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

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.
Type de document :
Article dans une revue
Publications Mathématiques de L'IHÉS, Springer Verlag, 2012, 116 (1), pp.69-114. <10.1007/s10240-012-0045-7>
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00628424
Contributeur : Kilian Raschel <>
Soumis le : dimanche 7 octobre 2012 - 12:51:16
Dernière modification le : jeudi 27 avril 2017 - 09:45:48
Document(s) archivé(s) le : mardi 8 janvier 2013 - 03:48:21

Fichier

NonHolonomy_12_.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

Collections

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>

Partager

Métriques

Consultations de
la notice

149

Téléchargements du document

50