Dynamical Localization for Random Band Matrices up to $W\ll N^{1\mathrm{/}4}$

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 $W\ll N^{1\mathrm{/}4}$. The proof uses the fractional moment method and an adaptive Mermin–Wagner style shift.