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.

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.

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.

scholarly journals Directed Animals, Quadratic Systems and Rewriting Systems

10.37236/2747 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Jean-François Marckert
Keyword(s):  
Strong Evidence ◽  
Percolation Cluster ◽  
Percolation Model ◽  
Square Lattice ◽  
Infinite Matrix ◽  
System Of Equations ◽  
Site Percolation ◽  
Quadratic Equations ◽  
Rewriting Systems ◽  
Strong Relation

A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function G of directed animals on the square lattice, counted according to the area and the perimeter, and the problem of solving a system of quadratic equations involving unknown matrices. We present some solid evidence that some infinite explicit matrices, the fixed points of a rewriting like system are the natural solutions to this system of equations: some strong evidence is given that the problem of finding G reduces to the problem of finding an eigenvector to an explicit infinite matrix.  Similar properties are shown for other combinatorial questions concerning directed animals, and for different lattices.

Weak Universality of a Site-Percolation Model with Two Different Sizes of Particles on a Square Lattice

1999 ◽  
Vol 68 (12) ◽  
pp. 3755-3758 ◽  
Author(s):  
Ryoji Sahara ◽  
Hiroshi Mizuseki ◽  
Kaoru Ohno ◽  
Yoshiyuki Kawazoe
Keyword(s):  
Percolation Model ◽  
Square Lattice ◽  
Site Percolation ◽  
A Site

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.

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