# On the functions counting walks with small steps in the quarter plane

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Abstract : Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the $51$ walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-Mélou and Mishna.
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Article dans une revue
Publications Mathématiques de L'IHÉS, Springer Verlag, 2012, 116 (1), pp.69-114. <10.1007/s10240-012-0045-7>
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https://hal.archives-ouvertes.fr/hal-00628424
Contributeur : Kilian Raschel <>
Soumis le : dimanche 7 octobre 2012 - 12:51:16
Dernière modification le : mercredi 12 octobre 2016 - 01:16:59
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NonHolonomy_12_.pdf
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### Citation

Irina Kurkova, Kilian Raschel. On the functions counting walks with small steps in the quarter plane. Publications Mathématiques de L'IHÉS, Springer Verlag, 2012, 116 (1), pp.69-114. <10.1007/s10240-012-0045-7>. <hal-00628424v2>

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