2,289 research outputs found
Exact mean first-passage time on the T-graph
We consider a simple random walk on the T-fractal and we calculate the exact
mean time to first reach the central node . The mean is performed
over the set of possible walks from a given origin and over the set of starting
points uniformly distributed throughout the sites of the graph, except .
By means of analytic techniques based on decimation procedures, we find the
explicit expression for as a function of the generation and of the
volume of the underlying fractal. Our results agree with the asymptotic
ones already known for diffusion on the T-fractal and, more generally, they are
consistent with the standard laws describing diffusion on low-dimensional
structures.Comment: 6 page
A Method of Intervals for the Study of Diffusion-Limited Annihilation, A + A --> 0
We introduce a method of intervals for the analysis of diffusion-limited
annihilation, A+A -> 0, on the line. The method leads to manageable diffusion
equations whose interpretation is intuitively clear. As an example, we treat
the following cases: (a) annihilation in the infinite line and in infinite
(discrete) chains; (b) annihilation with input of single particles, adjacent
particle pairs, and particle pairs separated by a given distance; (c)
annihilation, A+A -> 0, along with the birth reaction A -> 3A, on finite rings,
with and without diffusion.Comment: RevTeX, 13 pages, 4 figures, 1 table. References Added, and some
other minor changes, to conform with final for
Exact solution of the Nonconsensus Opinion Model on the line
The nonconcensus opinion model (NCO) introduced recently by Shao et al.,
[Phys. Rev. Lett.103, 018701 (2009)] is solved exactly on the line. Although,
as expected, the model exhibits no phase transition in one dimension, its study
is interesting because of the connection with invasion percolation with
trapping. The system evolves exponentially fast to the steady-state, rapidly
developing long-range correlations: The average cluster size in the steady
state scales as the square of the initial cluster size, of the (uncorrelated)
initial state. We also discuss briefly the NCO model on Bethe lattices, arguing
that its phase transition diagram is different than that of regular
percolation.Comment: New version corrects some spurious mistakes, conforms with version to
be published (PRE - Rapid Communication
Designer Nets from Local Strategies
We propose a local strategy for constructing scale-free networks of arbitrary
degree distributions, based on the redirection method of Krapivsky and Redner
[Phys. Rev. E 63, 066123 (2001)]. Our method includes a set of external
parameters that can be tuned at will to match detailed behavior at small degree
k, in addition to the scale-free power-law tail signature at large k. The
choice of parameters determines other network characteristics, such as the
degree of clustering. The method is local in that addition of a new node
requires knowledge of only the immediate environs of the (randomly selected)
node to which it is attached. (Global strategies require information on finite
fractions of the growing net.
Percolation in Hierarchical Scale-Free Nets
We study the percolation phase transition in hierarchical scale-free nets.
Depending on the method of construction, the nets can be fractal or small-world
(the diameter grows either algebraically or logarithmically with the net size),
assortative or disassortative (a measure of the tendency of like-degree nodes
to be connected to one another), or possess various degrees of clustering. The
percolation phase transition can be analyzed exactly in all these cases, due to
the self-similar structure of the hierarchical nets. We find different types of
criticality, illustrating the crucial effect of other structural properties
besides the scale-free degree distribution of the nets.Comment: 9 Pages, 11 figures. References added and minor corrections to
manuscript. In pres
Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
We consider a discrete-time random walk where the random increment at time
step depends on the full history of the process. We calculate exactly the
mean and variance of the position and discuss its dependence on the initial
condition and on the memory parameter . At a critical value
where memory effects vanish there is a transition from a weakly localized
regime (where the walker returns to its starting point) to an escape regime.
Inside the escape regime there is a second critical value where the random walk
becomes superdiffusive. The probability distribution is shown to be governed by
a non-Markovian Fokker-Planck equation with hopping rates that depend both on
time and on the starting position of the walk. On large scales the memory
organizes itself into an effective harmonic oscillator potential for the random
walker with a time-dependent spring constant . The solution of
this problem is a Gaussian distribution with time-dependent mean and variance
which both depend on the initiation of the process.Comment: 10 page
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