Proof that $$\delta =2$$ δ = 2 and $$\beta =1$$ β = 1 under the Triangle Condition

2017 ◽  
pp. 125-137
Author(s):  
Markus Heydenreich ◽  
Remco van der Hofstad
Keyword(s):  
Triangle Condition

scholarly journals Random subgraphs of finite graphs: I. The scaling window under the triangle condition

2005 ◽  
Vol 27 (2) ◽  
pp. 137-184 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Remco van der Hofstad ◽  
Gordon Slade ◽  
Joel Spencer
Keyword(s):  
Finite Graphs ◽  
Triangle Condition ◽  
Scaling Window ◽  
Random Subgraphs

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (3) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

scholarly journals Percolation Critical Exponents Under the Triangle Condition

1991 ◽  
Vol 19 (4) ◽  
pp. 1520-1536 ◽  
Author(s):  
D. J. Barsky ◽  
M. Aizenman
Keyword(s):  
Critical Exponents ◽  
Triangle Condition

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 Random subgraphs of finite graphs. II. The lace expansion and the triangle condition

2005 ◽  
Vol 33 (5) ◽  
pp. 1886-1944 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Remco van der Hofstad ◽  
Gordon Slade ◽  
Joel Spencer
Keyword(s):  
Lace Expansion ◽  
Finite Graphs ◽  
Triangle Condition ◽  
Random Subgraphs

scholarly journals The triangle condition for percolation

1989 ◽  
Vol 21 (2) ◽  
pp. 269-274 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade
Keyword(s):  
Triangle Condition

scholarly journals Triangle Condition for Oriented Percolation in High Dimensions

1993 ◽  
Vol 21 (4) ◽  
pp. 1809-1844 ◽  
Author(s):  
Bao Gia Nguyen ◽  
Wei-Shih Yang
Keyword(s):  
Oriented Percolation ◽  
High Dimensions ◽  
Triangle Condition

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (03) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point p c, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.

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