| Abstract: |
| Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we are able to derive the C-Toda lattice as well as its Lax pair by introducing time flows. Matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy. It is remarkable that the time-dependent partition function of the Cauchy two-matrix model is nothing but the $tau-$function of the CKP hierarchy. As a consequence, the correlation between the Cauchy two-matrix model and the CKP hierarchy is established by virtue of orthogonal polynomial theory and Toda-type equations. Moreover, the connection between the Cauchy two-matrix model and Bures ensemble is built from the viewpoint of integrable systems. |
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