Matrix Integrals and Several Integrable Differential-Difference Systems

Hu, Xing−Biao; Zhao, Jinping; Li, Chun−Xia

doi:10.1143/jpsj.75.054003

Cited by 9 publications

(12 citation statements)

References 21 publications

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“…As pointed out by Kakei in [12], once written in that form, the integrals in (1.2) in the cases β = 1, 2 can be interpreted as the continuum limits of N -soliton solutions. It can also be verified that the same observation applies to the integrals (1.3) studied in [18]. It is interesting to note that Tracy and Widom [3] used the identities (1.4), (1.5), and (1.6), to obtain Fredholm determinant expressions for expressions involved in the study of the spacings between consecutive eigenvalues.…”

Section: Introductionmentioning

confidence: 62%

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Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Lafortune

2016

J. Phys. A: Math. Theor.

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Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide τ -functions for several hierarchies of integrable equations. In this article, we extend this relation by showing that such integrals can also provide τ -functions for the discrete KP hierarchy and a coupled version of the same hierarchy obtained through the process of Pfaffianization.To do so, we consider the first equation of the discrete KP hierarchy, the Hirota-Miwa equation. We write the Wronskian determinant solutions to the Hirota-Miwa equation and consider a particular form of matrix integrals, which we show is an example of those Wronskian solutions. The argument is then generalized to the whole hierarchy. A similar strategy is used for the Pfaffianized version of the hierarchy except that in that case, the solutions are written in terms of Pfaffians rather than determinants.

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Section: Introductionmentioning

confidence: 62%

“…This connection of (1.1) with integrable systems was deepened by the authors of [18] who introduced dependence on the continuous variables t 1 , t 2 , and s and the discrete variables n 1 and n 2 in the following way…”

Section: Introductionmentioning

confidence: 99%

Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Lafortune

2016

J. Phys. A: Math. Theor.

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“…Matrix integral solutions (II). As was shown in [10,12], the partition function of the symplectic ensemble can also play a role as the tau function of the Pfaffianized system. Therefore, let's consider the case β = 4 with regarding to the equation (1.1).…”

Section: Matrix Integral Solutions (I)mentioning

confidence: 80%

“…The discrete integrable systems also play important roles in integrable theory and their connections with matrix models are interesting to be discussed. For example, in [12], the matrix integral solutions of several integrable differential-difference systems were considered, whose tau functions were shown to be related to the integrals of the eigenvalues in the form…”

Section: Introductionmentioning

confidence: 99%

Matrix integral solutions to the related Leznov lattice equations

Shen,

2019

Preprint

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Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide τ -functions for several hierarchies of integrable equations.In this paper, we construct the matrix integral solutions to the Leznov lattice equation, semidiscrete and full-discrete version and the Pfaffianized Leznov lattice systems, respectively. We demonstrate that the partition function of Jacobi unitary ensemble is a solution to the semidiscrete Leznov lattice and the partition function of Jacobi orthogonal/symplectic ensemble gives solutions of the Pfaffianized Leznov lattice.

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“…On the other hand, the time-dependent partition functions of matrix models provide us some explanations for integrable hierarchies. It has been realised that the partition function of coupled two-matrix models could be regarded as the tau function of 2d-Toda hierarchy [3], and the partition function of symplectic ensemble and orthogonal ensemble could be regarded as the tau functions of Pfaff lattice and some differential-difference equations [4,7,21]. It is also found that the partition function of Bures ensemble could be viewed as the tau function of BKP hierarchy [22,31] and that of Cauchy two-matrix model is related to the CKP hierarchy [26].…”

Section: Introductionmentioning

confidence: 99%

Rank shift conditions and reductions of 2d-Toda theory

Li¹,

Yu²

2019

Preprint

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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.

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Matrix Integrals and Several Integrable Differential-Difference Systems

Cited by 9 publications

References 21 publications

Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Matrix integral solutions to the related Leznov lattice equations

Rank shift conditions and reductions of 2d-Toda theory

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