and recovers individual volumes by a local limit theorem in the number of cusps. As an application, the spectral gap of a Weil--Petersson random surface with n≫√g cusps tends to zero in probability; this removes the restriction n/g→0 from a theorem of Shen and Wu.
ER -