We show that for any ε>0, α∈[0, 1 2 ), as g→∞ a generic finite-area genus g hyperbolic surface with n=O(gα) cusps, sampled with probability arising from the Weil–Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below 1 4 −( 2α+1 4 )2−ε. For α=0 this gives a spectral gap of size 3 16 −ε and for any α< 1 2 gives a uniform spectral gap of explicit size.
Hide, W. (2022). Spectral Gap for Weil–Petersson Random Surfaces with Cusps. International Mathematics Research Notices, https://doi.org/10.1093/imrn/rnac293