The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy

TL;DR

This paper explores the Cauchy two-matrix model's integrable hierarchy, deriving the C-Toda lattice and CKP hierarchy, and establishing their connections to matrix integrals and the Bures ensemble.

Contribution

It introduces the C-Toda lattice derived from Cauchy biorthogonal polynomials and links the matrix model to the CKP hierarchy and Bures ensemble.

Findings

Derived the CKP-type Toda equation and its Lax pair.

Connected the matrix integral solutions to the CKP hierarchy.

Established the relation between the Cauchy two-matrix model and Bures ensemble.

Abstract

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the {\tau}-function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.