An Improved Upper Bound for the Hexagonal Lattice Site Percolation Critical Probability

2002 ◽  
Vol 11 (6) ◽  
pp. 629-643 ◽  
Author(s):  
JOHN C. WIERMAN
Keyword(s):  
Upper Bound ◽  
Lattice Site ◽  
Hexagonal Lattice ◽  
Critical Probability ◽  
Site Percolation ◽  
Substitution Method ◽  
Site Model ◽  
Bond Model

The hexagonal lattice site percolation critical probability is shown to be at most 0.79472, improving the best previous mathematically rigorous upper bound. The bound is derived by using the substitution method to compare the site model with the bond model, the latter of which is exactly solved. Shortcuts which eliminate a substantial amount of computation make the derivation of the bound possible.

Substitution Method Critical Probability Bounds for the Square Lattice Site Percolation Model

1995 ◽  
Vol 4 (2) ◽  
pp. 181-188 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Upper Bound ◽  
Lattice Site ◽  
Percolation Model ◽  
Square Lattice ◽  
Critical Probability ◽  
Site Percolation ◽  
Substitution Method ◽  
Probability Bounds

The square lattice site percolation model critical probability is shown to be at most .679492, improving the best previous mathematically rigorous upper bound. This bound is derived by extending the substitution method to apply to site percolation models.

Upper bounds for the critical probability of oriented percolation in two dimensions

1993 ◽  
Vol 440 (1908) ◽  
pp. 201-220 ◽  
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.

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.

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.

An Improved Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

2005 ◽  
Vol 119 (1-2) ◽  
pp. 331-345 ◽  
Author(s):  
Élcio Lebensztayn ◽  
Fábio P. Machado ◽  
Serguei Popov
Keyword(s):  
Upper Bound ◽  
Critical Probability ◽  
Homogeneous Trees ◽  
Frog Model

ANTI-RED BOND CALCULATION ALGORITHM IN PERCOLATION

2009 ◽  
Vol 20 (02) ◽  
pp. 267-272 ◽  
Author(s):  
OLIVIER SCHOLDER
Keyword(s):  
Fractal Dimension ◽  
Percolation Threshold ◽  
Lattice Site ◽  
Calculation Algorithm ◽  
Site Percolation ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
The One

This paper presents an algorithm, which computes the number of anti-red bonds in a simple cubic lattice (site percolation) for different sizes and densities. Our interest was the fractal dimension of anti-red bonds at the percolation threshold. The value is found to be 1.18 ± 0.01. Two different theories proposed by Conigilio resp. Gouyet suggested a fractal dimension of 1.25 resp. 0.9. Thus, we can exclude the theory of Gouyet and are consistent with the one by Coniglio.

scholarly journals An Improved Upper Bound for Bootstrap Percolation in All Dimensions

2019 ◽  
Vol 28 (06) ◽  
pp. 936-960
Author(s):  
Andrew J. Uzzell
Keyword(s):  
Upper Bound ◽  
Bootstrap Percolation ◽  
Exact Constant ◽  
Critical Probability ◽  
Time Step ◽  
Natural Logarithm ◽  
Vertex Set ◽  
Image Position

AbstractIn r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then $$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.

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