6,712 research outputs found
Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets
We propose new summary statistics to quantify the association between the
components in coverage-reweighted moment stationary multivariate random sets
and measures. They are defined in terms of the coverage-reweighted cumulant
densities and extend classic functional statistics for stationary random closed
sets. We study the relations between these statistics and evaluate them
explicitly for a range of models. Unbiased estimators are given for all
statistics and applied to simulated examples.Comment: Added examples in version
Perfect simulation for interacting point processes, loss networks and Ising models
We present a perfect simulation algorithm for measures that are absolutely
continuous with respect to some Poisson process and can be obtained as
invariant measures of birth-and-death processes. Examples include area- and
perimeter-interacting point processes (with stochastic grains), invariant
measures of loss networks, and the Ising contour and random cluster models. The
algorithm does not involve couplings of the process with different initial
conditions and it is not tied up to monotonicity requirements. Furthermore, it
directly provides perfect samples of finite windows of the infinite-volume
measure, subjected to time and space ``user-impatience bias''. The algorithm is
based on a two-step procedure: (i) a perfect-simulation scheme for a (finite
and random) relevant portion of a (space-time) marked Poisson processes (free
birth-and-death process, free loss networks), and (ii) a ``cleaning'' algorithm
that trims out this process according to the interaction rules of the target
process. The first step involves the perfect generation of ``ancestors'' of a
given object, that is of predecessors that may have an influence on the
birth-rate under the target process. The second step, and hence the whole
procedure, is feasible if these ``ancestors'' form a finite set with
probability one. We present a sufficiency criteria for this condition, based on
the absence of infinite clusters for an associated (backwards) oriented
percolation model.Comment: Revised version after referee of SPA: 39 page
Variance Asymptotics and Scaling Limits for Random Polytopes
Let K be a convex set in R d and let K be the convex hull of a
homogeneous Poisson point process P of intensity on K. When
K is a simple polytope, we establish scaling limits as
for the boundary of K in a vicinity of a vertex of K and we
give variance asymptotics for the volume and k-face functional of K ,
k {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The
scaling limit of the boundary of K and the variance asymptotics are
described in terms of a germ-grain model consisting of cone-like grains pinned
to the extreme points of a Poisson point process on R d--1 R having
intensity \sqrt de dh dhdv
Computer simulation of crystallization kinetics with non-Poisson distributed nuclei
The influence of non-uniform distribution of nuclei on crystallization
kinetics of amorphous materials is investigated. This case cannot be described
by the well-known Johnson-Mehl-Avrami (JMA) equation, which is only valid under
the assumption of a spatially homogeneous nucleation probability. The results
of computer simulations of crystallization kinetics with nuclei distributed
according to a cluster and a hardcore distribution are compared with JMA
kinetics. The effects of the different distributions on the so-called Avrami
exponent are shown. Furthermore, we calculate the small-angle scattering
curves of the simulated structures which can be used to distinguish
experimentally between the three nucleation models under consideration.Comment: 14 pages including 7 postscript figures, uses epsf.sty and
ioplppt.st
Local central limit theorems in stochastic geometry
We give a general local central limit theorem for the sum of two independent
random variables, one of which satisfies a central limit theorem while the
other satisfies a local central limit theorem with the same order variance. We
apply this result to various quantities arising in stochastic geometry,
including: size of the largest component for percolation on a box; number of
components, number of edges, or number of isolated points, for random geometric
graphs; covered volume for germ-grain coverage models; number of accepted
points for finite-input random sequential adsorption; sum of nearest-neighbour
distances for a random sample from a continuous multidimensional distribution.Comment: V1: 31 pages. V2: 45 pages, with new results added in Section 5 and
extra explanation added elsewher
Marked Gibbs point processes with unbounded interaction: an existence result
We construct marked Gibbs point processes in under quite
general assumptions. Firstly, we allow for interaction functionals that may be
unbounded and whose range is not assumed to be uniformly bounded. Indeed, our
typical interaction admits an a.s. finite but random range. Secondly, the
random marks -- attached to the locations in -- belong to a
general normed space . They are not bounded, but their law should
admit a super-exponential moment. The approach used here relies on the
so-called entropy method and large-deviation tools in order to prove tightness
of a family of finite-volume Gibbs point processes. An application to
infinite-dimensional interacting diffusions is also presented
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