Matrix integral solutions to the related Leznov lattice equations

TL;DR

This paper constructs matrix integral solutions for the Leznov lattice equations and their variants, linking random matrix theory to integrable systems through partition functions of specific ensembles.

Contribution

It introduces explicit matrix integral solutions for the Leznov lattice and its Pfaffianized form, expanding the connection between random matrix ensembles and integrable equations.

Findings

Partition function of Jacobi unitary ensemble solves semi-discrete Leznov lattice.

Partition functions of Jacobi orthogonal and symplectic ensembles solve Pfaffianized Leznov lattice.

Matrix integrals serve as tau-functions for these integrable systems.

Abstract

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