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 [Symbol: see text] (0, ∞) if and only if ℓ ≥ 1, 1 ≤ γ ≤ N - 1, and β [Symbol: see text] (N, N (2)). Moreover, the infinite component is unique when α > α c .