Co-Existence of the occupied and vacant phase in boolean models in three or more dimensions

1997 ◽  
Vol 29 (04) ◽  
pp. 878-889 ◽  
Author(s):  
Anish Sarkar
Keyword(s):  
Poisson Process ◽  
Boolean Model ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Boolean Models ◽  
Whole Space ◽  
Critical Intensity ◽  
The Way

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d ≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

Co-Existence of the occupied and vacant phase in boolean models in three or more dimensions

1997 ◽  
Vol 29 (4) ◽  
pp. 878-889 ◽  
Author(s):  
Anish Sarkar
Keyword(s):  
Poisson Process ◽  
Boolean Model ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Boolean Models ◽  
Whole Space ◽  
Critical Intensity ◽  
The Way

Consider a continuum percolation model in which, at each point of ad-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for anyd≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

On a continuum percolation model

1991 ◽  
Vol 23 (3) ◽  
pp. 536-556 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Free Energy ◽  
Poisson Process ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Infinite Cluster ◽  
Cluster Density ◽  
Critical Intensity

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting ofkparticles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

On a continuum percolation model

1991 ◽  
Vol 23 (03) ◽  
pp. 536-556 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Free Energy ◽  
Poisson Process ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Infinite Cluster ◽  
Cluster Density ◽  
Critical Intensity

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (01) ◽  
pp. 48-57
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (04) ◽  
pp. 910-920
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λ E , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ E of eavesdropper nodes, there exists a critical intensity λ c < ∞ such that for all λ > λ c the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Clustering in a Continuum Percolation Model

1997 ◽  
Vol 29 (2) ◽  
pp. 327-336 ◽  
Author(s):  
J. Quintanilla ◽  
S. Torquato
Keyword(s):  
Poisson Process ◽  
Numerical Evaluation ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Average Volume ◽  
Integral Expression ◽  
Formal Expression

We study properties of the clusters of a system of fully penetrable balls, a model formed by centering equal-sized balls on the points of a Poisson process. We develop a formal expression for the density of connected clusters ofkballs (calledk-mers) in the system, first rigorously derived by Penrose [15]. Our integral expressions are free of inherent redundancies, making them more tractable for numerical evaluation. We also derive and evaluate an integral expression for the average volume ofk-mers.

scholarly journals The Boolean model in the Shannon regime: three thresholds and related asymptotics

2016 ◽  
Vol 53 (4) ◽  
pp. 1001-1018 ◽  
Author(s):  
Venkat Anantharam ◽  
François Baccelli
Keyword(s):  
Stochastic Geometry ◽  
Volume Fraction ◽  
Boolean Model ◽  
Boolean Models ◽  
Large Deviations Principle ◽  
Percolation Probability ◽  
Whole Space ◽  
Probability Threshold ◽  
The Mean ◽  
Model Features

Abstract Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅n√n. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (4) ◽  
pp. 910-920 ◽  
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Clustering in a Continuum Percolation Model

1997 ◽  
Vol 29 (02) ◽  
pp. 327-336 ◽  
Author(s):  
J. Quintanilla ◽  
S. Torquato
Keyword(s):  
Poisson Process ◽  
Numerical Evaluation ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Average Volume ◽  
Integral Expression ◽  
Formal Expression

We study properties of the clusters of a system of fully penetrable balls, a model formed by centering equal-sized balls on the points of a Poisson process. We develop a formal expression for the density of connected clusters of k balls (called k-mers) in the system, first rigorously derived by Penrose [15]. Our integral expressions are free of inherent redundancies, making them more tractable for numerical evaluation. We also derive and evaluate an integral expression for the average volume of k-mers.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (1) ◽  
pp. 48-57 ◽  
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

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