Publications mathématiques de l'IHÉS

, Volume 116, Issue 1, pp 69–114 | Cite as

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

  • Irina Kurkova
  • Kilian RaschelEmail author


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 xQ(x,0;z) and yQ(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 xQ(x,0;z) and yQ(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).


Riemann Surface Branch Point Universal Covering Group Case Algebraic Function 
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© IHES and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.Faculté des Sciences et TechniquesCNRS and Université de ToursToursFrance

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