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In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented., Comment: 24 pages, 2 figures. Comments are welcome

We investigate the integrability of the non-commutative leapfrog map in this paper. Firstly, we derive the explicit formula for the non-commutative leapfrog map and corresponding discrete zero-curvature equation by employing the concept of non-commutative cross-ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of non-commutative Laurent bi-orthogonal polynomials. Finally, the Poisson structure for this discrete non-commutative map is formulated with the help of a non-commutative network. We aim to enhance our understanding of the integrability properties of the non-commutative leapfrog map and its related mathematical structures through these analysis and constructions., Comment: 32 pages. Comments are welcome

This paper is devoted to revealing the relationship between matrix-valued $\theta$-deformed bi-orthogonal polynomials and non-commutative Toda-type hierarchies. In this procedure, Wronski quasi-determinants are widely used and play the role of non-commutative $\tau$-functions. At the same time, B\"acklund transformations are realized by using a moment modification method and non-commutative $\theta$-deformed Volterra hierarchies are obtained, which contain the known examples of the Itoh-Narita-Bogoyavlensky lattices and the fractional Volterra hierarchy., Comment: 30 pages. Comments are welcome

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff-Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials., Comment: 30 pages, comments are welcome!

Matrix-valued Cauchy bi-orthogonal polynomials were proposed in this paper, together with its quasideterminant expression. It is shown that the coefficients in four-term recurrence relation for matrix-valued Cauchy bi-orthogonal polynomials should satisfy a novel noncommutative integrable system, whose Lax pair is given by fractional differential operators with non-abelian variables., Comment: 18 pages. Comments are welcome

In a recent work the present authors have shown that the eigenvalue probability density function for Dyson Brownian motion from the identity on $U(N)$ is an example of a newly identified class of random unitary matrices called cyclic P\'olya ensembles. In general the latter exhibit a structured form of the correlation kernel. Specialising to the case of Dyson Brownian motion from the identity on $U(N)$ allows the moments of the spectral density, and the spectral form factor $S_N(k;t)$, to be evaluated explicitly in terms of a certain hypergeometric polynomial. Upon transformation, this can be identified in terms of a Jacobi polynomial with parameters $(N(\mu - 1),1)$, where $\mu = k/N$ and $k$ is the integer labelling the Fourier coefficients. From existing results in the literature for the asymptotics of the latter, the asymptotic forms of the moments of the spectral density can be specified, as can $\lim_{N \to \infty} {1 \over N} S_N(k;t) |_{\mu = k/N}$. These in turn allow us to give a quantitative description of the large $N$ behaviour of the average $ \langle | \sum_{l=1}^N e^{ i k x_l} |^2 \rangle$. The latter exhibits a dip-ramp-plateau effect, which is attracting recent interest from the viewpoints of many body quantum chaos, and the scrambling of information in black holes., Comment: 34 pages, 4 figures; v3 update following referee reports

Inspired by Aomoto's $q$-Selberg integral, the orthogonal ensemble in the exponential lattice is considered in this paper. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus, corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. Examples including Al-Salam & Carlitz, $q$-Laguerre, little $q$-Jacobi and big $q$-Jacobi cases are considered., Comment: 32 pages. Comments are welcome

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the $q$-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little $q$-Laguerre weight, a particular ${}_3 \phi_2$ basic hypergeometric polynomial is used to express density moments. The second approach is to study the $q$-Laplace transform of the un-normalised measure. Using integrability properties associated with the $q$-Pearson equation for the $q$-classical weights, a fourth order $q$-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases., Comment: 31 pages. Comments are welcome

A connection between matrix orthogonal polynomials and non-abelian integrable lattices is investigated in this paper. The normalization factors of matrix orthogonal polynomials expressed by quasi-determinant are shown to be solutions of non-abelian Toda lattice in semi-discrete and full-discrete cases. Moreover, with a moment modification method, we demonstrate that the B\"acklund transformation of non-abelian Toda given by Popowicz is equivalent to the non-abelian Volterra lattice, whose solutions could be expressed by quasi-determinants as well., Comment: 21 pages. Comments are welcome

The Bures-Hall ensemble is a unique measure of density matrices that satisfies various distinguished properties in quantum information processing. In this work, we study the statistical behavior of entanglement over the Bures-Hall ensemble as measured by the simplest form of an entanglement entropy - the quantum purity. The main results of this work are the exact second and third moment expressions of quantum purity valid for any subsystem dimensions, where the corresponding results in the literature are limited to the scenario of equal subsystem dimensions. In obtaining the results, we have derived recurrence relations of the underlying integrals over the Cauchy-Laguerre biorthogonal polynomials that may be of independent interest., Comment: 25 pages, 1 figure

This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and $q$-moment sequences., Comment: 14 pages

The framework of spherical transforms and P\'olya ensembles is of utility in deriving structured analytic results for sums and products of random matrices in a unified way. In the present work, we will carry over this framework to study products of unitary matrices. Those are not distributed via the Haar measure, but still are drawn from distributions where the eigenvalue and eigenvector statistics factorise. They include the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic P\'olya frequency functions and show their relation to the cyclic P\'olya ensembles, give a uniqueness statement for the corresponding weights, and derive the determinantal point processes of the eigenvalue statistics at fixed matrix dimension. An outline is given of problems one may encounter when investigating the local spectral statistics.

