Staircase polygons and recurrent lattice walks (original) (raw)

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

European Journal of Combinatorics, 2017

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N 2 , counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1-39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201-215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions.

Self-avoiding walks and polygons on the triangular lattice

Journal of Statistical Mechanics: Theory and Experiment, 2004

We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the meansquare radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate estimates for the connective constant of triangular self-avoiding walks, µ = 4.150797226(26), and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations.

Square Lattice Self-Avoiding Walks and Polygons

Annals of Combinatorics, 2001

We give an algorithm for the enumeration of self-avoiding walks on the (anisotropic) square lattice. Application of the algorithm on a 1024 processor Intel Paragon supercomputer resulted in a 51 term series. For (isotropic) square lattice self-avoiding polygons, a related algorithm has produced a 90 term series. Careful analysis provides compelling evidence for simple rational values of the exponents in both the dominant and subdominant terms in the asymptotic form of the coefficients. We also advance compelling arguments -but not a proof -that the generating function for SAW is not differentiably finite. The corresponding result for SAP has recently been proved.

Anisotropic step, surface contact, and area weighted directed walks on the triangular lattice

International Journal of Modern Physics B, 2002

We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

Lattice paths and multiple basic hypergeometric series

Pacific Journal of Mathematics, 1989

Certain basic hypergeometric series with multiple indices of summation are interpreted as generating functions for weighted lattice paths. The approach uses ideas of William Surge and gives rise to identities analogous to the Rogers-Ramanujan identities. LATTICE PATHS AND MULTIPLE BASIC HYPERGEOMETRIC SERIES 211 PLAIN: A section of path consisting of only horizontal steps which starts either on the y-axis or at a vertex preceded by a southeast step and ends at a vertex followed by a northeast step. EXAMPLE. The following path has five peaks, three valleys, three mountains, two ranges and one plain.

Polygonal polyominoes on the square lattice

Journal of Physics A: Mathematical and General, 2001

We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be self-avoiding polygons containing any number of holes, each of which is a self-avoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common vertices with each other, nor any common vertices with the surrounding polygon. There are no 'holes-within-holes'. We use the finite-lattice method to count the number of polygonal polyominoes on the square lattice. Series have been derived for both the perimeter and area generating functions. It is known that while the critical point is unchanged by a finite number of holes, when the number of holes is unrestricted the critical point changes. The area generating function coefficients grow exponentially, with a growth constant greater than that for polygons with a finite number of holes, but less than that of polyominoes. We provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter, the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.

The perimeter generating function of punctured staircase polygons

Journal of Physics A: Mathematical and General, 2006

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.

Anisotropic step, mutual contact and area weighted festoons and parallelogram polyominoes on the triangular lattice

Journal of Physics A: Mathematical and General, 2002

We present results for the generating functions of polygons and more general objects that can touch, constructed from two fully directed walks on the infinite triangular lattice, enumerated according to each type of step and weighted proportional to the area and the number of contacts between the directed sides of the objects. In general these directed objects are known as festoons, being constructed from the so-called friendly directed walks, while the subset constructed from vicious walks are staircase polygons, also known as parallelogram polyominoes. Additionally, we give explicit formulae for various first area-moment generating functions, that is when the area is summed over all configurations with a given perimeter. These results generalize and summarize nearly all known results on the square lattice, since such results can be obtained by setting one of the step weights to zero. All our results for the triangular lattice are new and hence provide the opportunity to study subtle changes in scaling between lattices. In most cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

Lattice Green function (at 0) for the 4D hypercubic lattice

Journal of Physics A: Mathematical and General, 1994

The generating function for recurrent Polya walks on the four dimensional hypercubic lattice is expressed as a Kampé-de-Fériet function. Various properties of the associated walks are enumerated.