Mathematics – Combinatorics
Scientific paper
2011-07-12
Mathematics
Combinatorics
39 pages, 17 figures
Scientific paper
Models of spatially homogeneous walks in the quarter plane with steps taken from a subset $\mathcal{S}$ of the set of steps to the eight nearest neighbors are considered. Counting 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-M\'elou and Mishna.
Kurkova Irina
Raschel Kilian
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