Typical near-optimal spectral gap conjecture for Weil–Petersson random hyperbolic surfaces

Let $\mathbb{P}_g^{\mathrm{\scriptsize{WP}}}$ denote the Weil–Petersson probability measure on compact hyperbolic surfaces of genus $g$, and let $\lambda_1>0$ be the smallest non-zero eigenvalue of their positive Laplace–Beltrami operator. Typical near-optimal spectral gap conjecture. For any $\epsilon>0$,

$$\lim_{g\rightarrow +\infty}\mathbb{P}_g^{\mathrm{\scriptsize{WP}}}\left(\lambda_1\geq\frac14-\epsilon\right)=1.$$

This conjectures that a near-optimal spectral gap is typical for large-genus Weil–Petersson random hyperbolic surfaces, strengthening the previously known probabilistic lower bounds $3/16-\epsilon$ and the theorem giving $2/9-\epsilon$.