Dynamical Localization for Random Band Matrices Up to W≪ N^1/4

We prove that a large class of N× N Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when W ≪ N^1/4 . The proof uses the fractional moment method (Aizenman and Molchanov in Commun Math Phys 157(2):245–278, 1993. https://projecteuclid.org/journals/communications-in- mathematical-physics/volume-157/issue-2/Localizationat-large-disorder-and-at-extreme- energies–an/cmp/1104253939.full ) and an adaptive Mermin–Wagner style shift.