A lower bound for the critical probability of the square lattice site percolation

1985 ◽  
Vol 69 (1) ◽  
pp. 19-22 ◽  
Author(s):  
B�lint T�th
Keyword(s):  
Lower Bound ◽  
Lattice Site ◽  
Square Lattice ◽  
Critical Probability ◽  
Site Percolation

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.

Exact site-percolation probability on the square lattice

2021 ◽  
Author(s):  
Stephan Mertens
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Percolation Cluster ◽  
Square Lattice ◽  
Site Percolation ◽  
Exact Probability ◽  
Percolation Probability ◽  
A Site ◽  
Time And Space Complexity

Abstract We present an algorithm to compute the exact probability $R_{n}(p)$ for a site percolation cluster to span an $n\times n$ square lattice at occupancy $p$. The algorithm has time and space complexity $O(\lambda^n)$ with $\lambda \approx 2.6$. It allows us to compute $R_{n}(p)$ up to $n=24$. We use the data to compute estimates for the percolation threshold $p_c$ that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.

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.

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

1980 ◽  
Vol 12 (04) ◽  
pp. 848-863 ◽  
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.

DIFFUSION UNDER CROSSED LOCAL AND GLOBAL BIASES IN DISORDERED SYSTEMS

2000 ◽  
Vol 11 (07) ◽  
pp. 1357-1369 ◽  
Author(s):  
SITANGSHU BIKAS SANTRA ◽  
WILLIAM A. SEITZ
Keyword(s):  
Relaxation Time ◽  
Disordered Systems ◽  
Maximum Rate ◽  
Rate Of Change ◽  
Square Lattice ◽  
Site Percolation ◽  
Local Bias ◽  
Random Walker ◽  
Percolation Clusters ◽  
Global Bias

Diffusion on 2D site percolation clusters at p = 0.7, 0.8, and 0.9 above pc on the square lattice in the presence of two crossed bias fields, a local bias B and a global bias E, has been investigated. The global bias E is applied in a fixed global direction whereas the local bias B imposes a rotational constraint on the motion of the diffusing particle. The rms displacement Rt ~ tk in the presence of both biases is studied. Depending on the strength of E and B, the behavior of the random walker changes from diffusion to drift to no-drift or trapping. There is always diffusion for finite B with no global bias. A crossover from drift to no-drift at a critical global bias Ec is observed in the presence of local bias B for all disordered lattices. At the crossover, value of the rms exponent changes from k = 1 to k < 1, the drift velocity vt changes from constant in time t to decreasing power law nature, and the "relaxation" time τ has a maximum rate of change with respect to the global bias E. The value of critical bias Ec depends on the disorder p as well as on the strength of local bias B. Phase diagrams for diffusion, drift, and no-drift are obtained as a function of bias fields E and B for these systems.

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