  We study Hermitian random matrix models with an external source matrix which
has equispaced eigenvalues, and with an external field such that the limiting
mean density of eigenvalues is supported on a single interval as the dimension
tends to infinity. We obtain strong asymptotics for the multiple orthogonal
polynomials associated to these models, and as a consequence for the average
characteristic polynomials. One feature of the multiple orthogonal polynomials
analyzed in this paper is that the number of orthogonality weights of the
polynomials grows with the degree. Nevertheless we are able to characterize
them in terms of a pair of 2 x 1 vector-valued Riemann-Hilbert problems, and to
perform an asymptotic analysis of the Riemann-Hilbert problems.
