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We consider certain functions of eigenvalues and eigenvectors (spectral statistics) of real symmetric and hermitian random matrices of large size. We first show that for these functions an analog of the Law of Large Numbers is valid as the size of matrices tends to infinity. We then discuss the scale and the form for limiting fluctuations laws of the statistics and show that the laws can be standard Gaussian (i.e., analogous to usual Central Limit Theorem for appropriately normalized sums of i.i.d. random variables) in non-standard asymptotic settings, certain non-Gaussian in seemingly standard asymptotic settings, and other non-Gaussian in non-standard asymptotic settings.

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Leonid Pastur, Institute for Low Temperatures, Kharkiv, Ukraine
Seminarraum Theoretische Physik
Contact: not specified