Random walk generated by random permutations of {1, 2, 3, ..., n + 1}

We study properties of a non-Markovian random walk X⁽ⁿ⁾_l, l = 0, 1, 2, ..., n, evolving in discrete time l on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the rise-and-descent sequences characterizing random permutations pgr of [n + 1] = {1, 2, 3, ..., n + 1}. We determine exactly the probability of finding the end-point X_n = X⁽ⁿ⁾_n of the trajectory of such a permutation-generated random walk (PGRW) at site X, and show that in the limit n rarr infin it converges to a normal distribution with a smaller, compared to the conventional Pólya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identical to the distribution of the intermediate points X⁽ⁿ⁾_l, l < n, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of 'turns' of the PGRW trajectories.