Iterated Random Walk

– The iterated random walk is a random process in which a random walker moves on a one-dimensional random walk which is itself taking place on a one-dimensional random walk, and so on. This process is investigated in the continuum limit using the method of moments. When the number of iterations n → ∞, a time-independent asymptotic density is obtained. It has a simple symmetric exponential form which is stable against the modification of a finite number of iterations. When n is large, the deviation from the stationary density is exponentially small in n. The continuum results are compared to Monte Carlo data for the discrete iterated random walk. Introduction. – When a walker moves at random along a straight support, its typical displacement at time t grows like l ∼ t 1/2. If the support of the walk is itself a random walk (RW), at time t the walker is typically located at the lth step of the walk on which it is moving. Thus the mean-square displacement of the walker behaves as X 2 (t) ∼ l ∼ t 1/2 and his actual typical displacement is reduced, growing as t 1/4 instead of t 1/2 for the conventional RW. This model of a RW on a RW has been studied for a one-dimensional support [1] and also in higher dimensions [2,3]. It has been used to discuss the diffusion of the stored length along a polymer chain entangled in a network [4, 5, 6] and the Brownian motion of charged particles in a turbulent plasma [7]. In this latter case, the particles are constrained to diffuse along the random magnetic field lines in the limit B → ∞. In this work we study the properties of the one-dimensional iterated random walk (IRW), i.e., the RW on a RW which is itself a RW on a RW, and so on. The first three steps of this iteration process are shown in fig. 1. Working in the continuum (Gaussian) limit, we first study the moments X p (t) of the probability density P (n) (X, t) to find the walker at (X, t) after the nth iteration. In the next section, we examine the limit n → ∞ for which the asymptotic probability density P (∞) (X, t) can be obtained explicitly. We also look at the way this asymptotic density is approached when n is large. Then the continuum …