Random walks with power-law fluctuations in the number of steps

Silvia Valeria Annibaldi
Dipartimento di Energetica, INFM - Politecnico di Torino, Italy

The dynamics leading to non-Gaussian transport are extremely various and complicated, and at present analytical predictions are not possible. But some characteristics can be reproduced by random walk models where correlations are introduced by letting the step number fluctuate. In our model, the step lengths have a distribution with all finite moments, and the step number fluctuates with a power law. When this random walk is isotropic, a power law decay for big values of the amplitude of the resultant emerges, which implies that it has a Levy distribution, a clear signature of non-Gaussian transport. When a small directional bias is added, the power law tail remains only in the direction of the bias, in all the other directions it is substituted by an exponential decay. Our model accounts then for the occurrence of power law tails and shows how strong the effect of anisotropy can be.