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

Abstract

Models of spatially homogeneous walks in the quarter plane ${\mathbf{Z}}_{+}^2$ 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 $(i,j)\in {\mathbf{Z}}_{+}^2$ 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 $\mathbf{C}$ 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 ${\mathbf{C}}^2$, 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.

40 pages, 17 figures

References in corpus (3)

• Walks with small steps in the quarter plane
• Two Non-holonomic Lattice Walks in the Quarter Plane
• Proof of Ira Gessel's Lattice Path Conjecture

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• The compensation approach for walks with small steps in the quarter plane
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• Green's Functions with Oblique Neumann Boundary Conditions in the Quadrant
• On the Kernel curves associated with walks in the quarter plane
• On the nature of four models of symmetric walks avoiding a quadrant
• Constructing discrete harmonic functions in wedges
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• Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models