Random Weil–Petersson surface eigenfunction sup-norm conjecture

Let $X$ be a compact hyperbolic surface of genus $g$ chosen uniformly at random with respect to Weil–Petersson volume. For $\varepsilon>0$, consider any Laplace eigenfunction $\psi_\lambda$ with eigenvalue

$$\lambda\in\left(\frac14+\varepsilon,+\infty\right).$$

Random-surface eigenfunction sup-norm conjecture. With probability tending to $1$ as $g\to+\infty$,

$$\\|\psi_\lambda\\|_\infty\lesssim \frac{(\log g)^{\alpha(\varepsilon)}}{\sqrt{g}}\\|\psi_\lambda\\|_2$$

for some function $\alpha(\varepsilon)>0$ depending on $\varepsilon$. This conjecture is motivated by analogous optimal sup-norm bounds for eigenvectors on large random regular graphs; it predicts near-volume-scale behavior for tempered eigenfunctions on large-genus random hyperbolic surfaces, but its resolution is not provided here.