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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 sites and bonds dilution. Very long range means that the probability $p_{ij}$ for a connection between two occupied sites $i,j$ at a distance $r_{ij}$ decays as a power law, i.e. $p_{ij} = ρ/[r_{ij}^αN^{1-α}]$ when $ 0 \le α< 1$, and $p_{ij} = ρ/[r_{ij} \ln(N)]$ when $α= 1$. Site dilution means that the occupancy probability of a site is $0 < p_s \le 1$. The behavior of this model results from the competition between long-range connectivity, which enhances the percolation, and site dilution, which weakens percolation. The case $α=0$ with $p_s =1 $ is well-known, being the exactly solvable mean-field model. The percolation order parameter $P_\infty$ is investigated numerically for different values of $α$, $p_s$ and $ρ$. We show that in the ranges $ 0 \le α\le 1$ and $0 < p_s \le 1$ the percolation order parameter $P_\infty$ depends only on the average connectivity $γ$ of sites, which can be explicitly computed in terms of the three parameters $α$, $p_s$ and $ρ$.