Restoring site percolation on a damaged square lattice

Abstract

We study how to restore site percolation on a damaged square lattice with nearest neighbor (N${}^2$) interactions. Two strategies are suggested for a density $x$ of destroyed sites by a random attack at $p_c$. In the first one, a density $y$ of new sites are created with longer range interactions, either next nearest neighbor (N${}^3$) or next next nearest neighbor (N${}^4$). In the second one, new longer range interactions N${}^3$ or N${}^4$ are created for a fraction $v$ of the remaining $(p_c-x)$ sites in addition to their N${}^2$ interactions. In both cases, the values of $y$ and $v$ are tuned in order to restore site percolation which then occurs at new percolation thresholds, respectively ${\pi }_3$, ${\pi }_4$, ${\pi }_{23}$ and ${\pi }_{24}$. Using Monte Carlo simulations the values of the pairs $\{y,{\pi }_3\}$, $\{y,{\pi }_4\}$ and $\{v,{\pi }_{23}\}$, $\{v,{\pi }_{24}\}$ are calculated for the whole range $0\le x\le p_c({\text{N}}^2)$. Our schemes are applicable to all regular lattices.

5 pages, revtex4

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• Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
• Site percolation thresholds on triangular lattice with complex neighborhoods
• Random site percolation on honeycomb lattices with complex neighborhoods
• Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone
• Universality of random-site percolation thresholds for two-dimensional complex non-compact neighborhoods
• Unusual percolation in simple small-world networks
• Influence of long-range interactions on strategy selection in crowd
• Extended-range percolation in five dimensions