Afanasiev I. An application of the Grassmann integration in random matrix theory problems

The thesis is devoted to the development of the Grassmann integration method in the random matrix theory. Two random matrix ensembles are considered: sparse hermitian random matrices and non-hermitian random matrices with independent entries (complex and real cases). The asymptotic behavior of the second correlation function of the characteristic polynomials of the sparse hermitian random matrices is established in the bulk of the spectrum. The asymptotic behavior of the second correlation function of the characteristic polynomials of the diluted hermitian random matrices is established at the edge of the spectrum and the asymptotic behavior of all correlation function of the characteristic polynomials of the diluted hermitian random matrices is established in the bulk of the spectrum. The asymptotic behavior of all correlation function of the characteristic polynomials of the complex random matrices with independent entries is established. The asymptotic behavior of all correlation function of the characteristic polynomials of the real random matrices with independent entries is established in the form of an integral. This integral is computed for the second correlation function.