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


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.


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