Universality in the Onset of Super-Diffusion in L\'evy 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 L'evy walk of order $1 < \beta < 2$. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as $\beta$ crosses the critical value ${\beta }_c=3\mathrm{/}2$. Above ${\beta }_c$, this correction scales as $\mid x\mid \sim t^{1\mathrm{/}2}$, describing simple diffusion. However, below ${\beta }_c$ it is instead found to remain superdiffusive, scaling as $\mid x\mid \sim t^{1\mathrm{/}(2\beta -1)}$. This transition is shown to be independent of the precise model details and is thus argued to be universal.