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.
A percolation system with extremely long range connections and node dilution
SERVA, Maurizio
2014
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.