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

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 [22], 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 taufunctions also admit the representations of multiple integrals. 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 random matrix 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 vector Padé approximants, for which Hietarinta, Joshi and Nijhoff pointed out that " This field remains largely to be explored. " in the recent monograph [27, Section 4.4] .

Degasperis–Procesi peakon dynamical system and finite Toda lattice of CKP type

“…The B-Toda lattice, especially the bilinear form, has also been discussed in some papers, e.g. [11,32,37]. The C-Toda lattice derived in the present paper looks novel and concise and remains further investigation.…”

An application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type

“…(2) If ∂ ∂x (i, j) = (a 0 , b 0 , i, j) and (a 0 , b 0 ) = 0, then ∂ ∂x (i 1 , i 2 , · · · , i 2N ) = (a 0 , b 0 , i 1 , i 2 , · · · , i 2N ), (A. 14) which gives as a special case ∂ ∂x (1, 2, · · · , 2N ) = (a 0 , b 0 , 1, 2, · · · , 2N ).…”

“…We remind the readers that one piece of work[14] has been done while the current paper was under review.It turns out that the related random matrix model is the Bures ensemble. And a family of so-called partial-skeworthogonal polynomials is introduced to act as a wave vector for an isospectral problem of the B-Toda lattice.…”

Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

“…Average of characteristic polynomials--θ-deformation partial-skew-orthogonal polynomials. This section relies on results from the recent work [8].…”

“…where the moments I B j,k and i B j are the same as defined in (3.2). This can be established by employing the Jacobi identity for determinants; see [8] for details. From the Pfaffian form, consideration of the de Bruijn formula [7] shows that…”