The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy
Abstract
This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomial theory and Toda-type equations. Starting from the symmetric reduction in 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 τ -function of the CKP hierarchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.
- Publication:
-
Journal of NonLinear Science
- Pub Date:
- February 2019
- DOI:
- 10.1007/s00332-018-9474-x
- arXiv:
- arXiv:1801.00538
- Bibcode:
- 2019JNS....29....3L
- Keywords:
-
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- High Energy Physics - Theory;
- Mathematical Physics;
- 37K10;
- 33C47;
- 15A15;
- 41A21
- E-Print:
- 19 pages