Arbitrarily Small Spectral Gaps for Random Hyperbolic Surfaces with Many Cusps

Abstract

Let \(\mathcal {M}_{g,n(g)}\) be the moduli space of hyperbolic surfaces of genus g with n(g) punctures endowed with the Weil–Petersson metric. In this paper we study the asymptotic behavior of the Cheeger constants and spectral gaps of random hyperbolic surfaces in \(\mathcal {M}_{g,n(g)}\), when n(g) grows slower than g as \(g\rightarrow \infty \).