Behavior of the supercritical phase of a continuum percolation model on ℝd

1993 ◽  
Vol 30 (2) ◽  
pp. 382-396 ◽  
Author(s):  
Hideki Tanemura
Keyword(s):  
Positive Constant ◽  
Effective Conductivity ◽  
Critical Value ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Supercritical Case ◽  
Supercritical Phase ◽  
Renormalization Technique

A continuum percolation model onis considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.

Behavior of the supercritical phase of a continuum percolation model on ℝd

1993 ◽  
Vol 30 (02) ◽  
pp. 382-396
Author(s):  
Hideki Tanemura
Keyword(s):  
Positive Constant ◽  
Effective Conductivity ◽  
Critical Value ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Supercritical Case ◽  
Supercritical Phase ◽  
Renormalization Technique

A continuum percolation model on is considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.

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.

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 Sharpness of the Phase Transition for the Orthant Model

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Thomas Beekenkamp
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
Critical Value ◽  
Percolation Model ◽  
Critical Threshold ◽  
Directed Percolation ◽  
Sharp Threshold ◽  
Shape Theorem ◽  
Exponentially Small

AbstractThe orthant model is a directed percolation model on $\mathbb {Z}^{d}$ ℤ d , in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

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

Kinetic Growth Aggregation (Leath Algorithm) Approach for Continuum Percolation Model

2009 ◽  
Author(s):  
Keizo Yamamoto ◽  
Hiroyuki Yoshinaga ◽  
Sasuke Miyazima ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Percolation Model ◽  
Continuum Percolation ◽  
Kinetic Growth
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