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

Abstract

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 \(\mathcal {M}_{g,n}\) 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 \(V_{g,n}\left( \ell _{1},\dots ,\ell _{k}\right) \) of the moduli space \(\mathcal {M}_{g,n}\left( \ell _{1},\dots ,\ell _{k}\right) \) 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].