Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1)

Murata, Mikio; Sakai, Harutaka; Yoneda, Jin

doi:10.1063/1.1531216

Cited by 37 publications

(85 citation statements)

References 11 publications

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“…, where: λ = cos(u) (92) § These kind of results should not be a surprise for the people working on integrable lattice models, or on Painlevé equations [35,36].…”

Section: Algebraic Solutions Of Pvi For λ = Cos(πm/n) and Modular Curvesmentioning

confidence: 99%

“…In order for these programs to be used to study the f N,N 's we need to efficiently produce large (up to several thousand terms) series expansions in t of the f (j) N,N 's. We have done this by use of both the integral representations (35), (36) and the representations of f (j) N,N in terms of theta functions of the nome of elliptic functions, presented in [14].…”

Section: Fuchsian Linear Differential Equations Formentioning

confidence: 99%

See 1 more Smart Citation

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

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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.

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“…, where: λ = cos(u) (92) § These kind of results should not be a surprise for the people working on integrable lattice models, or on Painlevé equations [35,36].…”

Section: Algebraic Solutions Of Pvi For λ = Cos(πm/n) and Modular Curvesmentioning

confidence: 99%

Section: Fuchsian Linear Differential Equations Formentioning

confidence: 99%

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

“…As to the description of (5.25) for w = T α 1 , several expressions are known in the literature [75,77,92]. Here we derive a new explicit expression based on the representation of the affine Weyl group W(E …”

Section: Elliptic Painlevé Equationmentioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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“…Integrable models on a lattice are probably deeper, and dressed with much more symmetries and remarkable structures 25 , than their scaling limits. Such an apparently paradoxical (for the field theory mainstream) conclusion is certainly not a surprise for Painlevé and (discrete) integrability specialists who are used to see, and understand, lattice equations as deeper, and more fundamental [85,106], than the differential equations.…”

Section: Resultsmentioning

confidence: 99%

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

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Abstract. We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a "Russian-doll" nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral E). The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure, the "scaled" linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the n-particle contributions χ (n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for n = 1, 2, 3, 4 and, only modulo a prime, for n = 5 and 6, thus providing a large set of (possible) new singularities of the χ (n) . We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w 2 = 0, that occurs in the linear differential equation of χ (3) , actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.

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Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1)

Cited by 37 publications

References 11 publications

Holonomy of the Ising model form factors

Holonomy of the Ising model form factors

Geometric aspects of Painlevé equations

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

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