On the localization length of finite-volume random block Schr\"odinger operators

We study a general class of random block Schr\"odinger operators (RBSOs) in dimensions 1 and 2, which naturally extend the Anderson model by replacing the random potential with a random block potential. Specifically, we focus on two RBSOs -- the block Anderson and Wegner orbital models -- defined on the $d$-dimensional torus $(\mathbb Z/L\mathbb Z)^d$. They take the form $H=V + \lambda \Psi$, where $V$ is a block potential with i.i.d. $W^d\times W^d$ Gaussian diagonal blocks, $\Psi$ describes interactions between neighboring blocks, and $\lambda>0$ is a coupling parameter. We normalize the blocks of $\Psi$ so that each block has a Hilbert-Schmidt norm of the same order as the blocks of $V$. Assuming $W\ge L^\delta$ for a small constant $\delta>0$ and $\lambda\gg W^{-d/2}$, we establish the following results. In dimension $d=2$, we prove delocalization and quantum unique ergodicity for bulk eigenvectors. Combined with the localization result from arXiv:1608.02922, which holds under the condition $\lambda\ll W^{-d/2}$, this provides a rigorous proof of the Anderson localization-delocalization transition as $\lambda$ crosses the critical threshold $W^{-d/2}$. In dimension $d=1$, we show that the localization length of bulk eigenvectors is at least of order $(W\lambda)^2$, which is believed to be the correct scaling.