  We prove an identity on Hermitian random matrix models with external source
relating the high rank cases to the rank 1 cases. This identity was proved and
used in a previous paper of ours to study the asymptotics of the top
eigenvalues. In this paper, we give an alternative, more conceptual proof of
this identity based on a connection between the Hermitian matrix models with
external source and the discrete KP hierarchy. This connection is obtained
using the vertex operator method of Adler and van Moerbeke. The desired
identity then follows from the Fay-like identity of the discrete KP tau vector.
