Large- 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- and cusps in the large- limit. We show that for a random hyperbolic surface in with large, the number of small Laplacian eigenvalues is linear in 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 scales are non-simple. Our main technical contribution is a novel large- asymptotic formula for the Weil-Petersson volume of the moduli space of genus- hyperbolic surfaces with geodesic boundary components and cusps with fixed, building on work of Manin and Zograf [31].