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 $g\to \infty$.

Communications in Mathematical Physics, to appear