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

TL;DR

This study investigates percolation thresholds on regular lattices with extended neighborhoods in 2D and 3D, revealing universal behaviors and confirming theoretical predictions through simulations and data analysis.

Contribution

It demonstrates the universality of the asymptotic behavior of percolation thresholds across different lattices and neighborhood ranges in two and three dimensions.

Findings

Threshold values collapse onto predicted lines for large coordination numbers.

Finite-z corrections for bond percolation follow predicted power laws.

Universal behavior depends only on dimension and percolation type, not lattice specifics.

Abstract

Extended-range percolation on various regular lattices, including all eleven Archimedean lattices in two dimensions, and the simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number $z$ and site thresholds $p_c$ for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting $z$ versus $1/p_{c}$ and $z$ versus $-1/\ln(1-p_c)$, using the data of d'Iribarne et al. [J. Phys. A 32:2611, 1999] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of $zp_{c} \sim 4 \eta_c = 4.51235$, where $\eta_c$ is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for BCC and FCC lattices with 2nd and 3rd NN, and bond thresholds for the SC lattice with up to the 13th NN, are obtained by Monte-Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of $zp_{c} \sim 8 \eta_c = 2.7351$, where $\eta_c$ is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior $p_c = 1/(z-1)$ is expected to hold for large $z$, and the finite-$z$ correction is confirmed to satisfy $zp_{c} - 1 \sim a_{1}z^{-x}$, with $x=2/3$ for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21:56, 2016] and by Xu et al. [Phys. Rev. E, 103:022127, 2021]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of $zp_{c}$ is universal, depending only upon the dimension of the system and whether site or bond percolation, but not upon the type of lattice.