3 problems

Spectral-gap, fluctuation, and localization conjecture. With high probability, $\lambda_2\in[a,b]$; the point measure

Eigenvector-localization conjecture. With high probability, every eigenvector of $\mathbf{P}\mathbf{D}^{\dagger}\mathbf{P}$ is localized in this sense.

Top-eigenvector localization conjecture. There exist random integers $K_n$ such that, for every $i\in\mathbb Z$,