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.

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.

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.

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.

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.

Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (01) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

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.

Scaling and Continuum Percolation Model for Enzyme-Catalyzed Gel Degradation

2007 ◽  
Vol 98 (22) ◽  
Author(s):  
D. Lairez ◽  
J.-P. Carton ◽  
G. Zalczer ◽  
J. Pelta
Keyword(s):  
Percolation Model ◽  
Continuum Percolation

scholarly journals Fractal structure of equipotential curves on a continuum percolation model

2012 ◽  
Vol 391 (23) ◽  
pp. 5802-5809 ◽  
Author(s):  
Shigeki Matsutani ◽  
Yoshiyuki Shimosako ◽  
Yunhong Wang
Keyword(s):  
Fractal Structure ◽  
Percolation Model ◽  
Continuum Percolation
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