Exact results for the generating function of directed column-convex animals on the square lattice

Joyce, G S; Guttman, A J

doi:10.1088/0305-4470/27/13/012

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…We have calculated the p-curvature of the corresponding linear differential operators and found that they are all globally nilpotent. A large set of other enumerative combinatorics problems for which a linear ODE exists for the generating function, but a representation of that generating function as a multiple integral has not yet been found, could be listed [98][99][100][101][102].…”

Section: Global Nilpotence Without (Known) Integral Representation: V...mentioning

confidence: 99%

Globally nilpotent differential operators and the square Ising model

Bostan¹,

Boukraa²,

Hassani³

et al. 2009

J. Phys. A: Math. Theor.

232

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We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their russian-doll and direct sum structures. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six, . . . , and even a remarkable weight-1 modular form emerging in the three-particle contribution χ (3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, ΦH , for the staircase polygons counting, and in Apéry's study of ζ(3). G-functions naturally occur as solutions of globally nilpotent operators. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or ∞) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property. PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14KxxGlobally nilpotent operators 2 Key-words: Globally nilpotent operators, p-curvature, G-functions, arithmetic Gevrey series, Form factors of the square Ising model, susceptibility of the Ising model, Fuchsian linear differential equations, moduli space of curves, two-point correlation functions of the lattice Ising model, complete elliptic integrals, scaling limit of the Ising model, apparent singularities, modular forms, Atkin-Lehmer involutions, Fricke involutions, Dedekind eta functions, Weber modular functions, Calabi-Yau manifolds, three-choice polygons, enumerative combinatorics. IntroductionGenerating large series expansions of physical quantities that are quite often defined as n-fold integrals is the bread and butter of lattice statistical mechanics, enumerative combinatorics, and more generally theoretical physics. The n-fold integrals...

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Section: Global Nilpotence Without (Known) Integral Representation: V...mentioning

confidence: 99%

Globally nilpotent differential operators and the square Ising model

Bostan¹,

Boukraa²,

Hassani³

et al. 2009

J. Phys. A: Math. Theor.

232

View full text Add to dashboard Cite

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“…Taking into account the combinatorial definition of the generalized Fibonacci numbers in (8), equations ( 15) and ( 16) yield…”

Section: Mean Heightmentioning

confidence: 99%

“…This led to the introduction of restricted classes of animals (Ferrers graphs, convex and/or directed animals) for which some exact results could be obtained, mainly in two dimensions (see for example [5][6][7][8][9][10][11][12]). …”

Section: Introductionmentioning

confidence: 99%

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Lattice animals on a staircase and Fibonacci numbers

Turban¹

2000

J. Phys. A: Math. Gen.

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Abstract. We study the statistics of column-convex lattice animals resulting from the stacking of squares on a single or double staircase. We obtain exact expressions for the number of animals with a given length and area, their mean length and their mean height. These objects are closely related to Fibonacci numbers. On a single staircase, the total number of animals with area k is given by the Fibonacci number F k .

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