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Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds $η_c$ for objects of those shapes. This mapping implies $zp_{c} \sim 4 η_c = 4.51235$ in 2D and $zp_{c} \sim 8 η_c = 2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c = c/(z+b)$ we find good agreement with $c = 2^d η_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.