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

TL;DR

This paper establishes a conceptual link between high rank and rank one cases in Hermitian random matrix models with external sources, using integrable systems theory to provide a new proof of an important identity.

Contribution

It offers a new, more conceptual proof of a key identity relating high rank and rank one cases in Hermitian models, via the discrete KP hierarchy and vertex operator methods.

Findings

Derived a new proof of the identity connecting high rank and rank one cases.

Linked Hermitian matrix models with external source to the discrete KP hierarchy.

Utilized vertex operator method and Fay-like identities for the proof.

Abstract

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.