21,330 research outputs found
Sharpness of the percolation transition in the two-dimensional contact process
For ordinary (independent) percolation on a large class of lattices it is
well known that below the critical percolation parameter the cluster size
distribution has exponential decay and that power-law behavior of this
distribution can only occur at . This behavior is often called ``sharpness
of the percolation transition.'' For theoretical reasons, as well as motivated
by applied research, there is an increasing interest in percolation models with
(weak) dependencies. For instance, biologists and agricultural researchers have
used (stationary distributions of) certain two-dimensional contact-like
processes to model vegetation patterns in an arid landscape (see [20]). In that
context occupied clusters are interpreted as patches of vegetation. For some of
these models it is reported in [20] that computer simulations indicate
power-law behavior in some interval of positive length of a model parameter.
This would mean that in these models the percolation transition is not sharp.
This motivated us to investigate similar questions for the ordinary (``basic'')
contact process with parameter . We show, using techniques from
Bollob\'{a}s and Riordan [8, 11], that for the upper invariant measure
of this process the percolation transition is sharp. If
is such that (-a.s.) there are no infinite
clusters, then for all parameter values below the cluster-size
distribution has exponential decay.Comment: Published in at http://dx.doi.org/10.1214/10-AAP702 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation
One of the most well-known classical results for site percolation on the
square lattice is the equation . In words, this equation means
that for all values of the parameter , the following holds:
either a.s. there is an infinite open cluster or a.s. there is an infinite
closed "star" cluster. This result is closely related to the percolation
transition being sharp: below , the size of the open cluster of a given
vertex is not only (a.s.) finite, but has a distribution with an exponential
tail. The analog of this result has been proven by Higuchi in 1993 for
two-dimensional Ising percolation (at fixed inverse temperature
) with external field , the parameter of the model. Using
sharp-threshold results (approximate zero-one laws) and a modification of an
RSW-like result by Bollob\'{a}s and Riordan, we show that these results hold
for a large class of percolation models where the vertex values can be "nicely"
represented (in a sense which will be defined precisely) by i.i.d. random
variables. We point out that the ordinary percolation model obviously belongs
to this class and we also show that the Ising model mentioned above belongs to
it.Comment: Published in at http://dx.doi.org/10.1214/07-AOP380 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The lowest crossing in 2D critical percolation
We study the following problem for critical site percolation on the
triangular lattice. Let A and B be sites on a horizontal line e separated by
distance n. Consider, in the half-plane above e, the lowest occupied crossing R
from the half-line left of A to the half-line right of B. We show that the
probability that R has a site at distance smaller than m from AB is of order
(log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be
carried out for other two-dimensional lattices as well.Comment: 16 pages, Latex, 2 eps figures, special macros: percmac.tex.
Submitted to Annals of Probabilit
Competitive exception learning using fuzzy frequency distributions
A competitive exception learning algorithm for finding a non-linear mapping is proposed which puts the emphasis on the discovery of the important exceptions rather than the main rules. To do so,we first cluster the output space using a competitive fuzzy clustering algorithm and derive a fuzzy frequency distribution describing the general, average system's output behavior. Next, we look for a fuzzy partitioning of the input space in such away that the corresponding fuzzy output frequency distributions `deviate at most' from the average one as found in the first step. In this way, the most important `exceptional regions' in the input-output relation are determined. Using the joint input-output fuzzy frequency distributions, the complete input-output function as extracted from the data, can be expressed mathematically. In addition, the exceptions encountered can be collected and described as a set of fuzzy if-then-else-rules. Besides presenting a theoretical description of the new exception learning algorithm, we report on the outcomes of certain practical simulations.competitive learning;exception learning;fuzzy pattern recognition
Genetic analysis of rare disorders: Bayesian estimation of twin concordance rates
Twin concordance rates provide insight into the possibility of a genetic background for a disease. These concordance rates are usually estimated within a frequentistic framework. Here we take a Bayesian approach. For rare diseases, estimation methods based on asymptotic theory cannot be applied due to very low cell probabilities. Moreover, a Bayesian approach allows a straightforward incorporation of prior information on disease prevalence coming from non-twin studies that is often available. An MCMC estimation procedure is tested using simulation and contrasted with frequentistic analyses. The Bayesian method is able to include prior information on both concordance rates and prevalence rates at the same time and is illustrated using twin data on cleft lip and rheumatoid arthritis
Non-Extensive Bose-Einstein Condensation Model
The imperfect Boson gas supplemented with a gentle repulsive interaction is
completely solved. In particular it is proved that it has non-extensive
Bose-Einstein condensation, i.e., there is condensation without macroscopic
occupation of the ground state (k=0) level
Financial Markets Analysis by Probabilistic Fuzzy Modelling
For successful trading in financial markets, it is important to develop financial models where one can identify different states of the market for modifying one???s actions. In this paper, we propose to use probabilistic fuzzy systems for this purpose. We concentrate on Takagi???Sugeno (TS) probabilistic fuzzy systems that combine interpretability of fuzzy systems with the statistical properties of probabilistic systems. We start by recapitulating the general architecture of TS probabilistic fuzzy rule-based systems and summarize the corresponding reasoning schemes. We mention how probabilities can be estimated from a given data set and how a probability distribution can be approximated by a fuzzy histogram. We apply our methodology for financial time series analysis and demonstrate how a probabilistic TS fuzzy system can be identified, assuming that a linguistic term set is given. We illustrate the interpretability of such a system by inspecting the rule bases of our models.time series analysis;data-driven design;fuzzy reasoning;fuzzy rule base;probabilistic fuzzy systems
Proof of a conjecture of N. Konno for the 1D contact process
Consider the one-dimensional contact process. About ten years ago, N. Konno
stated the conjecture that, for all positive integers , the upper
invariant measure has the following property: Conditioned on the event that
is infected, the events All sites are healthy and All
sites are healthy are negatively correlated. We prove (a stronger
version of) this conjecture, and explain that in some sense it is a dual
version of the planar case of one of our results in \citeBHK.Comment: Published at http://dx.doi.org/10.1214/074921706000000031 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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