We prove that a large class of \(N\times N\) Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when \(W \ll N^{1/4}\) . 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.
