Delocalization conjecture for the two-dimensional discrete random Schrödinger operator

Let $H_\omega$ be the two-dimensional discrete random Schrödinger operator

$$H_\omega=-\bigtriangleup+\sum_{i\in\mathbb{Z}^2}\omega_i\langle\cdot,\delta_i\rangle\delta_i$$

on $l^2(\mathbb{Z}^2)$, where the independent random variables $\omega_i$ are uniformly distributed in $[-c,c]$. Delocalization conjecture. For disorder $c\lesssim 0.7$, $H_\omega$ does not exhibit Anderson localization with positive probability; equivalently, it has nonzero absolutely continuous spectrum with positive probability. In particular, dynamical localization does not occur with positive probability for small disorder. The conjecture challenges the belief that arbitrary disorder produces localization in two dimensions; the paper supports it through a numerical approach, while no analytic proof or resolution is provided here.