On spàtial general epidemics and bond percolation processes

1984 ◽  
Vol 21 (4) ◽  
pp. 911-914 ◽  
Author(s):  
Kari Kuulasmaa ◽  
Stan Zachary
Keyword(s):  
Lower Bound ◽  
Bond Percolation ◽  
Percolation Process ◽  
Percolation Probability ◽  
Percolation Processes

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

On spàtial general epidemics and bond percolation processes

1984 ◽  
Vol 21 (04) ◽  
pp. 911-914 ◽  
Author(s):  
Kari Kuulasmaa ◽  
Stan Zachary
Keyword(s):  
Lower Bound ◽  
Bond Percolation ◽  
Percolation Process ◽  
Percolation Probability ◽  
Percolation Processes

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

Percolation on Penrose Tilings

1998 ◽  
Vol 41 (2) ◽  
pp. 166-177 ◽  
Author(s):  
A. Hof
Keyword(s):  
Large Class ◽  
Arbitrary Dimension ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Aperiodic Tilings ◽  
Infinite Clusters ◽  
Almost All ◽  
Percolation Processes ◽  
Penrose Tilings

AbstractIn Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ℤd, and to other percolation processes, including Bernoulli bond percolation.

Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (02) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

Perimeter polynomials for bond percolation processes

1981 ◽  
Vol 14 (1) ◽  
pp. 287-293 ◽  
Author(s):  
M F Sykes ◽  
D S Gaunt ◽  
M Glen
Keyword(s):  
Bond Percolation ◽  
Perimeter Polynomials ◽  
Percolation Processes

Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (2) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (3) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn+ is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ1, μ2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ2 × ··· × μn on ℝn+ is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds.Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (03) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn + is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ 1, μ 2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ 2 × ··· × μn on ℝn + is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds. Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

On some percolation results of J. M. Hammersley

1979 ◽  
Vol 16 (03) ◽  
pp. 526-540 ◽  
Author(s):  
J. G. Oxley ◽  
D. J. A. Welsh
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Percolation Theory ◽  
Fkg Inequality ◽  
Percolation Probability ◽  
Finite Set ◽  
The Fkg Inequality ◽  
Arbitrary Graphs

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.

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