Random patterns generated by random permutations of natural numbers
- 49 Downloads
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.
KeywordsRandom Walk European Physical Journal Special Topic Time Moment Random Permutation Local Height
Unable to display preview. Download preview PDF.
- E. Deutsch, A.J. Hildebrandt, H.S. Wilf, Elec. J. Combinatorics 9, R12 (2003) Google Scholar
- F. Hivert, S. Nechaev, G. Oshanin, O. Vasilyev, On the distribution of surface extrema in several one- and two-dimensional random landscapes, J. Stat. Phys., to appear; cond-mat/0509584 Google Scholar
- H. Cramér, Mathematical Methods of Statistics (Princeton University Press: Princeton, 1957) Google Scholar
- R.L Graham, D.E. Knuth, O. Patashnik, Eulerian Numbers, in: Concrete Mathematics: A Foundation for Computer Science (Addison-Wesley, Reading MA, 1994) Google Scholar
- B.D. Hughes, Random Walks and Random Environments, Vol. 1 (Oxford, Clarendon, 1995) Google Scholar
- J.M. Hammersley, Proc. 6th Berkeley Symp. Math. Statist. and Probability 1, 345 (University of California Press, 1972) Google Scholar
- P.A. Macmahon, Combinatorial Analysis (Cambridge University Press, 1915) Google Scholar
- B. Derrida, E. Gardner, J. Phys. (Paris) 47, 959 (1986) Google Scholar
- B. Derrida (unpublished) Google Scholar
- R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, New York, 1982) Google Scholar