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

By means of extensive Monte Carlo simulation, we study extended-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors up to the eighth nearest neighbors for the square lattice and the ninth nearest 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 of connected sites can be mapped to problems of lattice percolation of extended objects of a given shape, such as disks and spheres, and the thresholds can be related to the continuum thresholds eta(c) for objects of those shapes. This mapping implies zp(c) similar to 4 eta(c) = 4.51235 in two dimensions and zp(c) similar to 8 eta(c) = 2.7351 in three dimensions for large z for circular and spherical neighborhoods, respectively, where z is the coordination number. Fitting our data for compact neighborhoods to the form p(c) = c/(z + b) we find good agreement with this prediction, c = 2(d)eta(c), with the constant b representing a finite-z correction term. We also examined results from other studies using this fitting formula. A good fit of the large but finite-z behavior can also be made using the formula p(c) = 1 - exp(-2(d)eta(c)/z), a generalization of a formula of Koza, Kondrat, and Suszcaynski [J. Stat. Mech.: Theor. Exp. (2014) P11005]. We also study power-law fits which are applicable for the range of values of z considered here.

Automated summary

By conducting extensive Monte Carlo simulations on various lattices, the study identified precise thresholds for site percolation systems and confirmed theoretical predictions through fitting data for compact neighborhoods. Power-law fits were also examined for the range of coordination numbers considered.