  We study the averaged product of characteristic polynomials of large random
matrices in the Gaussian beta-ensemble perturbed by an external source of
finite rank. We prove that at the edge of the spectrum, the limiting
correlations involve two families of multivariate functions of Airy and
Gaussian types. The precise form of the limiting correlations depends on the
strength of the nonzero eigenvalues of the external source. A critical value
for the latter is obtained and a phase transition phenomenon similar to that of
arXiv:math/0403022 is established. The derivation of our results relies mainly
on previous articles by the authors, which deal with duality formulas
arXiv:0801.3438 and asymptotics for Selberg-type integrals arXiv:1112.1119v3.
