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.

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 &lt; ∞ such that for all λ &gt; 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 &lt; λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

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.

scholarly journals Line-of-Sight Percolation

2009 ◽  
Vol 18 (1-2) ◽  
pp. 83-106 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
SVANTE JANSON ◽  
OLIVER RIORDAN
Keyword(s):  
Random Walk ◽  
Higher Dimensions ◽  
Percolation Model ◽  
The Other ◽  
Branching Random Walk ◽  
Line Of Sight ◽  
Continuum Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Vertex Set

Given ω ≥ 1, let $\Z^2_{(\omega)}$ be the graph with vertex set $\Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus $\Z^2_{(1)}$ is precisely $\Z^2$.) Let pc(ω) be the critical probability for site percolation on $\Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

scholarly journals Percolation and Connectivity in AB Random Geometric Graphs

2012 ◽  
Vol 44 (01) ◽  
pp. 21-41 ◽  
Author(s):  
Srikanth K. Iyer ◽  
D. Yogeshwaran
Keyword(s):  
Point Processes ◽  
Nearest Neighbor ◽  
Geometric Graph ◽  
Boolean Model ◽  
Unit Cube ◽  
Poisson Point Processes ◽  
Random Geometric Graphs ◽  
Asymptotic Bounds ◽  
Discrete Lattices ◽  
Critical Intensity

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

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.

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 and Connectivity in AB Random Geometric Graphs

2012 ◽  
Vol 44 (1) ◽  
pp. 21-41 ◽  
Author(s):  
Srikanth K. Iyer ◽  
D. Yogeshwaran
Keyword(s):  
Point Processes ◽  
Nearest Neighbor ◽  
Geometric Graph ◽  
Boolean Model ◽  
Unit Cube ◽  
Poisson Point Processes ◽  
Random Geometric Graphs ◽  
Asymptotic Bounds ◽  
Discrete Lattices ◽  
Critical Intensity

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

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.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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