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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
Article

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 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).

Keywords

Riemann Surface Branch Point Universal Covering Group Case Algebraic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Bostan and M. Kauers, The complete generating function for Gessel walks is algebraic, Proc. Am. Math. Soc., 432 (2010), 3063–3078. MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, Contemp. Math., 520 (2010), 1–40. CrossRefGoogle Scholar
  3. 3.
    M. Bousquet-Mélou and M. Petkovsek, Walks confined in a quadrant are not always D-finite, Theor. Comput. Sci., 307 (2003), 257–276. zbMATHCrossRefGoogle Scholar
  4. 4.
    G. Fayolle and R. Iasnogorodski, Two coupled processors: the reduction to a Riemann-Hilbert problem, Z. Wahrscheinlichkeitstheor. Verw. Geb., 47 (1979), 325–351. MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. Fayolle and K. Raschel, On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane, Markov Process. Relat. Fields, 16 (2010), 485–496. MathSciNetzbMATHGoogle Scholar
  6. 6.
    G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random Walks in the Quarter-Plane, Springer, Berlin, 1999. zbMATHCrossRefGoogle Scholar
  7. 7.
    P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. zbMATHCrossRefGoogle Scholar
  8. 8.
    J. Gerretsen and G. Sansone, Lectures on the Theory of Functions of a Complex Variable II: Geometric Theory, Wolters-Noordhoff Publishing, Groningen, 1969. zbMATHGoogle Scholar
  9. 9.
    I. Gessel, A probabilistic method for lattice path enumeration, J. Stat. Plan. Inference, 14 (1986), 49–58. MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    G. Jones and D. Singerman, Complex Functions, Cambridge University Press, Cambridge, 1987. zbMATHGoogle Scholar
  11. 11.
    M. Kauers, C. Koutschan, and D. Zeilberger, Proof of Ira Gessel’s lattice path conjecture, Proc. Natl. Acad. Sci. USA, 106 (2009), 11502–11505. MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    I. Kurkova and K. Raschel, Explicit expression for the generating function counting Gessel’s walks, Adv. Appl. Math., 47 (2011), 414–433. MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    V. Malyshev, Random Walks, Wiener-Hopf Equations in the Quarter Plane, Galois Automorphisms, Lomonossov Moscow University Press, Moscow, 1970 (in Russian). Google Scholar
  14. 14.
    V. Malyshev, Positive random walks and Galois theory, Usp. Mat. Nauk, 26 (1971), 227–228. zbMATHGoogle Scholar
  15. 15.
    V. Malyshev, An analytical method in the theory of two-dimensional positive random walks, Sib. Math. J., 13 (1972), 1314–1329. Google Scholar
  16. 16.
    S. Melczer and M. Mishna, Singularity analysis via the iterated kernel method (2011, in preparation). Google Scholar
  17. 17.
    M. Mishna and A. Rechnitzer, Two non-holonomic lattice walks in the quarter plane, Theor. Comput. Sci., 410 (2009), 3616–3630. MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    K. Raschel, Counting walks in a quadrant: a unified approach via boundary value problems, J. Eur. Math. Soc., 14 (2012), 749–777. MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999. CrossRefGoogle Scholar
  20. 20.
    G. Watson and E. Whittaker, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1962. zbMATHGoogle Scholar

Copyright information

© 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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