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2012 Long-range percolation on the hierarchical lattice
Vyacheslav Koval, Ronald Meester, Pieter Trapman
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Electron. J. Probab. 17: 1-21 (2012). DOI: 10.1214/EJP.v17-1977

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.

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Vyacheslav Koval. Ronald Meester. Pieter Trapman. "Long-range percolation on the hierarchical lattice." Electron. J. Probab. 17 1 - 21, 2012. https://doi.org/10.1214/EJP.v17-1977

Information

Accepted: 23 July 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1260.60183
MathSciNet: MR2955049
Digital Object Identifier: 10.1214/EJP.v17-1977

Subjects:
Primary: 60K35
Secondary: 37F25 , 47A35

Keywords: ergodicity , Long-range percolation , renormalisation

JOURNAL ARTICLE
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