Correlated random walks with a finite memory range

摘要

We study a family of correlated one-dimensional random walks with a finite memory range ni. These walks are extensions of the Taylor's walk as investigated by Goldstein, which has a memory range equal to one. At each step, with a probability p(1) the random walker moves either to the right or to the left with equal probabilities, or with a probability q = 1 - p, performs a move, which is a stochastic Boolean function of the nl previous steps. We first derive the most general form of this stochastic Boolean function, and study some typical cases which ensure that the average value < Rn > of the walker's location after n steps is zero for all values of n. In each case, using a matrix technique, we provide a general method for constructing the generating function of the probability distribution of R-n; we also establish directly an exact analytic expression for the step-step correlations and the variance , which is not straightforward to derive from the probability distribution, we show that, for n approaching infinity, the variance of any of these walks behaves as n, provided p > 0. Moreover, in many cases, for a very small fixed value of p, the variance exhibits a crossover phenomenon as n increases from a not too large value. The crossover takes place for values of n around 1/p. This feature may mimic the existence of a nontrivial Hurst exponent, and induce a misleading analysis of numerical data issued from mathematical or natural sciences experiments. [References: 17]