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;1g, a ≥ 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 α c(β) is non-trivial, i.e., α c(β) ∈ (0,∞), if and only if ß ∈ (N,N 2). © 2012 Verlag der Zeitschrift für Naturforschung, Tübingen.