Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-ε$

Abstract: Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+\epsilon)\ln(g)$ for large genus.