1,283 research outputs found
Shape-Driven Nested Markov Tessellations
A new and rather broad class of stationary (i.e. stochastically translation
invariant) random tessellations of the -dimensional Euclidean space is
introduced, which are called shape-driven nested Markov tessellations. Locally,
these tessellations are constructed by means of a spatio-temporal random
recursive split dynamics governed by a family of Markovian split kernel,
generalizing thereby the -- by now classical -- construction of iteration
stable random tessellations. By providing an explicit global construction of
the tessellations, it is shown that under suitable assumptions on the split
kernels (shape-driven), there exists a unique time-consistent whole-space
tessellation-valued Markov process of stationary random tessellations
compatible with the given split kernels. Beside the existence and uniqueness
result, the typical cell and some aspects of the first-order geometry of these
tessellations are in the focus of our discussion
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
Continuous cellular automata on irregular tessellations : mimicking steady-state heat flow
Leaving a few exceptions aside, cellular automata (CA) and the intimately related coupled-map lattices (CML), commonly known as continuous cellular automata (CCA), as well as models that are based upon one of these paradigms, employ a regular tessellation of an Euclidean space in spite of the various drawbacks this kind of tessellation entails such as its inability to cover surfaces with an intricate geometry, or the anisotropy it causes in the simulation results. Recently, a CCA-based model describing steady-state heat flow has been proposed as an alternative to Laplace's equation that is, among other things, commonly used to describe this process, yet, also this model suffers from the aforementioned drawbacks since it is based on the classical CCA paradigm. To overcome these problems, we first conceive CCA on irregular tessellations of an Euclidean space after which we show how the presented approach allows a straightforward simulation of steady-state heat flow on surfaces with an intricate geometry, and, as such, constitutes an full-fledged alternative for the commonly used and easy-to-implement finite difference method, and the more intricate finite element method
Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations
In this work, we study a new model for continuum line-of-sight percolation in
a random environment driven by the Poisson-Voronoi tessellation in the
-dimensional Euclidean space. The edges (one-dimensional facets, or simply
1-facets) of this tessellation are the support of a Cox point process, while
the vertices (zero-dimensional facets or simply 0-facets) are the support of a
Bernoulli point process. Taking the superposition of these two processes,
two points of are linked by an edge if and only if they are sufficiently
close and located on the same edge (1-facet) of the supporting tessellation. We
study the percolation of the random graph arising from this construction and
prove that a 0-1 law, a subcritical phase as well as a supercritical phase
exist under general assumptions. Our proofs are based on a coarse-graining
argument with some notion of stabilization and asymptotic essential
connectedness to investigate continuum percolation for Cox point processes. We
also give numerical estimates of the critical parameters of the model in the
planar case, where our model is intended to represent telecommunications
networks in a random environment with obstructive conditions for signal
propagation.Comment: 30 pages, 4 figures. Accepted for publication in Advances in Applied
Probabilit
Voronoi Tessellations and the Cosmic Web: Spatial Patterns and Clustering across the Universe
The spatial cosmic matter distribution on scales of a few up to more than a
hundred Megaparsec displays a salient and pervasive foamlike pattern. Voronoi
tessellations are a versatile and flexible mathematical model for such weblike
spatial patterns. They would be the natural asymptotic result of an evolution
in which low-density expanding void regions dictate the spatial organization of
the Megaparsec Universe, while matter assembles in high-density filamentary and
wall-like interstices between the voids. We describe the results of ongoing
investigations of a variety of aspects of cosmologically relevant spatial
distributions and statistics within the framework of Voronoi tessellations.
Particularly enticing is the finding of a profound scaling of both clustering
strength and clustering extent for the distribution of tessellation nodes,
suggestive for the clustering properties of galaxy clusters. Cellular patterns
may be the source of an intrinsic ``geometrically biased'' clustering.Comment: 10 pages, 9 figures, accepted for publication as long paper in
proceedings Fourth International Symposium on Voronoi Diagrams in Science and
Engineering (ISVD 2007), ed. C. Gold, IEEE Computer Society, July 2007. For
high-res version see
http://www.astro.rug.nl/~weygaert/tim1publication/vorwey.isvd07.pd
Geometry of iteration stable tessellations: Connection with Poisson hyperplanes
Since the seminal work by Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible, yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. We provide in this paper a
fundamental link between typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations using martingale
techniques and general theory of piecewise deterministic Markov processes
(PDMPs). As applications, new mean values and new distributional results for
the STIT model are obtained.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ424 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:1001.099
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