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $\beta$ generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices $\beta = 1,2$ and 4, and also to the spectral density in the corresponding $\beta$-ensemble with $\beta$ even. The former requires an analysis involving the Routh-Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation., Comment: 21 pages

The theory of P\'olya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involving the hard edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel -- the Bessel kernel -- is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is $O(1/N^2)$., Comment: 21 pages

In this article, we consider the Christoffel transformations for skew-orthogonal polynomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel transformations can act as spectral problems for discrete integrable hierarchies, and therefore we derive certain integrable hierarchies from these transformations. Some reductional cases are also considered., Comment: 21 pages; comments are welcome

In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence relations, which provide us spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest example with exact solvability is demonstrated., Comment: 15 pages; comments are welcome

Inspired by the connection between the Dodgson's condensation algorithm and Hirota's difference equation, we consider condensation algorithms for Pfaffians from the perspectives of discrete integrable systems. The discretisation of Pfaffian elements demonstrate its effectiveness to the Pfaffian $\tau$-functions and discrete integrable systems. The free parameter in the discretisation allows us in particular to obtain explicit, one-parameter condensation algorithms for the Pfaffians., Comment: 14 pages; comments are welcome!

There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$ invariant random matrix ensembles. The solvability of this log-gas system relies on the construction of particular skew orthogonal polynomials, with the skew inner product a linear combination of the $\beta = 1$ and $\beta = 4$ inner products, each involving weight functions. For suitably related classical weight functions, we seek to express the skew orthogonal polynomials as linear combinations of the underlying orthogonal polynomials. It is found that in each case (Gaussian, Laguerre, Jacobi and generalised Cauchy) the coefficients can be expressed in terms of hypergeometric polynomials with argument relating to the fugacity. In the Jacobi case, for example, these are a special case of the Wilson polynomials., Comment: 21 pages

This paper focuses on different reductions of 2-dimensional (2d-)Toda hierarchy. Symmetric and skew symmetric moment matrices are firstly considered, resulting in the differential relations between symmetric/skew symmetric tau functions and 2d-Toda's tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank one shift condition in symmetric case and rank two shift condition in skew symmetric case, from which the C-Toda hierarchy and B-Toda hierarchy are found respectively, together with their special Lax matrices and integrable structures., Comment: 34 pages, comments are welcome!

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalised to discrete settings involving either a linear or exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete, and $q$, skew orthogonal polynomials respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases that the weights for the corresponding discrete unitary ensembles are classical. Crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones, and have a tridiagonal action on the corresponding (discrete or $q$) orthogonal polynomials., Comment: 31 pages; comments are welcome

A $\theta$-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib-Borodin ensemble. The $\theta$-deformed Cauchy-Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written as the Fox H-functions, generalising the Meijer G-function class known from the study of the case $\theta= 1$. In the $\theta=1$ case, it is shown Laguerre-Bures ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding the Bures ensemble corresponds to a Pfaffian point process. This carries over to the $\theta$-deformed case, allowing explicit expressions involving Fox H-functions for the correlation kernel, and its hard edge scaling limit, to be obtained., Comment: 24 pages

Inspired by Okounkov's work [\emph{Selecta Mathematica}, 7(1):57--81, 2001] which relates KP hierarchy to determinant point process, we establish a relationship between BKP hierarchy and Pfaffian point process. We prove that the correlation function of the shifted Schur measures on strict partitions can be expressed as a Pfaffian of skew symmetric matrix kernel, whose elememts are certain vacuum expectations of neutral fermions. We further show that the matrix integrals solution of BKP hierarchy can also induce a certain Pfaffian point process., Comment: 17 pages

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of 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 hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems., Comment: 19 pages

In this paper, we propose a finite Toda lattice of CKP type (C-Toda) together with a Lax pair. Our motivation is based on the fact that the Camassa-Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows in some sense. As an intriguing analogue to the CH equation, the Degasperis-Procesi (DP) equation also supports the presence of peakon solutions. Noticing that the peakon solution to the DP equation is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equation. The corresponding quartic representation is shown to be a continuum limit of a discrete CKP equation, due to which we call the obtained equation finite Toda lattice of CKP type. Then, a nonlinear version of the C-Toda lattice together with a Lax pair is derived. As a result, it is shown that the DP peakon lattice and the finite C-Toda lattice form opposite flows under certain transformation., Comment: 28 pages

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 tau-functions 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 Pad\'e approximants. This yields the first example regarding the connection between integrable lattices and vector Pad\'e 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] ., Comment: 47 pages

The Novikov equation is an integrable analogue of the Camassa-Holm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians., Comment: 33pages