Large- n asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces

Journal article

We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-g and n cusps in the large-n limit. We show that for a random hyperbolic surface in Mg, n with n large, the number of small Laplacian eigenvalues is linear in n with high probability. By work of Otal and Rosas [42], this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to log(n) scales are non-simple. Our main technical contribution is a novel large-n asymptotic formula for the Weil-Petersson volume Vg, nℓ1, ⋯, ℓk of the moduli space Mg, nℓ1, ⋯, ℓk of genus-g hyperbolic surfaces with k geodesic boundary components and n-k cusps with k fixed, building on work of Manin and Zograf [31].

Authors: Hide, W; Thomas, J

• Alternative title: Large- n asymptotics for Weil-Petersson..
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Communications in Mathematical Physics
• Volume: 406
• Issue: 9
• Article number: 203
• Publication date: 2025-08-01
• Acceptance date: 2025-06-03
• DOI: 10.1007/s00220-025-05369-4
• EISSN: 1432-0916
• ISSN: 0010-3616
• Language: English