Random processes generated by random permutations

In this talk, we overview recent results obtained for an exactly solvable model of a non-markovian random walk generated by random permutations of natural series [1,2,3,...,n]. In this model, a random walker moves on a lattice of integers and its moves to the right and to the left are prescribed the sequence of rises and descents characterizing each given permutation of [n]. We determine exactly the probability distribution function of the end-point of the trajectory, its moments, the probability measure of different excursions, as well as different characteristics showing how scrambled the trajectories are. In addition, we discuss properties of 1d and 2d surfaces associated with random permutations and calculate the distribution function of the number of local extrema. As a by-product, we obtain many novel results on intrinsic features of random permutations (statistics of rises, descents, peaks and throughs).