Inhomogeneous ensembles of correlated random walkers

Discrete time random walks, in which a step of random sign but constant length δx is performed after each time interval δt, are widely used models for stochastic processes. In the case of a correlated random walk, the next step has the same sign as the previous one with a probability q 6= 1 2 . We extend this model to an inhomogeneous ensemble of random walkers with a given distribution of persistence probabilites p(q) and show that remarkable statistical properties can result from this inhomogenity: Depending on the distribution p(q), we find that the probability density p(∆x,∆t) for a displacement ∆x after lagtime ∆t can have a leptocurtic shape and that mean squared displacements can increase approximately like a fractional powerlaw with ∆t. For the special case of persistence parameters distributed equally in the full range q∈ [0, 1], the mean squared displacement is derived analytically. The model is further extended by allowing different step lengths δxj for each member j of the ensemble. We show that two ensembles [δt, {(qj , δxj)}] and [ δt′, { (q′ j , δx ′ j) }] defined at different time intervals δt 6= δt′ can have the same statistical properties at long lagtimes ∆t, if their parameters are related by a certain scaling transformation. Finally, we argue that similar statistical properties are expected for homogeneous ensembles, in which the parameters (qj(t), δxj(t)) of each individual walker fluctuate temporarily, provided the parameters can be considered constant for time periods T ∆t longer than the considered lagtime ∆t. Similar models are applicable to many complex systems in which the individual agents undergo distinct yet aysnchronous behavioural phases, so that the statistics of the ensemble as a whole can still be considered as stationary.