Levy walks and scaling in quenched disordered media

We study Levy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites, in analogy with recent experiments. Our results are compared with numerical simulations, with excellent agreement.

Levy-type diffusion on one-dimensional directed Cantor Graphs

Levy-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.