Why Are So Many Problems Unsolved?

Guttmann, A J

doi:10.1007/978-1-4020-9927-4_4

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…The singularity closest to the origin on the positive real axis is at x = 1/4, the same location as for staircase polygons. From the analysis of its series expansion [28], it is also expected that the anisotropic perimeter generating function is both solvable and D-finite (see page 85 of [10] for the mention, see ref. [4] of [10], of an unpublished proof that it is Dfinite).…”

Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

“…From the analysis of its series expansion [28], it is also expected that the anisotropic perimeter generating function is both solvable and D-finite (see page 85 of [10] for the mention, see ref. [4] of [10], of an unpublished proof that it is Dfinite). Furthermore, the generating function for the area and anisotropic perimeter was shown in [7] to satisfy self-reciprocity and inversion relations.…”

Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

“…In the case of column-convex but not row-convex SAPs, the area generating functions are simpler than the corresponding perimeter generating function, being rational functions [4]. However, all known cases of area-perimeter generating functions involve q-series [4].Three-choice and 1-punctured staircase polygons are two classes of SAPs well studied in the literature [6,7], known to be D-finite functions [8,9] but whose perimeter generating functions have resisted closed-form solutions [10]. We here provide hypergeometric solutions to the operators appearing in their linear ODEs.…”

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confidence: 99%

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The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

Assis¹,

Hoeij²,

Maillard³

2016

J. Phys. A: Math. Theor.

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We consider the isotropic perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian ODEs are previously known. We derive simple relationships between the three generating functions, and show that all three generating functions are joint solutions of a common 12th order Fuchsian linear ODE. We find that the 8th order differential operators can each be rewritten as a direct sum of a direct product, with operators no larger than 3rd order. We give closed-form expressions for all the solutions of these operators in terms of 2 F 1 hypergeometric functions with rational and algebraic arguments. The solutions of these linear differential operators can in fact be expressed in terms of two modular forms, since these Self-avoiding walks (SAWs) and self-avoiding polygons (SAPs) have long been studied in enumerative combinatorics as models of percolation, polymers, surface roughness, and more [1], although both their generating functions remain unsolved to this day. Several classes of SAWs and SAPs have been solved by imposing either convexity or directedness constraints, or both. Within a class, walks and polygons usually havePerimeter g.f. of 3-choice, imperfect, 1-punctured staircase polygons 2 the same growth constant, also known as the connective constant, although recently prudent polygons have been shown to be exponentially sparse among prudent walks [2].The study of SAPs and its sub-categories involves the search for exact expressions of their generating functions as a function of various parameters of interest. These include the perimeter, width, height, site perimeter, left and right corners, and area, and for certain classes of SAPs, a generating function has been found which include all of these parameters explicitly, e.g. [3]. Among the known generating functions, rational, algebraic, D-finite, non D-finite, and natural boundaries have been derived (see [4] for a good review). Furthermore, among still unsolved classes, it is possible to prove results concerning the nature of the unsolved generating function. For example, the anisotropic perimeter generating function for the full SAPs class has been proven to not be a D-finite function in [5]. The wide variety of types of functions which arise in the study of SAPs offers an intriguing source of knowledge for what constitutes exact solutions in statistical mechanics.Among known perimeter generating functions are rational functions, algebraic functions, q-series, and natural boundaries [4], and quite generically the nature of the isotropic and anisotropic perimeter generating functions are the same. In the case of column-convex but not row-convex SAPs, the area generating functions are simpler than the corresponding perimeter generating function, being rational functions [4]. However, all known cases of area-perimeter generating functions involve q-series [4].Three-choice and 1-punctured staircase polygons are two classes of SAPs well studied in the literature [6,7], known to be D-finite functions...

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Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

Section: Three-choice and Imperfect Staircase Polygonsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

Assis¹,

Hoeij²,

Maillard³

2016

J. Phys. A: Math. Theor.

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“…By using their result in[90] there was found some relation to the cyclotomic polynomials. The cyclotomic polynomials also appear in the context of integrable models of statistical mechanics[91,92].…”

mentioning

confidence: 99%

Integrability from supersymmetric duality: a short review

Gahramanov¹

2022

Preprint

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Integrable models of statistical mechanics play a prominent role in modern mathematical physics, especially in conformal field theory, knot theory, combinatorics, topology, etc. In this brief review, we discuss a program of constructing integrable lattice spin models with the nearest neighbor interaction using methods inspired by the supersymmetric gauge theory computations, called gauge/YBE correspondence. After a brief introduction to the topic, we review some recent examples of this correspondence and the role of special functions and symmetries. Finally, we discuss future directions of research.

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Synergistic development of differential approximants and the finite lattice method in lattice statistics

Enting¹

2017

J. Phys. A: Math. Theor.

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Several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena. These studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series. One of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as D-finite functions. Somewhat ironically, given power and success of differential approximants in analysing series, neither the assumed functional form, nor any finite generalisation thereof can fit such cases exactly.

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Why Are So Many Problems Unsolved?

Cited by 3 publications

References 20 publications

The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

Integrability from supersymmetric duality: a short review

Synergistic development of differential approximants and the finite lattice method in lattice statistics

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