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Walks on the slit plane
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  • Published: November 2002

Walks on the slit plane

  • Mireille Bousquet-Mélou1 &
  • Gilles Schaeffer2 

Probability Theory and Related Fields volume 124, pages 305–344 (2002)Cite this article

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Abstract.

 In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line . We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable is algebraic of degree 8. Moreover, for any point (i, j), the length for walks of this type ending at (i, j) is also algebraic, of degree 2 or 4, and involves the famous Catalan numbers.

Our method is based on the solution of a functional equation, established via a simple combinatorial argument. It actually works for more general models, in which walks take their steps in a finite subset of ℤ2 satisfying two simple conditions. The corresponding are always algebraic.

In the second part of the paper, we derive from our enumerative results a number of probabilistic corollaries. For instance, we can compute exactly the probability that an ordinary random walk starting from (i, j) hits for the first time the half-line at position (k, 0), for any triple (i, j, k). This generalizes a question raised by R. Kenyon, which was the starting point of this paper.

Taking uniformly at random all n-step walks on the slit plane, we also compute the probability that they visit a given point (k, 0), and the average number of visits to this point. In other words, we quantify the transience of the walks. Finally, we derive an explicit limit law for the coordinates of their endpoint.

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Authors and Affiliations

  1. CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France. e-mail: mireille.bousquet@labri.fr, , , , , , FR

    Mireille Bousquet-Mélou

  2. CNRS, LORIA, Campus Scientifique, 615 rue du Jardin Botanique – B.P. 101, 54602 Villers–lès–Nancy Cedex, France. e-mail: Gilles.Schaeffer@loria.fr, , , , , , FR

    Gilles Schaeffer

Authors
  1. Mireille Bousquet-Mélou
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  2. Gilles Schaeffer
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Additional information

Received: 22 December 2001 / Revised version: 19 February 2002 / Published online: 30 September 2002

Both authors were partially supported by the INRIA, via the cooperative research action Alcophys.

Mathematics Subject Classification (2000): O5A15-60C05

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Bousquet-Mélou, M., Schaeffer, G. Walks on the slit plane. Probab Theory Relat Fields 124, 305–344 (2002). https://doi.org/10.1007/s004400200205

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  • Issue Date: November 2002

  • DOI: https://doi.org/10.1007/s004400200205

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Keywords

  • Endpoint
  • General Model
  • Random Walk
  • Functional Equation
  • Simple Condition
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