6,712 research outputs found

    Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

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    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

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    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

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    Let K be a convex set in R d and let K λ\lambda be the convex hull of a homogeneous Poisson point process P λ\lambda of intensity λ\lambda on K. When K is a simple polytope, we establish scaling limits as λ\lambda \rightarrow \infty for the boundary of K λ\lambda in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K λ\lambda, k \in {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K λ\lambda 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 ×\times R having intensity \sqrt de dh dhdv

    Computer simulation of crystallization kinetics with non-Poisson distributed nuclei

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    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 nn 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

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    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

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    We construct marked Gibbs point processes in Rd\mathbb{R}^d 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 Rd\mathbb{R}^d -- belong to a general normed space S\mathcal{S}. 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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