The eight-vertex model and Painlevé VI

Bazhanov, Vladimir V.; Mangazeev, Vladimir V.

doi:10.1088/0305-4470/39/39/s15

Cited by 46 publications

(103 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This deep relation between elliptic curves and Painlevé VI explains the occurrence of Painlevé VI on the Ising model, and on other lattice Yang-Baxter integrable models which are canonically parametrized in term of elliptic functions (like the eight-vertex Baxter model, the RSOS models, see for instance [34]). We will see, in sec.…”

Section: The Elliptic Representation Of Painlevé VImentioning

confidence: 81%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

View full text Add to dashboard Cite

Abstract. We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f N,N is expressed polynomially in terms of the complete elliptic integrals E and K. The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure. The previous λ-extensions, C(N, N ; λ) are, for singledout values λ = cos(πm/n) (m, n integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

show abstract

Section: The Elliptic Representation Of Painlevé VImentioning

confidence: 81%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

View full text Add to dashboard Cite

show abstract

“…This is the potential of the Painlevé VI equation in the elliptic form [46,47,48,30] (see also [31,49] and [50]). We remark that the non-stationary Lamé equation in connection with the P VI equation (and with the 8-vertex model) was discussed in [51]. Recently, the non-stationary Lamé equation has appeared [19,20], [21,22,23] in the context of the AGT conjecture.…”

Section: (614)mentioning

confidence: 94%

Classical-Quantum Correspondence and Functional Relations for Painlevé Equations

Zabrodin

Zotov

2015

Constr Approx

View full text Add to dashboard Cite

In the light of the Quantum Painlevé-Calogero Correspondence established in our previous papers [1, 2], we investigate the inverse problem. We imply that this type of the correspondence (Classical-Quantum Correspondence) holds true and find out what kind of potentials arise from the compatibility conditions of the related linear problems. The latter conditions are written as functional equations for the potentials depending on a choice of a single function -the left-upper element of the Lax connection. The conditions of the Correspondence impose restrictions on this function. In particular, it satisfies the heat equation. It is shown that all natural choices of this function (rational, hyperbolic and elliptic) reproduce exactly the Painlevé list of equations. In this sense the Classical-Quantum Correspondence can be regarded as an alternative definition of the Painlevé equations.

show abstract

“…9] for a text book treatment. At this point it is worth mentioning other remarkable apperances of Painlevé equations in the theory of integrable systems [10][11][12]. The first study in random matrix theory to give rise to a kernel of the form (1.4) with r = 3 was that of the so-called Pearcey kernel [13][14][15].…”

Section: 2)mentioning

confidence: 97%

Integrable structure of products of finite complex Ginibre random matrices

Mangazeev

Forrester

2018

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We consider the squared singular values of the product of M standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov [1] that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we obtain the Hamiltonian structure for a finite size matrices and formulate it in terms of a (M + 1) × (M + 1) matrix Schlesinger system. The case M = 1 reproduces the Tracy and Widom theory which results in the Painlevé V equation for the (0, s) gap probability. Some integrals of motion for M = 2 are identified, and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding (0, s) gap probability.

show abstract

The eight-vertex model and Painlevé VI

Cited by 46 publications

References 38 publications

Holonomy of the Ising model form factors

Holonomy of the Ising model form factors

Classical-Quantum Correspondence and Functional Relations for Painlevé Equations

Integrable structure of products of finite complex Ginibre random matrices

Contact Info

Product

Resources

About