L\'evy-type diffusion on one-dimensional directed Cantor Graphs

TL;DR

This paper investigates Levy-type walks on one-dimensional Cantor graphs, deriving exact scaling exponents for key properties and revealing a transition from superdiffusive to diffusive behavior based on fractal filling.

Contribution

It provides exact analytical results for Levy walks on deterministic fractal graphs, highlighting the impact of topology and initial conditions on diffusion behavior.

Findings

Exact scaling exponents for return probability, resistivity, and mean square displacement.

Transition from superdiffusive to diffusive behavior as fractal filling varies.

Differences between local and average measurements in asymptotic behavior.

Abstract

L\'evy-type walks with correlated jumps, induced by the topology of the medium, are studied on a class of one-dimensional deterministic graphs built from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a standard random walk on the sets but is also allowed to move ballistically throughout the empty regions. Using scaling relations and the mapping onto the electric network problem, we obtain the exact values of the scaling exponents for the asymptotic return probability, the resistivity and the mean square displacement as a function of the topological parameters of the sets. Interestingly, the systems undergoes a transition from superdiffusive to diffusive behavior as a function of the filling of the fractal. The deterministic topology also allows us to discuss the importance of the choice of the initial condition. In particular, we demonstrate that local and average measurements can display different asymptotic behavior. The analytic results are compared with the numerical solution of the master equation of the process.