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Long-range percolation on the hierarchical lattice


 
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1. Title Title of document Long-range percolation on the hierarchical lattice
 
2. Creator Author's name, affiliation, country Vyacheslav Koval; Utrecht University; Netherlands
 
2. Creator Author's name, affiliation, country Ronald Meester; VU University Amsterdam; Netherlands
 
2. Creator Author's name, affiliation, country Pieter Trapman; Stockholm University; Sweden
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) long-range percolation; renormalisation; ergodicity
 
3. Subject Subject classification 60K35; 37F25; 47A35
 
4. Description Abstract We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that $\alpha_c(\beta)$ (the infimum of those $\alpha$ for which an infinite cluster exists a.s.) is non-trivial if and only if $N < \beta < N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $\alpha_c(\beta)$ as a function of $\beta$. This means that the phase diagram of this model is well understood.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Dutch research council (NWO), Riksbanken
 
7. Date (YYYY-MM-DD) 2012-07-23
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/1977
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-1977
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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