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The European Physical Journal Special Topics

, Volume 143, Issue 1, pp 143–157 | Cite as

Random patterns generated by random permutations of natural numbers

  • G. Oshanin
  • R. Voituriez
  • S. Nechaev
  • O. Vasilyev
  • F. Hivert
Article

Abstract.

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of “U-turns" of the trajectories - permutation “peaks" and “through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1,2,3, ...,L on sites of a 1d or 2d lattices containing L sites. We calculate the distribution function of the number of local “peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss surprising collective behavior emerging in this model.

Keywords

Random Walk European Physical Journal Special Topic Time Moment Random Permutation Local Height 
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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Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • G. Oshanin
    • 1
    • 2
  • R. Voituriez
    • 1
  • S. Nechaev
    • 3
  • O. Vasilyev
    • 2
  • F. Hivert
    • 4
  1. 1.Physique Théorique de la Matière Condensée (UMR 7600), Université Pierre et Marie Curie – Paris 6ParisFrance
  2. 2.Department of Inhomogeneous Condensed Matter TheoryMax-Planck-Institute für MetallforschungStuttgartGermany
  3. 3.LPTMS, Université Paris SudOrsay CedexFrance
  4. 4.LITIS/LIFAR, Université de RouenSaint Étienne du RouvrayFrance

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