Eigenvector statistics of random band matrices

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Abstract
The statistics of eigenvector elements is studied for random band matrices as a function of band and matrix sizes. It is shown that the statistics obey a scaling law similar to that found for the mean localization length of eigenvectors. To describe the statistics we propose an expression for the probability distribution of elements based on the assumption of an exponential form of eigenvectors. We demonstrate a fundamental role of fluctuations of the localization length.