18,946 research outputs found
L\'evy-type diffusion on one-dimensional directed Cantor Graphs
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.Comment: 9 pages, 9 figure
Local and average behavior in inhomogeneous superdiffusive media
We consider a random walk on one-dimensional inhomogeneous graphs built from
Cantor fractals. Our study is motivated by recent experiments that demonstrated
superdiffusion of light in complex disordered materials, thereby termed L\'evy
glasses. We introduce a geometric parameter which plays a role
analogous to the exponent characterizing the step length distribution in random
systems. We study the large-time behavior of both local and average
observables; for the latter case, we distinguish two different types of
averages, respectively over the set of all initial sites and over the
scattering sites only. The "single long jump approximation" is applied to
analytically determine the different asymptotic behaviours as a function of
and to understand their origin. We also discuss the possibility that
the root of the mean square displacement and the characteristic length of the
walker distribution may grow according to different power laws; this anomalous
behaviour is typical of processes characterized by L\'evy statistics and here,
in particular, it is shown to influence average quantities
Dynamics of continuous-time quantum walks in restricted geometries
We study quantum transport on finite discrete structures and we model the
process by means of continuous-time quantum walks. A direct and effective
comparison between quantum and classical walks can be attained based on the
average displacement of the walker as a function of time. Indeed, a fast growth
of the average displacement can be advantageously exploited to build up
efficient search algorithms. By means of analytical and numerical
investigations, we show that the finiteness and the inhomogeneity of the
substrate jointly weaken the quantum walk performance. We further highlight the
interplay between the quantum-walk dynamics and the underlying topology by
studying the temporal evolution of the transfer probability distribution and
the lower bound of long time averages.Comment: 25 pages, 13 figure
Dynamics around the Site Percolation Threshold on High-Dimensional Hypercubic Lattices
Recent advances on the glass problem motivate reexamining classical models of
percolation. Here, we consider the displacement of an ant in a labyrinth near
the percolation threshold on cubic lattices both below and above the upper
critical dimension of simple percolation, d_u=6. Using theory and simulations,
we consider the scaling regime part, and obtain that both caging and
subdiffusion scale logarithmically for d >= d_u. The theoretical derivation
considers Bethe lattices with generalized connectivity and a random graph
model, and employs a scaling analysis to confirm that logarithmic scalings
should persist in the infinite dimension limit. The computational validation
employs accelerated random walk simulations with a transfer-matrix description
of diffusion to evaluate directly the dynamical critical exponents below d_u as
well as their logarithmic scaling above d_u. Our numerical results improve
various earlier estimates and are fully consistent with our theoretical
predictions.Comment: 12 pages, 6 figure
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