489 research outputs found
Robust nonparametric detection of objects in noisy images
We propose a novel statistical hypothesis testing method for detection of
objects in noisy images. The method uses results from percolation theory and
random graph theory. We present an algorithm that allows to detect objects of
unknown shapes in the presence of nonparametric noise of unknown level and of
unknown distribution. No boundary shape constraints are imposed on the object,
only a weak bulk condition for the object's interior is required. The algorithm
has linear complexity and exponential accuracy and is appropriate for real-time
systems. In this paper, we develop further the mathematical formalism of our
method and explore important connections to the mathematical theory of
percolation and statistical physics. We prove results on consistency and
algorithmic complexity of our testing procedure. In addition, we address not
only an asymptotic behavior of the method, but also a finite sample performance
of our test.Comment: This paper initially appeared in 2010 as EURANDOM Report 2010-049.
Link to the abstract at EURANDOM repository:
http://www.eurandom.tue.nl/reports/2010/049-abstract.pdf Link to the paper at
EURANDOM repository: http://www.eurandom.tue.nl/reports/2010/049-report.pd
Directed percolation effects emerging from superadditivity of quantum networks
Entanglement indcued non--additivity of classical communication capacity in
networks consisting of quantum channels is considered. Communication lattices
consisiting of butterfly-type entanglement breaking channels augmented, with
some probability, by identity channels are analyzed. The capacity
superadditivity in the network is manifested in directed correlated bond
percolation which we consider in two flavours: simply directed and randomly
oriented. The obtained percolation properties show that high capacity
information transfer sets in much faster in the regime of superadditive
communication capacity than otherwise possible. As a byproduct, this sheds
light on a new type of entanglement based quantum capacity percolation
phenomenon.Comment: 6 pages, 4 figure
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
We consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
Random-cluster representation of the Blume-Capel model
The so-called diluted-random-cluster model may be viewed as a random-cluster
representation of the Blume--Capel model. It has three parameters, a vertex
parameter , an edge parameter , and a cluster weighting factor .
Stochastic comparisons of measures are developed for the `vertex marginal' when
, and the `edge marginal' when q\in[1,\oo). Taken in conjunction
with arguments used earlier for the random-cluster model, these permit a
rigorous study of part of the phase diagram of the Blume--Capel model
Population Dynamics in Spatially Heterogeneous Systems with Drift: the generalized contact process
We investigate the time evolution and stationary states of a stochastic,
spatially discrete, population model (contact process) with spatial
heterogeneity and imposed drift (wind) in one- and two-dimensions. We consider
in particular a situation in which space is divided into two regions: an oasis
and a desert (low and high death rates). Carrying out computer simulations we
find that the population in the (quasi) stationary state will be zero,
localized, or delocalized, depending on the values of the drift and other
parameters. The phase diagram is similar to that obtained by Nelson and
coworkers from a deterministic, spatially continuous model of a bacterial
population undergoing convection in a heterogeneous medium.Comment: 8 papes, 12 figure
Empires and Percolation: Stochastic Merging of Adjacent Regions
We introduce a stochastic model in which adjacent planar regions merge
stochastically at some rate , and observe analogies with the
well-studied topics of mean-field coagulation and of bond percolation. Do
infinite regions appear in finite time? We give a simple condition on
for this {\em hegemony} property to hold, and another simple condition for it
to not hold, but there is a large gap between these conditions, which includes
the case . For this case, a non-rigorous analytic
argument and simulations suggest hegemony.Comment: 13 page
Strict inequalities of critical values in continuum percolation
We consider the supercritical finite-range random connection model where the
points of a homogeneous planar Poisson process are connected with
probability for a given . Performing percolation on the resulting
graph, we show that the critical probabilities for site and bond percolation
satisfy the strict inequality . We also show
that reducing the connection function strictly increases the critical
Poisson intensity. Finally, we deduce that performing a spreading
transformation on (thereby allowing connections over greater distances but
with lower probabilities, leaving average degrees unchanged) {\em strictly}
reduces the critical Poisson intensity. This is of practical relevance,
indicating that in many real networks it is in principle possible to exploit
the presence of spread-out, long range connections, to achieve connectivity at
a strictly lower density value.Comment: 38 pages, 8 figure
Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation
Consider a cellular automaton with state space
where the initial configuration is chosen according to a Bernoulli
product measure, 1's are stable, and 0's become 1's if they are surrounded by
at least three neighboring 1's. In this paper we show that the configuration
at time n converges exponentially fast to a final configuration
, and that the limiting measure corresponding to is in
the universality class of Bernoulli (independent) percolation.
More precisely, assuming the existence of the critical exponents ,
, and , and of the continuum scaling limit of crossing
probabilities for independent site percolation on the close-packed version of
(i.e., for independent -percolation on ), we
prove that the bootstrapped percolation model has the same scaling limit and
critical exponents.
This type of bootstrap percolation can be seen as a paradigm for a class of
cellular automata whose evolution is given, at each time step, by a monotonic
and nonessential enhancement.Comment: 15 page
First Passage Properties of the Erdos-Renyi Random Graph
We study the mean time for a random walk to traverse between two arbitrary
sites of the Erdos-Renyi random graph. We develop an effective medium
approximation that predicts that the mean first-passage time between pairs of
nodes, as well as all moments of this first-passage time, are insensitive to
the fraction p of occupied links. This prediction qualitatively agrees with
numerical simulations away from the percolation threshold. Near the percolation
threshold, the statistically meaningful quantity is the mean transit rate,
namely, the inverse of the first-passage time. This rate varies
non-monotonically with p near the percolation transition. Much of this behavior
can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma
Form factor for large quantum graphs: evaluating orbits with time-reversal
It has been shown that for a certain special type of quantum graphs the
random-matrix form factor can be recovered to at least third order in the
scaled time \tau using periodic-orbit theory. Two types of contributing pairs
of orbits were identified, those which require time-reversal symmetry and those
which do not. We present a new technique of dealing with contribution from the
former type of orbits.
The technique allows us to derive the third order term of the expansion for
general graphs. Although the derivation is rather technical, the advantages of
the technique are obvious: it makes the derivation tractable, it identifies
explicitly the orbit configurations which give the correct contribution, it is
more algorithmical and more system-independent, making possible future
applications of the technique to systems other than quantum graphs.Comment: 25 pages, 14 figures, accepted to Waves in Random Media (special
issue on Quantum Graphs and their Applications). Fixed typos, removed an
overly restrictive condition (appendix), shortened introductory section
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