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.
Citation
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
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