Universality in the Onset of Superdiffusion in Lévy Walks

Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic superdiffusive behavior is explored using the 1D Lévy walk of order 1<β<2. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as β crosses the critical value β_{c}=3/2. Above β_{c}, this correction scales as |x|∼t^{1/2}, describing simple diffusion. However, below β_{c} it is instead found to remain superdiffusive, scaling as |x|∼t^{1/(2β-1)}. This transition is shown to be independent of the precise model details and is thus argued to be universal.