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Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide $τ$-functions for several hierarchies of integrable equations. In this article, we extend this relation by showing that such integrals can also provide $τ$-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.