Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices
We propose a family of bipartite hierarchical lattice of order N governed by a pair of parameters ℓ and γ. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability p k = 1 − exp(...
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| Format: | Online |
| Language: | English |
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Hindawi Publishing Corporation
2013
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| Online Access: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3830813/ |
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pubmed-38308132013-11-28 Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices Shang, Yilun Research Article We propose a family of bipartite hierarchical lattice of order N governed by a pair of parameters ℓ and γ. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability p k = 1 − exp(−αβ −k), independently of all other edges. The parameter α is the percolation parameter, while β describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value α c ∈ (0, ∞) if and only if ℓ ≥ 1, 1 ≤ γ ≤ N − 1, and β ∈ (N, N 2). Moreover, the infinite component is unique when α > α c. Hindawi Publishing Corporation 2013-10-29 /pmc/articles/PMC3830813/ /pubmed/24288461 http://dx.doi.org/10.1155/2013/172393 Text en Copyright © 2013 Yilun Shang. https://creativecommons.org/licenses/by/3.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
| repository_type |
Open Access Journal |
| institution_category |
Foreign Institution |
| institution |
US National Center for Biotechnology Information |
| building |
NCBI PubMed |
| collection |
Online Access |
| language |
English |
| format |
Online |
| author |
Shang, Yilun |
| spellingShingle |
Shang, Yilun Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| author_facet |
Shang, Yilun |
| author_sort |
Shang, Yilun |
| title |
Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| title_short |
Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| title_full |
Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| title_fullStr |
Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| title_full_unstemmed |
Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices |
| title_sort |
phase transition in long-range percolation on bipartite hierarchical lattices |
| description |
We propose a family of bipartite hierarchical lattice of order N governed by a pair of parameters ℓ and γ. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability p
k = 1 − exp(−αβ
−k), independently of all other edges. The parameter α is the percolation parameter, while β describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value α
c ∈ (0, ∞) if and only if ℓ ≥ 1, 1 ≤ γ ≤ N − 1, and β ∈ (N, N
2). Moreover, the infinite component is unique when α > α
c. |
| publisher |
Hindawi Publishing Corporation |
| publishDate |
2013 |
| url |
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3830813/ |
| _version_ |
1612027658065936384 |