Phase transition in long-range percolation on bipartite hierarchical lattices (Q904587)

Summary: We propose a family of bipartite hierarchical lattice of order \(N\) governed by a pair of parameters \(\ell\) and \(\gamma\). 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 \(p_k=1-\exp (-\alpha\beta^{-k})\), independently of all other edges. The parameter \(\alpha\) is the percolation parameter, while \(\beta\) describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value \(\alpha_c \in (0, \infty)\) if and only if \(\ell \geq 1\), \(1 \leq \gamma \leq N-1\), and \(\beta \in (N,N^2)\). Moreover, the infinite component is unique when \(\alpha>\alpha_c\).