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Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions
Chang, Xiang-Ke1,2; He, Yi3; Hu, Xing-Biao1,2; Li, Shi-Hao1,2
AbstractSkew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in Forrester and Kieburg (Commun Math Phys 342(1):151-187, 2016), we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the multiple integral representations. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures ensemble. Besides, we derive a discrete integrable lattice which can be used to compute certain vector Pade approximants. This yields the first example regarding the connection between integrable lattices and generalised inverse vector-valued Pade approximants, about which Hietarinta, Joshi, and Nijhoff pointed out that, This field remains largely to be explored, in the recent monograph (Hietarinta etal. in Discrete systems and integrability, vol 54. Cambridge University Press, Cambridge, 2016, [Section 4.4]).
Funding ProjectNational Natural Science Foundation of China ; Youth Innovation Promotion Association CAS ; [11701550] ; [11731014] ; [11571358] ; [11331008]
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000451726400005
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Document Type期刊论文
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, POB 2719, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.Chinese Acad Sci, Wuhan Inst Phys & Math, Wuhan 430071, Hubei, Peoples R China
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GB/T 7714
Chang, Xiang-Ke,He, Yi,Hu, Xing-Biao,et al. Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS,2018,364(3):1069-1119.
APA Chang, Xiang-Ke,He, Yi,Hu, Xing-Biao,&Li, Shi-Hao.(2018).Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions.COMMUNICATIONS IN MATHEMATICAL PHYSICS,364(3),1069-1119.
MLA Chang, Xiang-Ke,et al."Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions".COMMUNICATIONS IN MATHEMATICAL PHYSICS 364.3(2018):1069-1119.
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