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

This is the authors' abstract. We don't add key points for this paper.

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 ηc for objects of those shapes. This mapping implies zpc∼4ηc=4.51235 in two dimensions and zpc∼8ηc=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 pc=c/(z+b) we find good agreement with this prediction, c=2ᵈηc, 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 pc=1-exp(-2ᵈηc/z), a generalization of a formula of Koza, Kondrat, and Suszcayński [J. Stat. Mech.: Theor. Exp. (2014) P110051742-546810.1088/1742-5468/2014/11/P11005]. We also study power-law fits which are applicable for the range of values of z considered here.