A percolation system with extremely long range connections and node dilution

We study the very long-range bond-percolation problem on a linear chain with both node and bond dilution. Very long-range means that the probability pijpij for a connection between two nodes i,ji,j at a distance rijrij decays as a power-law, i.e. View the MathML sourcepij=ρ/[rijαN1−α] when α∈[0,1)α∈[0,1), and pij=ρ/[rijln(N)]pij=ρ/[rijln(N)] when α=1α=1. Node dilution means that the probability that a node is present in a site is ps∈(0,1]ps∈(0,1]. The behavior of this model results from the competition between long-range connectivity which enhances the percolation, and node dilution which weakens percolation. The case α=0α=0 with ps=1ps=1 is well-known, being the exactly solvable mean-field model. The percolation order parameter P∞P∞ is investigated numerically for different values of αα, psps and ρρ. We show that in all range α∈[0,1]α∈[0,1] the percolation order parameter P∞P∞ depends only on the average connectivity γγ of the nodes, which can be explicitly computed in terms of the three parameters αα, psps and ρρ.