  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.
