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arXiv · 1601.05316

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

Stéphane Lafortune·Chun-Xia Li
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Abstract

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.

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Stéphane Lafortune, Chun-Xia Li. 2016-10-30. Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version. https://doi.org/10.1088/1751-8113%2F49%2F47%2F475202

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