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Random patterns generated by random permutations of natural numbers

G. Oshanin·R. Voituriez·S. Nechaev·O. Vasilyev·F. Hivert
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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 induced 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 square 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 some surprising collective behavior emerging in this model.

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G. Oshanin, R. Voituriez, S. Nechaev, O. Vasilyev, F. Hivert. 2006-09-27. Random patterns generated by random permutations of natural numbers. https://doi.org/10.1140/epjst%2Fe2007-00082-2

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