Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version Stéphane Lafortune1 and Chun-Xia Li2,1 6 1 1 Department of Mathematics, College of Charleston, Charleston, SC 29401, USA 0 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048, CHINA 2 t c O Abstract 0 Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles 3 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 I S 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 n i write the Wronskian determinant solutions to theHirota-Miwa equation and consider a particular form l n of matrix integrals, which we show is an example of those Wronskian solutions. The argument is then [ generalizedtothewholehierarchy. AsimilarstrategyisusedforthePfaffianizedversionofthehierarchy 2 except that in that case, thesolutions are written in terms of Pfaffians rather than determinants. v 6 1 3 Key words. Matrix integrals, Hirota-Miwa equation, Wronskian determinant solutions, Pfaffianized 5 0 Hirota-Miwa system, Pfaffian solutions. . 1 0 1 Introduction 6 1 : Inrandommatrixtheory,thanks to a trick introducedby Mehta [1,2], the matrix integralsused to compute v the probability that the eigenvalues of the matrices in an Hermitian ensemble belong to a certain interval i X can be written as integrals over the eigenvalues. These integrals take the form r a N ··· |x −x |βexp f(x ) dx dx ... dx , (1.1) j k j 1 2 N   Z Z 1≤j

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

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