Limiting Laws for Spectral Statistics of Large Random Matrices

Leonid Pastur (Institute of Low Temperature Physics, Kharkov, Ukraine)
We consider certain functions of eigenvalues and eigenvectors (spectral statistics) of real symmetric and hermitian random matrices of large size. We first explain that an analog of the Law of Large Numbers is valid for these functions as the size of matrices tends to infinity. We then discuss the scale and the form for limiting fluctuation laws of the statistics and show that the laws can the standard Gaussian (i.e., analogous to usual Central Limit Theorem for appropriately normalized sums of independent or weakly dependent random variables) in non-standard asymptotic settings, certain non Gaussian in seemingly standard asymptotic settings, and other non Gaussian in non-standard asymptotic settings.