scholarly journals A lower bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

2016 ◽  
Vol 21 (0) ◽  
Author(s):  
Spencer Frei ◽  
Edwin Perkins
Keyword(s):  
Lower Bound ◽  
Three Dimensions ◽  
Bond Percolation

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.

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.

scholarly journals Expansion of Percolation Critical Points for Hamming Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 68-100
Author(s):  
Lorenzo Federico ◽  
Remco Van Der Hofstad ◽  
Frank Den Hollander ◽  
Tim Hulshof
Keyword(s):  
Lower Bound ◽  
Cartesian Product ◽  
Branching Random Walk ◽  
Bond Percolation ◽  
Complete Graphs ◽  
Lace Expansion ◽  
Critical Window ◽  
Hamming Graphs ◽  
Image Position ◽  
Crucial Ingredient

AbstractThe Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$, $$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$ which extends the asymptotics found in [10] by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random graph.

scholarly journals Study of droplets for correlated site-bond percolation in three dimensions

1982 ◽  
Vol 43 (20) ◽  
pp. 703-709 ◽  
Author(s):  
J. Roussenq ◽  
A. Coniglio ◽  
D. Stauffer
Keyword(s):  
Three Dimensions ◽  
Bond Percolation
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