Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Asymptotics of lattice walks via analytic combinatorics in several variables
Stephen Melczer  1, 2  , Mark C. Wilson  3  
1 : Laboratoire de l'Informatique du Parallélisme  (LIP)  -  Website
École Normale Supérieure - Lyon, Université Claude Bernard Lyon 1, Institut National de Recherche en Informatique et en Automatique, Centre National de la Recherche Scientifique : UMR5668
46 Allée dÍtalie 69364 LYON CEDEX 07 -  France
2 : Cheriton School of Computer Science [Waterloo]  (CS)  -  Website
200 University Avenue West Waterloo, ON, N2L 3G1 -  Canada
3 : Department of Computer Science [Auckland]  -  Website
The University of Auckland Private Bag 92019 Auckland 1142 New Zealand -  Nouvelle-Zélande

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. 

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