Restoring site percolation on a damaged square lattice

TL;DR

This paper investigates methods to restore site percolation on a damaged square lattice by adding new sites or interactions, using Monte Carlo simulations to determine the thresholds for percolation under various strategies.

Contribution

It introduces two novel strategies for restoring percolation on damaged lattices by incorporating longer-range interactions and quantifies their effectiveness through simulation.

Findings

Percolation thresholds are shifted by adding longer-range interactions.

Restoration strategies depend on the fraction of new sites or interactions added.

Monte Carlo simulations provide quantitative thresholds for different scenarios.

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 $0\leq x \leq p_c(\text{N}^2)$. Our schemes are applicable to all regular lattices.