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.Comment: 16 pages, 5 figures; submitted to European Physical Journal,
proceedings of the conference "Stochastic and Complex Systems: New Trends and
Expectations" Santander, Spai