Matrix integral solutions to the discrete and coupled Leznov lattice equations

Abstract Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide τ-functions for several hierarchies of integrable equations. In this paper, we construct the matrix integral solutions to the Leznov lattice equation, semi-discrete and fully-discrete version and the Pfaffianized Leznov lattice systems, respectively. We demonstrate that the partition function of the Jacobi type unitary ensemble is a solution to the semi-discrete Leznov lattice and the partition function of the Jacobi type orthogonal/symplectic ensemble solves the Pfaffianized Leznov lattice.

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