Uniform spectral gaps for random hyperbolic surfaces with not many cusps

In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^α)$ where $α\in \left[0,\frac{1}{2}\right)$, then for any $ε>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-α)}\right)^2-ε\right)$. If $α$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-ε$ for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation".

💡 Research Summary

The paper studies the spectral gap of random finite‑area hyperbolic surfaces with cusps, where randomness is given by the Weil–Petersson (WP) probability measure on the moduli space (\mathcal M_{g,n}). For a surface (X_{g,n}) of genus (g) and (n) cusps, the Laplace–Beltrami operator has continuous spectrum (