Abstract
We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.
Highlights
The study of two-dimensional lattice walks restricted to the non-negative quadrant has been an active topic of interest in several sub-areas of combinatorics, with applications in branches of applied mathematics including queuing theory and the study of linear polymers
They used the kernel method to prove that the GFs corresponding to 22 of the 79 non-equivalent two-dimensional models are D-finite
One can verify that (x1, y1, t1) minimizes the height function—as the minimum must occur at a critical point — so the points contributing to the dominant asymptotics are (x1, y1, t1) and any other points on V3 with the same modulus
Summary
The study of two-dimensional lattice walks restricted to the non-negative quadrant has been an active topic of interest in several sub-areas of combinatorics (see, for instance, [12, 8, 23, 24, 18, 4, 5, 11, 30, 16, 20, 7, 6, 10]), with applications in branches of applied mathematics including queuing theory and the study of linear polymers. The seminal work of Mishna and Bousquet-Melou [11] gave a uniform approach to several enumerative questions, including the nature of a model’s GF(i) (algebraic, D-finite, etc.) and the determination of exact or asymptotic counting formulas They used the kernel method to prove that the GFs corresponding to 22 of the 79 non-equivalent two-dimensional models are D-finite. Ongoing work of Bostan, Chyzak, van Hoeij, Kauers, and Pech [3] attempts to get around this problem by using creative telescoping techniques combined with the kernel method to represent the walk GFs explicitly in terms of hypergeometric functions Such a representation should, in principle, allow one to rigorously determine asymptotics, in practice this depends on computing integrals of hypergeometric functions which those authors have only been able to numerically approximate(ii). (i) We abbreviate ‘generating function’ as GF throughout. (ii) For some models, such integrals need to be rigorously determined to show the asymptotic constant of growth but even its exponential growth
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