For one-dimensional block band matrices with W > N^{1/2+c}, the paper proves the local semicircle law, eigenvector delocalization, quantum unique ergodicity, and GUE universality.
Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$
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abstract
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 and an adaptive Mermin--Wagner style shift.
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Delocalization of One-Dimensional Random Band Matrices
For one-dimensional block band matrices with W > N^{1/2+c}, the paper proves the local semicircle law, eigenvector delocalization, quantum unique ergodicity, and GUE universality.