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 $π_3$, $π_4$, $π_{23}$ and $π_{24}$. Using Monte Carlo simulations the values of the pairs $\{y, π_3 \}$, $\{y, π_4\}$ and $\{v, π_{23}\}$, $\{v, π_{24}\}$ are calculated for the whole range $0\leq x \leq p_c(\text{N}^2)$. Our schemes are applicable to all regular lattices.