Improved Lower Bounds for the Critical Probability of Oriented Bond Percolation in Two Dimensions

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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Bond percolation on honeycomb and triangular lattices

Advances in Applied Probability ◽

10.2307/1426685 ◽

1981 ◽

Vol 13 (2) ◽

pp. 298-313 ◽

Cited By ~ 54

Author(s):

John C. Wierman

Keyword(s):

Lower Bounds ◽

Triangular Lattice ◽

Cluster Size ◽

Honeycomb Lattice ◽

Bond Percolation ◽

Critical Probability ◽

Lattice Graph ◽

The Mean ◽

Triangular Lattices ◽

Crossing Probabilities

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

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Bond percolation on honeycomb and triangular lattices

Advances in Applied Probability ◽

10.1017/s0001867800036028 ◽

1981 ◽

Vol 13 (02) ◽

pp. 298-313 ◽

Cited By ~ 23

Author(s):

John C. Wierman

Keyword(s):

Lower Bounds ◽

Triangular Lattice ◽

Cluster Size ◽

Honeycomb Lattice ◽

Bond Percolation ◽

Critical Probability ◽

Lattice Graph ◽

The Mean ◽

Triangular Lattices ◽

Crossing Probabilities

The two common critical probabilities for a lattice graphLare the cluster size critical probabilitypH(L) and the mean cluster size critical probabilitypT(L). The values for the honeycomb latticeHand the triangular latticeTare proved to bepH(H) =pT(H) = 1–2 sin (π/18) andPH(T) =pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

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Lower bounds for the critical probability in percolation models with oriented bonds

Journal of Applied Probability ◽

10.1017/s0021900200097266 ◽

1980 ◽

Vol 17 (04) ◽

pp. 979-986 ◽

Cited By ~ 4

Author(s):

Lawrence Gray ◽

John C. Wierman ◽

R. T. Smythe

Keyword(s):

Lower Bounds ◽

Oriented Percolation ◽

Critical Probability ◽

Critical Percolation ◽

Simple Method

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

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Percolation on Penrose Tilings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1998-026-0 ◽

1998 ◽

Vol 41 (2) ◽

pp. 166-177 ◽

Cited By ~ 6

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.

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Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200101573 ◽

1997 ◽

Vol 34 (04) ◽

pp. 859-867

Author(s):

Béla Bollabás ◽

Alan Stacey

Keyword(s):

Monte Carlo ◽

Markov Chains ◽

Monte Carlo Simulations ◽

Statistical Tests ◽

Upper Bounds ◽

Two Dimensions ◽

Oriented Percolation ◽

Critical Probability ◽

Confidence Levels

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

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On the time constant and path length of first-passage percolation

Advances in Applied Probability ◽

10.1017/s0001867800020127 ◽

1980 ◽

Vol 12 (04) ◽

pp. 848-863 ◽

Cited By ~ 6

Author(s):

Harry Kesten

Keyword(s):

Distribution Function ◽

Time Constant ◽

Path Length ◽

Passage Time ◽

Square Lattice ◽

Bond Percolation ◽

First Passage Percolation ◽

Critical Probability ◽

First Passage ◽

Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT , then there exist constants 0 < a, C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C 1 n). From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn (c) <∞, where Nn (c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

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