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

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.