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

On a relationship between high rank cases and rank one cases of Hermitian random matrix models with external source