Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices

Abstract EN

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 pk=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 αc∈0,∞ if and only if ℓ≥1, 1≤γ≤N-1, and β∈N,N2.

Moreover, the infinite component is unique when α>αc.