scholarly journals Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Yilun Shang
Keyword(s):  
Long Range ◽  
Hierarchical Lattice ◽  
Infinite Component

We study a long-range percolation in the hierarchical lattice of order where probability of connection between two nodes separated by distance is of the form min, and . We show the uniqueness of the infinite component for this model.

Percolation in a Hierarchical Lattice

2012 ◽  
Vol 67 (5) ◽  
pp. 225-229 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Critical Percolation ◽  
Hierarchical Lattice ◽  
Vertex Degrees ◽  
Infinite Component

We study the percolation in the hierarchical lattice of order N where the probability of connection between two nodes separated by a distance k is of the form min{αβ-k,1},α ≥ 0 and β > 0. We focus on the vertex degrees of the resulting percolation graph and on whether there exists an infinite component. For fixed β, we show that the critical percolation value αc(β) is non-trivial, i.e., αc(β) ∊ (0;∞), if and only if β ∊ (N,N2).

A Note on the Perturbation of Mixed Percolation on the Hierarchical Group

2013 ◽  
Vol 68 (6-7) ◽  
pp. 475-478 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Long Range ◽  
Infinite Component ◽  
Hierarchical Group

We study mixed percolation on the hierarchical group of order N where each node is open withprobability 1-g, 0 ≤ g ≤ 1, and the probability of connection between two open nodes separatedby a distance k is of the form 1-exp(-αβ-k), α ≥ 0, and β > 0. The parameters α and γ are thepercolation parameters, while b describes the long-range nature of the model. In terms of parametersα,β, and γ, we show some perturbation results for the percolation function θ(α,β, γ), which is theprobability of existing an infinite component containing a prescribed node.

scholarly journals Long-range percolation on the hierarchical lattice

2012 ◽  
Vol 17 (0) ◽  
Author(s):  
Vyacheslav Koval ◽  
Ronald Meester ◽  
Pieter Trapman
Keyword(s):  
Long Range ◽  
Hierarchical Lattice
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