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

Understand

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.