Onset of anomalous diffusion from local motion rules

Anomalous diffusion processes, in particular superdiffusive ones, are known to be efficient strategies for searching and navigation by animals and also in human mobility. One way to create such regimes are Levy flights, where the walkers are allowed to perform jumps, the flights, that can eventually be very long as their length distribution is asymptotically power-law distributed. In our work, we present a model in which walkers are allowed to perform, on a 1D lattice, cascades of n unitary steps instead of one jump of a randomly generated length, as in the Levy case. Instead of imposing a length distribution, we thus define our process by its cascade distribution pn. We first derive the connections between the two distributions and show that this local mechanism may give rise to superdiffusion or normal diffusion when pn is distributed as a power law. We also investigate the interplay of this process with the possibility to be stuck on a node, introducing waiting times that are power-law distributed as well. In this case, the competition of the two processes extends the palette of the reachable diffusion regimes and, again, this switch relies on the two PDF's power-law exponents. As a perspective, our approach may engender a possible generalization of anomalous diffusion in context where distances are difficult to define, as in the case of complex networks, and also provide an interesting model for diffusion in temporal networks.