Bond percolation on simple cubic lattices with extended neighborhoods

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 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 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 ηc for objects of those shapes. This mapping implies zpc ∼ 4ηc = 4.51235 in 2D and zpc ∼ 8ηc = 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 pc = c/(z + b) we find good agreement with c = 2 d ηc, where the constant b represents a finite-z correction term. We also study power-law fits of the thresholds.

Percolation and the pandemic

“…The study of extended-range non-planar percolation goes back to the equivalent-neighbor model of Domb and Dalton [25], where site percolation was studied with several extended neighborhoods on a variety of lattices. Other site percolation models were studied by Gouker and Family [26], Galam and Malarz [27], d'Iribarne et al [28], and many others [29,30,31,32,33,34].…”

“…Here we are more interested in bond percolation, where an occupied bond represents the spread of an infection to a susceptible individual within the allowed neighborhood. Bond percolation on lattices with extended neighborhoods has been studied more recently [35,36,34,37,38]. Another related model is long-range percolation which has an infinite range and a power-law bond probability distribution [39,40,41].…”

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

“…The values for sc-1,2, sc-1,2,3 and sc-1,2,3,4 lattices come from Ref. [50], while the results for other lattices were determined here. The only other previous value seems to be 0.0991 (5) for bcc-1,2 of Ref.…”

“…Here we carried out extensive Monte-Carlo simulations, using a single-cluster growth method [49,50,60]. In this method, many individual clusters are generated from a seeded site on the lattice, and clusters with different sizes s are put in different bins with the range of (2 n , 2 n+1 − 1) for n = 0, 1, 2, .…”

“…Koza and collaborators [25,26] studied percolation of overlapping shapes on a lattice, which can be mapped to extended-range site percolation. While much of the earlier work concerned site percolation, bond percolation on extended lattices has been studied more extensively recently [9,[47][48][49][50].…”