Punctured polygons and polyominoes on the square lattice

Guttmann, A J; Jensen, Iwan; Wong, Ling Heng; Enting, I. G.

doi:10.1088/0305-4470/33/9/303

Cited by 17 publications

(56 citation statements)

References 23 publications

(55 reference statements)

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“…We here provide hypergeometric solutions to the operators appearing in their linear ODEs. It has long been suspected that their generating functions are related to each other [7,8,9,11], and indeed we here show that they are equal up to the sum of an algebraic factor. Our hypergeometric solutions constitute the first example of a SAP generating function which is D-finite but not algebraic.We begin by reviewing the literature of staircase, three-choice, and punctured staircase polygons in section 2, followed by an analysis of the linear differential operators of the three-choice and punctured staircase polygon linear ODEs in section 3.…”

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confidence: 65%

“…Finally, it is possible to consider the effect of increasing the number of punctures for punctured staircase polygons. In [11], the effect of increasing the number of punctures was considered: it was found that as the number of punctures increases, the perimeter generating function critical exponent increases by 3/2 per puncture, while the area generating function critical exponent increases by 1 per puncture. In both cases, the critical point was found to be unchanged by a finite number of punctures.…”

Section: Towards Generalizations Of the Resultsmentioning

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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confidence: 65%

Section: Towards Generalizations Of the Resultsmentioning

confidence: 99%

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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“…Using the CLN 5 library for c++, we computed a c (p) to 300 signiÿcant digits for p from 10 to 400, and then to 1000 digits for p from 1000 to 1100. Plotting this data and using the techniques described in [8,17], we reached the following hypothesis for the asymptotic behaviour of a c (p):…”

Section: Analysis Of {Ba P−1 } * Bmentioning

confidence: 88%

Exchange relations, Dyck paths and copolymer adsorption

Rechnitzer

Rensburg

2004

Discrete Applied Mathematics

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We consider a lattice model of fully directed copolymer adsorption equivalent to the enumeration of vertex-coloured Dyck paths. For two inÿnite families of periodic colourings we are able to solve the model exactly using a type of symmetry we call an exchange relation. For one of these families we are able to ÿnd an asymptotic expression for the location of the critical adsorption point as a function of the period of the colouring. This expression describes the e ect of a regular inhomogeneity in the polymer on the adsorption transition. ?

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“…The generating function for staircase polygons with a staircase hole is also not known in closed form. Its properties are expected to be similar in many respects to the generating function for three-choice polygons [11]. We shall be concerned with self-reciprocity properties of the generating functions H m (y, q) that count polygons of width m. We first give two examples.…”

Section: Self-reciprocity In Polyomino Enumerationmentioning

confidence: 99%

Inversion relations, reciprocity and polyominoes

Bousquet-Mélou

Guttmann

Orrick

et al. 1999

Annals of Combinatorics

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We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.

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Punctured polygons and polyominoes on the square lattice

Cited by 17 publications

References 23 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

Exchange relations, Dyck paths and copolymer adsorption

Inversion relations, reciprocity and polyominoes

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