Journal Article

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

He, Yi, Li, Shi-Hao, Hu, Xing-Biao, Chang, Xiang-Ke

2018

Overview

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]).

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V
• Subject: Approximants / Classical and Quantum Gravitation / Complex Systems / Deformation / Digital Object Identifier / Discrete systems / Distribution functions / Integral calculus / Lattices / Lattices (mathematics) / Mathematical and Computational Physics / Mathematical Physics / Matrices (mathematics) / Pade approximation / Partitions (mathematics) / Physics / Physics and Astronomy / Polynomials / Quantum Physics / Relativity Theory / Theoretical / Weight / Weighting functions