@article{EJP1977,
author = {Vyacheslav Koval and Ronald Meester and Pieter Trapman},
title = {Long-range percolation on the hierarchical lattice},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {17},
year = {2012},
keywords = {long-range percolation; renormalisation; ergodicity},
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.},
pages = {no. 57, 1-21},
issn = {1083-6489},
doi = {10.1214/EJP.v17-1977},
url = {http://ejp.ejpecp.org/article/view/1977}}