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 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 τ-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.

19 pages

References in corpus (4)

• Degasperis-Procesi peakons and the discrete cubic string
• The Cauchy two-matrix model
• Peakons and Cauchy Biorthogonal Polynomials
• Quantum torus algebras and B(C) type Toda systems

Cited by in corpus (7)

• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• CKP hierarchy and free Bosons
• Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
• Rank shift conditions and reductions of 2d-Toda theory
• Discrete integrable systems and condensation algorithms for Pfaffians
• Matrix integral solutions to the related Leznov lattice equations
• Two-parameter generalisations of Cauchy bi-orthogonal polynomials and integrable lattices