  It has been shown by Strahov and Fyodorov that averages of products and
ratios of characteristic polynomials corresponding to Hermitian matrices of a
unitary ensemble, involve kernels related to orthogonal polynomials and their
Cauchy transforms. We will show that, for the unitary ensemble $\frac{1}{\hat
Z_n}|\det M|^{2\alpha}e^{-nV(M)}dM$ of $n\times n$ Hermitian matrices, these
kernels have universal behavior at the origin of the spectrum, as $n\to\infty$,
in terms of Bessel functions. Our approach is based on the characterization of
orthogonal polynomials together with their Cauchy transforms via a matrix
Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of
the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to
obtain the asymptotic behavior of the Riemann-Hilbert problem.
