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

TL;DR

This study uses Monte Carlo simulations to determine site percolation thresholds on square and cubic lattices with extended neighborhoods, establishing a connection to continuum percolation thresholds and analyzing finite-size effects.

Contribution

It provides precise percolation thresholds for various extended neighborhoods and relates lattice thresholds to continuum thresholds, including finite-$z$ corrections.

Findings

Thresholds for 23 lattice systems determined

Mapping lattice percolation to continuum shapes established

Finite-$z$ correction term identified

Abstract

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 $\eta_c$ for objects of those shapes. This mapping implies $zp_{c} \sim 4 \eta_c = 4.51235$ in 2D and $zp_{c} \sim 8 \eta_c = 2.73512$ 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 $c = 2^d \eta_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.