Inhomogeneous ensembles of correlated random walkers

Discrete time random walks, in which a step of random sign but constant length $\delta x$ is performed after each time interval $\delta t$, 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 $q \neq 1/2$. 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 $p(\Delta x, \Delta t)$ for a displacement $\Delta x$ after lagtime $\Delta t$ can have a leptocurtic shape and that mean squared displacements can increase approximately like a fractional powerlaw with $\Delta t$. For the special case of persistence parameters distributed equally in the full range $q \in [0,1]$, the mean squared displacement is derived analytically. The model is further extended by allowing different step lengths $\delta x_j$ for each member $j$ of the ensemble. We show that two ensembles $[\delta t, {(q_j,\delta x_j)}]$ and $[\delta t^{\prime}, {(q^{\prime}_j,\delta x^{\prime}_j)}]$ defined at different time intervals $\delta t\neq\delta t^{\prime}$ can have the same statistical properties at long lagtimes $\Delta t$, 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 $(q_j(t),\delta x_j(t))$ of each individual walker fluctuate temporarily, provided the parameters can be considered constant for time periods $T\gg\Delta t$ longer than the considered lagtime $\Delta t$.