Restoring site percolation on a damaged square lattice

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We study how to restore site percolation on a damaged square lattice with nearest neighbor (N²) interactions. Two strategies are suggested for a density x of destroyed sites by a random attack at pc. In the first one, a density y of new sites are created with longer range interactions, either next nearest neighbor (N³) or next next nearest neighbor (N⁴). In the second one, new longer range interactions N³ or N⁴ are created for a fraction v of the remaining (pc-x) sites in addition to their N² 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 π₃, π₄, π₂₃ and π₂₄. Using Monte Carlo simulations the values of the pairs {y, π₃ }, {y, π₄} and {v, π₂₃}, {v, π₂₄} are calculated for the whole range 0≤ x ≤ pc(N²). Our schemes are applicable to all regular lattices.