The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

Li, Chunxia; Li, Shihao

doi:10.1007/s00332-018-9474-x

Cited by 19 publications

(31 citation statements)

References 30 publications

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“…Remark 3.6. The appearance of the evolution relation (3.19) is slightly different from that in [43]. The formulae here are neat and helpful to calculate the compatibility condition.…”

Section: C-toda Lattice and Its Lax Pair Iḟmentioning

confidence: 82%

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

Chang

2018

Nonlinearity

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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.2010 Mathematics Subject Classification. 37K10, 35Q51, 15A15.

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“…Remark 3.6. The appearance of the evolution relation (3.19) is slightly different from that in [43]. The formulae here are neat and helpful to calculate the compatibility condition.…”

Section: C-toda Lattice and Its Lax Pair Iḟmentioning

confidence: 82%

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

Chang

2018

Nonlinearity

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“…, m. This ensemble has been recently studied in [40] in connection to a θ-deformed Cauchy-Laguerre two-matrix model. In the case when θ = 1, the corresponding ensembles have been studied in [33][34][35][36][37]. Among other results in [40], the k-point (1 ≤ k ≤ m) correlation function of the unconstrained ensemble is shown to follow a Pfaffian point process…”

Section: Average Entropies Over Interpolating Ensemblementioning

confidence: 99%

“…= B q = 1, the Fox H-function reduces to the Meijer G-function [41]. The integral forms of the kernel functions (17a)-(17d) are useful for the computation of mean entropies, whereas several other distinct representations of the kernel functions exist [32,[35][36][37]40]. In particular, we present the following biorthogonal polynomial forms [40] useful for a later discussion…”

Section: Average Entropies Over Interpolating Ensemblementioning

confidence: 99%

“…The relatively less-studied Bures-Hall ensemble [3,4,[24][25][26][27][28][29] gains renewed interest very recently [30][31][32]. This is owing to the recent breakthrough in probability theory in understanding various aspects of the Bures-Hall ensemble [33][34][35][36][37]. Despite the distinct behavior of Hilbert-Schmidt measure and Bures-Hall measure, an interesting question is whether one could propose a measure that interpolates between the two measures.…”

Section: Introduction and Quantum Interpolating Ensemblementioning

confidence: 99%

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Quantum interpolating ensemble: Average entropies and orthogonal polynomials

Wei,

Witte

2021

Preprint

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The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known and physically relevant measures are the Hilbert-Schmidt ensemble and the Bures-Hall ensemble. In this work, we propose a generalized ensemble of density matrices, termed quantum interpolating ensemble, which is able to interpolate between these two seemingly unrelated ensembles. As a first step to understand the proposed ensemble, we derive the exact mean formulas of entanglement entropies over such an ensemble generalizing several recent results in the literature. We also derive some key properties of the corresponding orthogonal polynomials relevant to obtaining other statistical information of the entropies. Numerical results demonstrate the usefulness of the proposed ensemble in estimating the degree of entanglement of quantum states.

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“…The Cauchy two-matrix model has attracted attention due to its applications in multiple aspects of mathematical physics. It is not only significant in studies of classical integrable systems theory like Degasperis-Procesi peakons [27] and Toda lattice of CKP type [26], but important too in random matrix theory [3,5]. This model is defined on some subset of the configuration space R N × R N , as specified by an eigenvalue probability density function (PDF) of the form…”

Section: Introductionmentioning

confidence: 99%

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

Forrester

2019

International Mathematics Research Notices

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A θ-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 θ-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 θ = 1. In the θ = 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 θ-deformed case, allowing explicit expressions involving Fox H-functions for the correlation kernel, and its hard edge scaling limit, to be obtained.2010 Mathematics Subject Classification. 66B20, 15A15, 33E20.Through the variable transformation x → x/t and y → y/t, we see that the dependence on t can be written as J j,k (t) = t −(1+a+b+θ(j+k−2)) I j,k (a, b; θ) and d dt J j,k (t) = −t −(2+a+b+θ(j+k−2)) (1 + a + b + θ(j + k − 2))I j,k (a, b; θ).

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The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

Cited by 19 publications

References 30 publications

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

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

Quantum interpolating ensemble: Average entropies and orthogonal polynomials

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

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