G. Oshanin 1, 2, R. Voituriez 1, S. Nechaev 3, O. Vasilyev 2, Florent Hivert 4
The European Physical Journal Special Topics 143 (2007) 143-157
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.
- 1. Laboratoire de Physique Théorique de la Matière Condensée (LPTMC),
CNRS : UMR7600 – Université Paris VI – Pierre et Marie Curie - 2. Department of Inhomogeneous Condensed Matter Theory,
Max-Planck-Institut - 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI – Paris Sud - 4. Laboratoire d’Informatique, de Traitement de l’Information et des Systèmes (LITIS),
Institut National des Sciences Appliquées (INSA) – Rouen – Université du Havre – Université de Rouen : EA4108