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

Abstract

Starting from the symmetric reduction of Cauchy bi-orthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing the time flow. Determinant solutions of the C-Toda lattice are expressed in terms of matrix integrals which can be extended to give matrix integral solutions of the CKP hierarchy. It is remarkable that the time-dependent partition function of the Cauchy two-matrix model is nothing but the matrix integral solution of the CKP hierarchy. As the by-product, the connection between the Cauchy two-matrix model and Bures ensemble is established from the viewpoint of integrable systems, which gives an explanation for the question ‘The relationship between the two models does not seem to go much further in the sense that there is no direct and simple relationship between the correlation functions of the two models .... We leave it as an open problem to establish a connection between these models on the level of the correlation functions’ proposed by Bertola et al.