Correlation functions of characteristic polynomials as determinants of integrable kernels: universality in the Dyson limit

We consider correlation functions of both ratios and products of characteristic polynomials of NxN Hermitian random matrices taken from unitary invariant ensembles. We demonstrate that the correlation functions are generally governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study large-N asymptotics of correlation functions of characteristic polynomials via Deift - Zhou steepest-descent/stationary phase method for Riemann - Hilbert problems. In particular, we reveal the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β=2 symmetry class.