Random patterns generated by random permutations of natural numbers

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.