He, Yukun; Knowles, Antti Mesoscopic eigenvalue statistics of Wigner matrices. (English) Zbl 1375.15055 Ann. Appl. Probab. 27, No. 3, 1510-1550 (2017). 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. Reviewer: Andreas Arvanitoyeorgos (Patras) Cited in 47 Documents MSC: 15B52 Random matrices (algebraic aspects) 60B20 Random matrices (probabilistic aspects) Keywords:Wigner matrices; mesoscopic eigenvalue distribution; linear statistics; universality PDFBibTeX XMLCite \textit{Y. He} and \textit{A. Knowles}, Ann. Appl. Probab. 27, No. 3, 1510--1550 (2017; Zbl 1375.15055) Full Text: DOI arXiv