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.

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.

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.

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 Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

2011 ◽  
Vol 48 (03) ◽  
pp. 597-610 ◽  
Author(s):  
Jianping Jiang ◽  
Sanguo Zhang ◽  
Tiande Guo
Keyword(s):  
Free Energy ◽  
Euclidean Space ◽  
Continuum Percolation ◽  
Site Percolation ◽  
Number Of Clusters ◽  
Infinite Cluster ◽  
The Mean ◽  
New Formula

A new formula for continuum percolation on the Euclidean space R d (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

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.

scholarly journals Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

2011 ◽  
Vol 48 (3) ◽  
pp. 597-610
Author(s):  
Jianping Jiang ◽  
Sanguo Zhang ◽  
Tiande Guo
Keyword(s):  
Free Energy ◽  
Euclidean Space ◽  
Continuum Percolation ◽  
Site Percolation ◽  
Number Of Clusters ◽  
Infinite Cluster ◽  
The Mean ◽  
New Formula

A new formula for continuum percolation on the Euclidean space Rd (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

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