Abstract
It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width W raised to a power μ remains smaller than the matrix size N. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate μ ≤ 8 holds.
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Schenker, J. Eigenvector Localization for Random Band Matrices with Power Law Band Width. Commun. Math. Phys. 290, 1065–1097 (2009). https://doi.org/10.1007/s00220-009-0798-0
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DOI: https://doi.org/10.1007/s00220-009-0798-0