Lower Limit of Percolation Threshold on Square Lattice with Complex Neighborhoods

In this paper, the 60-year-old concept of long-range interaction in percolation problems introduced by Dalton, Domb and Sykes is reconsidered. With Monte Carlo simulation-based on the Newman-Ziff algorithm and the finite-size scaling hypothesis-we estimate 64 percolation thresholds for a random site percolation problem on a square lattice with neighborhoods that contain sites from the seventh coordination zone. The percolation thresholds obtained range from 0.27013 (for the neighborhood that contains only sites from the seventh coordination zone) to 0.11535 (for the neighborhood that contains all sites from the first to the seventh coordination zone). Similarly to neighborhoods with smaller ranges, the power-law dependence of the percolation threshold on the effective coordination number with an exponent close to -1/2 is observed. Finally, we empirically determine the limit of the percolation threshold on square lattices with complex neighborhoods. This limit scales with the inverse square of the mean radius of the neighborhood. The boundary of this limit is touched for threshold values associated with extended (compact) neighborhoods.