Journal of Nonlinear Science

, Volume 29, Issue 1, pp 3–27 | Cite as

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

  • Chunxia Li
  • Shi-Hao LiEmail author


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 \(\tau \)-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.


Matrix models Cauchy biorthogonal polynomials C-Toda lattice CKP hierarchy \(\tau \)-Function theory 

Mathematics Subject Classification

37K10 15A15 42C05 35C15 



This work was supported by National Natural Science Foundation of China (Grant Nos. 11271266, 11705284 and 11701550) and Beijing Natural Science Foundation (Grant No. 1162003). The authors would like to thank the referee for valuable suggestions and bringing the article (Bertola et al. 2006) into our horizons, which is of great help for further study. Dr. C. X. Li would like to thank for the hospitality of School of Mathematics and Science during her visit to Fudan University. S. H. Li would like to thank Dr. X. K. Chang and Mr. B. Wang for helpful discussions and thank Prof. X. B. Hu for his attentive guidance.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  2. 2.LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSSChinese Academy of SciencesBeijingChina
  3. 3.Department of Mathematical SciencesUniversity of the Chinese Academy of SciencesBeijingChina

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