Uniqueness of the infinite component for percolation on a hierarchical lattice (Q456611)

Summary: We study a long-range percolation in the hierarchical lattice \(\Omega_N\) of order \(N\), where the probability of connection between two nodes separated by a distance \(k\) is of the form \(\min\{\alpha \beta^{-k}, 1\}\), \( \alpha \geq 0\) and \(\beta > 0\). We prove the uniqueness of the infinite component for this model.