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We prove a 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-$g$ hyperbolicsurfaces with $k$ geodesic boundary components and $n-k$ cusps when $k$ isfixed, building on work of Manin and Zograf [MZ00]. As a first application weshow 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 highprobability. As a second application, we gain insight into the relativefrequency of simple and non-simple closed geodesics on random surfaces withmany cusps.

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