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. $p_{ij} = rho/[r_{ij}^alpha N^{1-alpha}]$ when $ 0 le alpha < 1$, and $p_{ij} = rho/[r_{ij} ln(N)]$ when $alpha = 1$. 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 $alpha=0$ 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 $alpha$, $p_s$ and $rho$. We show that in the ranges $ 0 le alpha le 1$ and $0 < p_s le 1$ the percolation order parameter $P_infty$ depends only on the average connectivity $gamma$ of sites, which can be explicitly computed in terms of the three parameters $alpha$, $p_s$ and $rho$.