Shcherbyna T. Distribution of eigenvalues of hermitian random matrices of large dimensions

Purpose of the work is the analysis of the local eigenvalue distribution of some ensembles of random matrices in the bulk and at the edge of the spectrum. The object of research is deformed Gaussian unitary ensemble, hermitian sample covarince matrices and hermitian Wigner matrices with symmetrical distribution of entries. The results obtained are new. The main result are the following. We prove the universality conjecture in the bulk and at the edge of the spectrum for the deformed Gaussian unitary ensemble. We obtain also the determinant formulas for the correlation functions and the universality of the local bulk regime for the hermitian sample covariance matrices. We obtain the asymptotic integral representation for the mixed moments of the characteristic polynomials for the hermitian Wigner matrices and prove that the asymptotic behavior of these functions depends only on first four moments of the distribution of entries.