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Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

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Abstract

Skew-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 Padé approximants. This yields the first example regarding the connection between integrable lattices and generalised inverse vector-valued Padé approximants, about which Hietarinta, Joshi, and Nijhoff pointed out that, “This field remains largely to be explored”, in the recent monograph (Hietarinta et al. in Discrete systems and integrability, vol 54. Cambridge University Press, Cambridge, 2016, [Section 4.4]).

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Acknowledgments

We would like to thank late Professors Jonathan Nimmo and Dr. Junxiao Zhao for their helpful discussions on full-discrete B-Toda lattices and the referee for helpful suggestions. This work was supported in part by the National Natural Science Foundation of China. X. C. was supported under Grant Nos. 11701550 and 11731014. Y. H. was supported under Grant Nos. 11571358 and the Youth Innovation Promotion Association CAS. X. H. was supported under Grant Nos. 11331008 and 11571358.

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O.Box 2719, Beijing, 100190, People’s Republic of China

    Xiang-Ke Chang, Xing-Biao Hu & Shi-Hao Li

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Xiang-Ke Chang, Xing-Biao Hu & Shi-Hao Li

  3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Yi He

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  1. Xiang-Ke Chang
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Correspondence to Shi-Hao Li.

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Communicated by P. Deift

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Chang, XK., He, Y., Hu, XB. et al. Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions. Commun. Math. Phys. 364, 1069–1119 (2018). https://doi.org/10.1007/s00220-018-3273-y

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