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

Abstract

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...

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…One easily verifies that this is the case for the previous modular form example where W (x) reads (93), as well as for all the other modular forms emerging in physics or enumerative combinatorics we mentioned in previous papers [29,30,31,35,37].…”

“…along with the existence of n F n−1 analogues of the previous relation. Such remarkable identities correspond to modular forms that emerged in the analysis of multiple integrals related to the square Ising model [29,30,31,35] or in other enumerative combinatorics context [37]. They can be seen as a simple occurence of infinite order † † covariance symmetries in physics [2] or enumerative combinatorics.…”

“…Therefore, if one restricts oneself to W (x) =W (x) which identifies the two Schwarzian conditions (37) and (38), one sees that condition (42) is automatically verified: condition W (x) =W (x) is thus a sufficient condition for the Calabi-Yau condition (32). The analysis of pullback symmetry on reducible linear differential operators is clearly an interesting and challenging problem in physics.…”

“…We have considered in [1,29,30,31,37] many examples of modular forms represented as pullbacked 2 F 1 hypergeometric functions. Each time the one-parameter commuting series combined with the modular correspondences [8] series yields one-parameter series of the form y n (x) = a n · x n + · · · , n = 2, 3, 4, · · · that are solutions of the Schwarzian equation (90).…”

“…The pullback symmetry up to conjugation studied in sections 2, 3, 4, 5, 6 is appropriate for modular forms [29,30,31,37], but not for derivatives of modular forms that also occur in physics (see for instance the previous relation (119) on the square Ising model).…”