Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices

Figure 1

An illustration of bipartite hierarchical lattice (Ω¹_𝑁(ℓ,𝛾),Ω²_𝑁(ℓ,𝛾),𝑑) of order 𝑁=3, ℓ=2, and 𝛾=2. Vertices of type 1 are represented by solid points while those of type 2 hollow points. The distances between three vertices 𝟎=(0,0,0,…)∈Ω¹_𝑁, 𝐱=(1,0,0,…)∈Ω¹_𝑁, and 𝐲=(0,2,0,…)∈Ω²_𝑁 are 𝑑(𝟎,𝐱)=1 and 𝑑(𝟎,𝐲)=𝑑(𝐱,𝐲)=2.