Long-range percolation on the hierarchical lattice

TL;DR

This paper analyzes long-range percolation on a hierarchical lattice, establishing the conditions for phase transition, uniqueness of the infinite cluster, and continuity of critical parameters, thus providing a comprehensive understanding of the model's phase diagram.

Contribution

It characterizes the critical threshold and phase diagram for long-range percolation on the hierarchical lattice, including conditions for the existence and uniqueness of the infinite cluster.

Findings

Critical value (eta) is non-trivial if and only if N <  < N^2.

Uniqueness of the infinite component is proven.

Percolation probability and (eta) are continuous functions of .

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.