Correlated random walks with a finite memory range

Abstract

We study a family of correlated one-dimensional random walks with a finite memory range M.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, 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 M 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 <R_n> 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 <R_n^2> of the walk. From the expression of <R_n^2>, which is not straightforward to derive from the probability distribution, we show that, for n going to 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 non-trivial Hurst exponent, and induce a misleading analysis of numerical data issued from mathematical or natural sciences experiments.