Long-range percolation on the hierarchical lattice

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 the critical value ${\alpha }_c(\beta )$ 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.

24 pages

References in corpus (2)

• On the scaling of the chemical distance in long-range percolation models
• The growth of the infinite long-range percolation cluster

Cited by in corpus (4)

• Number variance for hierarchical random walks and related fluctuations
• The critical two-point function for long-range percolation on the hierarchical lattice
• A percolation system with extremely long range connections and node dilution
• Transience and recurrence of random walks on percolation clusters in an ultrametric space