20 research outputs found
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
Shearer's point process, the hard-sphere model and a continuum Lov\'asz Local Lemma
A point process is R-dependent, if it behaves independently beyond the
minimum distance R. This work investigates uniform positive lower bounds on the
avoidance functions of R-dependent simple point processes with a common
intensity. Intensities with such bounds are described by the existence of
Shearer's point process, the unique R-dependent and R-hard-core point process
with a given intensity. This work also presents several extensions of the
Lov\'asz Local Lemma, a sufficient condition on the intensity and R to
guarantee the existence of Shearer's point process and exponential lower
bounds. Shearer's point process shares combinatorial structure with the
hard-sphere model with radius R, the unique R-hard-core Markov point process.
Bounds from the Lov\'asz Local Lemma convert into lower bounds on the radius of
convergence of a high-temperature cluster expansion of the hard-sphere model.
This recovers a classic result of Ruelle on the uniqueness of the Gibbs measure
of the hard-sphere model via an inductive approach \`a la Dobrushin
On comparison of clustering properties of point processes
In this paper, we propose a new comparison tool for spatial homogeneity of
point processes, based on the joint examination of void probabilities and
factorial moment measures. We prove that determinantal and permanental
processes, as well as, more generally, negatively and positively associated
point processes are comparable in this sense to the Poisson point process of
the same mean measure. We provide some motivating results and preview further
ones, showing that the new tool is relevant in the study of macroscopic,
percolative properties of point processes. This new comparison is also implied
by the directionally convex ( ordering of point processes, which has
already been shown to be relevant to comparison of spatial homogeneity of point
processes. For this latter ordering, using a notion of lattice perturbation, we
provide a large monotone spectrum of comparable point processes, ranging from
periodic grids to Cox processes, and encompassing Poisson point process as
well. They are intended to serve as a platform for further theoretical and
numerical studies of clustering, as well as simple models of random point
patterns to be used in applications where neither complete regularity northe
total independence property are not realistic assumptions.Comment: 23 pages, 1 figure. This submission revisits and adds to ideas
concerning clustering and ordering presented in arXiv:1105.4293.
Results on associated point process in Section 3.3 are new. arXiv admin note:
substantial text overlap with arXiv:1105.429
Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma
A point process is R-dependent if it behaves independently beyond the minimum
distance R. In this paper we investigate uniform positive lower bounds on the avoidance
functions of R-dependent simple point processes with a common intensity. Intensities
with such bounds are characterised by the existence of Shearer’s point process, the unique
R-dependent and R-hard-core point process with a given intensity. We also present
several extensions of the Lovász local lemma, a sufficient condition on the intensity
andR to guarantee the existence of Shearer’s point process and exponential lower bounds.
Shearer’s point process shares a combinatorial structure with the hard-sphere model with
radius R, the unique R-hard-core Markov point process. Bounds from the Lovász local
lemma convert into lower bounds on the radius of convergence of a high-temperature
cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle
(1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive
approach of Dobrushin (1996)
Clustering, percolation and directionally convex ordering of point processes
Heuristics indicate that point processes exhibiting clustering of points have
larger critical radius for the percolation of their continuum percolation
models than spatially homogeneous point processes. It has already been shown,
and we reaffirm it in this paper, that the ordering of point processes is
suitable to compare their clustering tendencies. Hence, it was tempting to
conjecture that is increasing in order. Some numerical evidences
support this conjecture for a special class of point processes, called
perturbed lattices, which are "toy models" for determinantal and permanental
point processes. However, the conjecture is not true in full generality, since
one can construct a Cox point process with degenerate critical radius ,
that is larger than a given homogeneous Poisson point process.
Nevertheless, we are able to compare some nonstandard critical radii related,
respectively, to the finiteness of the expected number of void circuits around
the origin and asymptotic of the expected number of long occupied paths from
the origin in suitable discrete approximations of the continuum model. These
new critical radii sandwich the "true" one. Surprisingly, the inequalities for
them go in opposite directions, which gives uniform lower and upper bounds on
for all processes smaller than some given process. In fact, the
above results hold under weaker assumptions on the ordering of void
probabilities or factorial moment measures only. Examples of point processes
comparable to Poisson processes in this weaker sense include determinantal and
permanental processes. More generally, we show that point processes
smaller than homogeneous Poisson processes exhibit phase transitions in certain
percolation models based on the level-sets of additive shot-noise fields, as
e.g. -percolation and SINR-percolation.Comment: 48 pages, 6 figure
Clustering and percolation of point processes
We are interested in phase transitions in certain percolation models on point
processes and their dependence on clustering properties of the point processes.
We show that point processes with smaller void probabilities and factorial
moment measures than the stationary Poisson point process exhibit non-trivial
phase transition in the percolation of some coverage models based on level-sets
of additive functionals of the point process. Examples of such point processes
are determinantal point processes, some perturbed lattices, and more generally,
negatively associated point processes. Examples of such coverage models are
-coverage in the Boolean model (coverage by at least grains) and
SINR-coverage (coverage if the signal-to-interference-and-noise ratio is
large). In particular, we answer in affirmative the hypothesis of existence of
phase transition in the percolation of -faces in the \v{C}ech simplicial
complex (called also clique percolation) on point processes which cluster less
than the Poisson process. We also construct a Cox point process, which is "more
clustered" than the Poisson point process and whose Boolean model percolates
for arbitrarily small radius. This shows that clustering (at least, as detected
by our specific tools) does not always "worsen" percolation, as well as that
upper-bounding this clustering by a Poisson process is a consequential
assumption for the phase transition to hold.Comment: 25 pages, 1 figure. This paper complements arXiv:1111.6017. arXiv
admin note: substantial text overlap with arXiv:1105.429
Multiconfigurational Hartree-Fock theory for identical bosons in a double well
Multiconfigurational Hartree-Fock theory is presented and implemented in an
investigation of the fragmentation of a Bose-Einstein condensate made of
identical bosonic atoms in a double well potential at zero temperature. The
approach builds in the effects of the condensate mean field and of atomic
correlations by describing generalized many-body states that are composed of
multiple configurations which incorporate atomic interactions. Nonlinear and
linear optimization is utilized in conjunction with the variational and
Hylleraas-Undheim theorems to find the optimal ground and excited states of the
interacting system. The resulting energy spectrum and associated eigenstates
are presented as a function of double well barrier height. Delocalized and
localized single configurational states are found in the extreme limits of the
simple and fragmented condensate ground states, while multiconfigurational
states and macroscopic quantum superposition states are revealed throughout the
full extent of barrier heights. Comparison is made to existing theories that
either neglect mean field or correlation effects and it is found that
contributions from both interactions are essential in order to obtain a robust
microscopic understanding of the condensate's atomic structure throughout the
fragmentation process.Comment: 21 pages, 13 figure