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.
Random band matrices in the delocalized phase, I: Quantum unique ergodicity and universality
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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.