scholarly journals Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice

1998 ◽  
Vol 3 (0) ◽  
Author(s):  
Harry Kesten ◽  
Vladas Sidoravicius ◽  
Yu Zhang
Keyword(s):  
Triangular Lattice ◽  
Site Percolation ◽  
Almost All ◽  
Critical Site

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 Percolation of Words on Zd with Long-Range Connections

2011 ◽  
Vol 48 (4) ◽  
pp. 1152-1162 ◽  
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model on Zd, with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1}N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

scholarly journals Critical Site Percolation in High Dimension

2020 ◽  
Vol 181 (3) ◽  
pp. 816-853
Author(s):  
Markus Heydenreich ◽  
Kilian Matzke
Keyword(s):  
High Dimension ◽  
Critical Exponents ◽  
Mean Field ◽  
High Dimensions ◽  
Lace Expansion ◽  
Site Percolation ◽  
Triangle Condition ◽  
Hypercubic Lattice ◽  
Infra Red ◽  
Critical Site

Abstract We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

scholarly journals Percolation of Words on Z d with Long-Range Connections

2011 ◽  
Vol 48 (04) ◽  
pp. 1152-1162
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model onZd, with parameterp∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameterp, there exists an integerK(p) such that all binary sequences (words) ξ ∈ {0, 1}Ncan be seen simultaneously, almost surely, even if all connections with length larger thanK(p) are suppressed. We also show some results concerning howK(p) should scale withpaspgoes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

scholarly journals Forward Clusters for Degenerate Random Environments

2016 ◽  
Vol 25 (5) ◽  
pp. 744-765
Author(s):  
MARK HOLMES ◽  
THOMAS S. SALISBURY
Keyword(s):  
Lower Bound ◽  
Triangular Lattice ◽  
Directed Graphs ◽  
Two Dimensions ◽  
Occupation Probability ◽  
Random Environments ◽  
Connected Component ◽  
Site Percolation ◽  
Set Of Points

We consider connectivity properties and asymptotic slopes for certain random directed graphs on ℤ2in which the set of points$\mathcal{C}_o$that the origin connects to is always infinite. We obtain conditions under which the complement of$\mathcal{C}_o$has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in two dimensions.

Sign in / Sign up

Export Citation Format

Share Document