42 research outputs found
Winding clusters in percolation on the Torus and the Moebius strip
Using a simulation technique introduced recently, we study winding clusters
in percolation on the torus and the Moebius strip for different aspect ratios.
The asynchronous parallelization of the simulation makes very large system and
sample sizes possible. Our high accuracy results are fully consistent with
predictions from conformal field theory. The numerical results for the Moebius
strip and the number distribution of winding clusters on the torus await
theoretical explanation. To our knowledge, this study is the first of its kind.Comment: 20 pages, 9 figures, submitted to J. Stat. Phy
Functional-integral based perturbation theory for the Malthus-Verhulst process
We apply a functional-integral formalism for Markovian birth and death
processes to determine asymptotic corrections to mean-field theory in the
Malthus-Verhulst process (MVP). Expanding about the stationary mean-field
solution, we identify an expansion parameter that is small in the limit of
large mean population, and derive a diagrammatic expansion in powers of this
parameter. The series is evaluated to fifth order using computational
enumeration of diagrams. Although the MVP has no stationary state, we obtain
good agreement with the associated {\it quasi-stationary} values for the
moments of the population size, provided the mean population size is not small.
We compare our results with those of van Kampen's -expansion, and apply
our method to the MVP with input, for which a stationary state does exist.Comment: 24 pages, 15 figure
Comment on: "Superscaling of Percolation on Rectangular Domains"
In [Watanabe et al., Phys. Rev. Lett. 93 190601 (2004)], the authors show
numerically that spanning and percolation probabilities in two-dimensional
systems with different aspect ratios obey a form of "superscaling". In this
comment, we would like to point out some difficulties with their proposed
scaling ansatz and suggest why this remained undetected in their numerical
analysis.Comment: 1 page + references, 1 figur
Data-Driven Prediction of Thresholded Time Series of Rainfall and SOC models
We study the occurrence of events, subject to threshold, in a representative
SOC sandpile model and in high-resolution rainfall data. The predictability in
both systems is analyzed by means of a decision variable sensitive to event
clustering, and the quality of the predictions is evaluated by the receiver
operating characteristics (ROC) method. In the case of the SOC sandpile model,
the scaling of quiet-time distributions with increasing threshold leads to
increased predictability of extreme events. A scaling theory allows us to
understand all the details of the prediction procedure and to extrapolate the
shape of the ROC curves for the most extreme events. For rainfall data, the
quiet-time distributions do not scale for high thresholds, which means that the
corresponding ROC curves cannot be straightforwardly related to those for lower
thresholds.Comment: 19 pages, 10 figure
Analysis of an information-theoretic model for communication
We study the cost-minimization problem posed by Ferrer i Cancho and Sol\'e in
their model of communication that aimed at explaining the origin of Zipf's law
[PNAS 100, 788 (2003)]. Direct analysis shows that the minimum cost is , where determines the relative weights of
speaker's and hearer's costs in the total, as shown in several previous works
using different approaches. The nature and multiplicity of the minimizing
solution changes discontinuously at , being qualitatively
different for , and . Zipf's law
is found only in a vanishing fraction of the minimum-cost solutions at and therefore is not explained by this model. Imposing the further
condition of equal costs yields distributions substantially closer to Zipf's
law, but significant differences persist. We also investigate the solutions
reached by the previously used minimization algorithm and find that they
correctly recover global minimum states at the transition.Comment: 19 pages, 4 figures. Important references and new results adde
The perils of thresholding
The thresholding of time series of activity or intensity is frequently used
to define and differentiate events. This is either implicit, for example due to
resolution limits, or explicit, in order to filter certain small scale physics
from the supposed true asymptotic events. Thresholding the birth-death process,
however, introduces a scaling region into the event size distribution, which is
characterised by an exponent that is unrelated to the actual asymptote and is
rather an artefact of thresholding. As a result, numerical fits of simulation
data produce a range of exponents, with the true asymptote visible only in the
tail of the distribution. This tail is increasingly difficult to sample as the
threshold is increased. In the present case, the exponents and the spurious
nature of the scaling region can be determined analytically, thus demonstrating
the way in which thresholding conceals the true asymptote. The analysis also
suggests a procedure for detecting the influence of the threshold by means of a
data collapse involving the threshold-imposed scale.Comment: 16 pages, 10 figure
Avalanche Behavior in an Absorbing State Oslo Model
Self-organized criticality can be translated into the language of absorbing
state phase transitions. Most models for which this analogy is established have
been investigated for their absorbing state characteristics. In this article,
we transform the self-organized critical Oslo model into an absorbing state
Oslo model and analyze the avalanche behavior. We find that the resulting gap
exponent, D, is consistent with its value in the self-organized critical model.
For the avalanche size exponent, \tau, an analysis of the effect of the
external drive and the boundary conditions is required.Comment: 4 pages, 2 figures, REVTeX 4, submitted to PRE Brief Reports; added
reference and some extra information in V
Extreme value and record statistics in heavy-tailed processes with long-range memory
Extreme events are an important theme in various areas of science because of
their typically devastating effects on society and their scientific
complexities. The latter is particularly true if the underlying dynamics does
not lead to independent extreme events as often observed in natural systems.
Here, we focus on this case and consider stationary stochastic processes that
are characterized by long-range memory and heavy-tailed distributions, often
called fractional L\'evy noise. While the size distribution of extreme events
is not affected by the long-range memory in the asymptotic limit and remains a
Fr\'echet distribution, there are strong finite-size effects if the memory
leads to persistence in the underlying dynamics. Moreover, we show that this
persistence is also present in the extreme events, which allows one to make a
time-dependent hazard assessment of future extreme events based on events
observed in the past. This has direct applications in the field of space
weather as we discuss specifically for the case of the solar power influx into
the magnetosphere. Finally, we show how the statistics of records, or
record-breaking extreme events, is affected by the presence of long-range
memory.Comment: 15 pages, 20 figures, accepted for publication in AGU Monographs:
Complexity and Extreme Events in Geoscienc