Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices

We propose a family of bipartite hierarchical lattice of order N governed by a pair of parameters ℓ and γ. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability p (k) = 1 − exp(−αβ (−k)), independently of all other edges. The parameter α is the percolation parameter, while β describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value α (c) ∈ (0, ∞) if and only if ℓ ≥ 1, 1 ≤ γ ≤ N − 1, and β ∈ (N, N (2)). Moreover, the infinite component is unique when α > α (c).