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

Kurkova, Irina¹  ; Raschel, Kilian² 

¹ Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie 4 Place Jussieu, 75252, Paris Cedex 05 France
² Faculté des Sciences et Techniques, CNRS and Université de Tours Parc de Grandmont, 37200, Tours France

Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 69-114

Résumé

Models of spatially homogeneous walks in the quarter plane ${𝐙}_{+}^2$ with steps taken from a subset $𝒮$ 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 of such walks starting at the origin and ending at $(i,j)\in {𝐙}_{+}^2$ after $n$ steps is studied. For all non-singular models of walks, the functions $x\mapsto Q(x,0;z)$ and $y\mapsto Q(x,0;z)$ are continued as multi-valued functions on $𝐂$ 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 ${𝐂}^2$, the interval $]0,1/|𝒮\left|\right[$ 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 in Contemp. Math. 520:1–40(2010).

Détail

DOI : 10.1007/s10240-012-0045-7