  Matrix integrals used in random matrix theory for the study of eigenvalues of
Hermitian ensembles have been shown to provide $\tau$-functions for several
hierarchies of integrable equations. In this article, we extend this relation
by showing that such integrals can also provide $\tau$-functions for the
discrete KP hierarchy and a coupled version of the same hierarchy obtained
through the process of Pfaffianization. To do so, we consider the first
equation of the discrete KP hierarchy, the Hirota-Miwa equation. We write the
Wronskian determinant solutions to the Hirota-Miwa equation and consider a
particular form of matrix integrals, which we show is an example of those
Wronskian solutions. The argument is then generalized to the whole hierarchy. A
similar strategy is used for the Pfaffianized version of the hierarchy except
that in that case, the solutions are written in terms of Pfaffians rather than
determinants.

Exactly Solvable and Integrable Systems