Lévy-type diffusion on one-dimensional directed Cantor graphs

Lévy-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 system 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 to the numerical solution of the master equation of the process.

Figure 1

(Color online) (Upper figure) The discrete Cantor set in gray (fuchsia); the sites which have been removed (voids). (Lower figure) The Cantor graph associated to the Cantor set at generation $G=3$ where undirectional bubbles have been introduced to describe ballistic motion in the voids. The site $i=1$ is the origin of the graph.

Figure 2

Three examples of Cantor graphs of generation 2 with $n_r=2,3,4$.

Figure 3

The generation $G=3$ of the graph with $n_u=3,4,5$ in the $n_r=2$ case.

Figure 4

(Color online) Dynamical scaling of the probability for initial site $i=1$. The sizes of the lattices correspond to generations $G=9,9,8$, respectively.

Figure 5

(Color online) Dynamical scaling of the probability for initial site $i=1$. Lattices of size $N_{tot}$ correspond to $G=9,7,8$, respectively. To avoid the big jumps and zeros, what is actually plotted is a coarse-grained probability ${\sum }_{t=t_{k-1}}^{t_k}{P}_{ii}\left(t\right)/\left[t_k-t_{k-1}\right]$.

Figure 6

(Color online) Growth of the mean-square displacements for initial site $i=1$. Lattices of size $N$ corresponding to $G=9,7,8$, respectively.

Figure 7

(Color online) Average mean-square displacement, as defined in Eq. (18), for the fat fractals with $n_r=4$, $n_u=3$ and $n_r=5$, $n_u=2$. Solid and dashed lines are the superdiffusive, Eq. (25), and diffusive behaviors expected in the two structures, respectively.

Figure 8

(Color online) Dynamic scaling of the averaged probability $\overline{P}$ with $n_r=4$ and $n_u=3$ lattices of size $N_{tot}=58976$ $\left(G=7\right)$. The average is on all initial sites.