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Mesoscopic eigenvalue statistics of Wigner matrices. (English) Zbl 1375.15055

Let \(H\) be an \(N\times N\) Wigner matrix normalized so that as \(N\to\infty\) its spectrum converges to the interval \([-2, 2]\). The authors study linear eigenvalue statistics of \(H\) of the form \(\operatorname{Tr} f(\frac{H-E}{\eta})\), where \(f\) is a test function, \(E\in (-2, 2)\) a fixed reference energy inside the bulk spectrum, and \(\eta\) an \(N\)-independent spectral scale. The mesoscopic regime is when \(N^{-1}<<\eta<<1\). It is shown that the linear statistics of the eigenvalues of a Wigner matrix converge to a universal Gaussian process on all mesoscopic spectral scales, that is, scales larger than the typical eigenvalue spacing and smaller than the global extent of the spectrum.

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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