A Two‐Term Empirical Approximation for Site‐Percolation Thresholds on Hypercubic Lattices
Abstract
We investigate a compact two-term empirical approximation for the site-percolation threshold on (d)-dimensional hypercubic lattices:
[
p_0(d)=\frac{1}{2d-1}+\frac{7(d-1)}{(2d-1)^3}.
]
The approximation combines the leading Bethe-lattice contribution with a dimension-dependent correction motivated by the local coordination structure and short loops of hypercubic lattices. Its single adjustable coefficient is fixed by imposing the rational approximation (p_0(2)=16/27), rather than by fitting the known critical threshold directly.
The resulting expression predicts (p_0(2)=0.59259259) and (p_0(3)=0.312), differing from established numerical site-percolation thresholds by approximately 0.026% and 0.126%, respectively. For dimensions 4–13, the relative error reaches approximately 8% before decreasing with increasing dimension.
Although the formula reproduces the correct leading high-dimensional asymptotic behavior, it does not recover the known second-order coefficient. Its accuracy in low dimensions therefore remains an empirical observation rather than evidence of a universal scaling law.
The approximation is presented as a simple analytical benchmark, illustrating both the potential and the limitations of low-parameter expressions for percolation thresholds across spatial dimensions.
Related
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
- Strongly-connected percolation on directed lattices
- Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps
- Seed thresholds and degree variance in heterogeneous bootstrap percolation with growing degrees
- Site and bond percolation thresholds of all k-uniform tilings with k≤ 10
BibTeX
@misc{kaczorowski2026twoterm,
title = {{A Two‐Term Empirical Approximation for Site‐Percolation Thresholds on Hypercubic Lattices}},
author = {Jarosław Kaczorowski},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23435460},
url = {https://zenodo.org/records/23435460},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Two‐Term Empirical Approximation for Site‐Percolation Thresholds on Hypercubic Lattices
AU - Jarosław Kaczorowski
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23435460
UR - https://zenodo.org/records/23435460
AB - We investigate a compact two-term empirical approximation for the site-percolation threshold on (d)-dimensional hypercubic lattices: [ p_0(d)=\frac{1}{2d-1}+\frac{7(d-1)}{(2d-1)^3}. ] The approximation combines the leading Bethe-lattice contribution with a dimension-dependent correction motivated by the local coordination structure and short loops of hypercubic lattices. Its single adjustable coefficient is fixed by imposing the rational approximation (p_0(2)=16/27), rather than by fitting the known critical threshold directly. The resulting expression predicts (p_0(2)=0.59259259) and (p_0(3)=0.312), differing from established numerical site-percolation thresholds by approximately 0.026% and 0.126%, respectively. For dimensions 4–13, the relative error reaches approximately 8% before decreasing with increasing dimension. Although the formula reproduces the correct leading high-dimensional asymptotic behavior, it does not recover the known second-order coefficient. Its accuracy in low dimensions therefore remains an empirical observation rather than evidence of a universal scaling law. The approximation is presented as a simple analytical benchmark, illustrating both the potential and the limitations of low-parameter expressions for percolation thresholds across spatial dimensions.
ER -