Percolation in a Hierarchical Lattice

Abstract

We study the percolation in the hierarchical lattice of order N where the probability of connection between two nodes separated by a distance k is of the form min{αβ-k,1},α ≥ 0 and β > 0. We focus on the vertex degrees of the resulting percolation graph and on whether there exists an infinite component. For fixed β, we show that the critical percolation value αc(β) is non-trivial, i.e., αc(β) ∊ (0;∞), if and only if β ∊ (N,N2).