  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 $g\to \infty$.
