  Random Hermitian matrices are used to model complex systems without
time-reversal invariance. Adding an external source to the model can have the
effect of shifting some of the matrix eigenvalues, which corresponds to
shifting some of the energy levels of the physical system. We consider the case
when the $n\times n$ external source matrix has two distinct real eigenvalues:
$a$ with multiplicity $r$ and zero with multiplicity $n-r$. For a Gaussian
potential, it was shown by P\'ech\'e \cite{Peche:2006} that when $r$ is fixed
or grows sufficiently slowly with $n$ (a small-rank source), $r$ eigenvalues
are expected to exit the main bulk for $|a|$ large enough. Furthermore, at the
critical value of $a$ when the outliers are at the edge of a band, the
eigenvalues at the edge are described by the $r$-Airy kernel. We establish the
universality of the $r$-Airy kernel for a general class of analytic potentials
for $r=\mathcal{O}(n^\gamma)$ for $0\leq\gamma<1/12$.
