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 [41], 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 [30].

33 pages