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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 \(\mathbf{ Z}_{+}^{2}\) 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)↦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↦Q(x,0;z) and y↦Q(0,y;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 \(]0,1/|\mathcal{S}|[\) of variation of z splits into two dense subsets such that the functions x↦Q(x,0;z) and y↦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)↦Q(x,y;z), and therefore proves a conjecture of Bousquet-Mélou and Mishna in Contemp. Math. 520:1–40 (2010).

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Irina Kurkova

  2. Faculté des Sciences et Techniques, CNRS and Université de Tours, Parc de Grandmont, 37200, Tours, France

    Kilian Raschel

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  1. Irina Kurkova
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  2. Kilian Raschel
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Correspondence to Kilian Raschel.

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Kurkova, I., Raschel, K. On the functions counting walks with small steps in the quarter plane. Publ.math.IHES 116, 69–114 (2012). https://doi.org/10.1007/s10240-012-0045-7

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  • DOI: https://doi.org/10.1007/s10240-012-0045-7

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