35 research outputs found
Limit theory for planar Gilbert tessellations
A Gilbert tessellation arises by letting linear segments (cracks) in the
plane unfold in time with constant speed, starting from a homogeneous Poisson
point process of germs in randomly chosen directions. Whenever a growing edge
hits an already existing one, it stops growing in this direction. The resulting
process tessellates the plane. The purpose of the present paper is to establish
law of large numbers, variance asymptotics and a central limit theorem for
geometric functionals of such tessellations. The main tool applied is the
stabilization theory for geometric functionals.Comment: 12 page
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
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