Extended-range percolation on lattices with cubic and non-compact neighborhoods in three dimensions
Abstract
Using MC simulations based on an efficient single-cluster growth algorithm, we investigate two types of three-dimensional extended-range percolation models, namely cubic and complex (non-compact) neighborhoods. We obtain high-precision percolation thresholds pc for approximately 100 distinct structures and further explore the relationship between these thresholds and the topology of the lattices, which includes the total coordination number z, the number of sites zi and these sites' distance to the central site ri of the given neighborhood in the ith coordination zone. Both bond and site percolation are analyzed for cubic neighborhoods on the SC lattice. For bond percolation, the asymptotic value of zpc tends to Bethe-lattice behavior (zpc∼1), and finite-z correction is found to be consistent with zpc−1∼ a1z−x, where x=(d−1)/d. For site percolation, the value of zpc varies non-monotonically, exhibiting an initial increase followed by a decrease with the coordination number z. Linear fitting leads to zpc∼2.5987(1) for large z, predicting a value of continuum percolation threshold of aligned cubes ηc=0.324838(12) using zpc=2dηc. We then examine site percolation for non-compact neighborhoods extending to the fifth coordination zone on SC, FCC, and BCC lattices. Through a comparison of the three indices of z, ξ=∑iziri2/i, and ζ=∑iziri, we find that ζ is effective in removing the degeneracy. The results obtained show power-law dependence pc∝(ζ/K)−γ with γSCζ=0.703(12), γFCCζ=0.728(10), and γBCCζ=0.699(12) for SC, FCC, and BCC lattices, and pc∝ζ−γ with γζ=0.719(9) independent of the underlying lattice. The number K indicates lattice connectivity.
Related
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
- Strongly-connected percolation on directed lattices
- Towards the spin-glass transition in finite dimensions via blue percolation
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BibTeX
@misc{zhang2026extendedrange,
title = {{Extended-range percolation on lattices with cubic and non-compact neighborhoods in three dimensions}},
author = {Zhe Zhang and Yuxuan Fan and Yang Yang and Yusong Tu and Zhipeng Xun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15441},
eprint = {2610.15441},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15441},
}
RIS
TY - UNPB
TI - Extended-range percolation on lattices with cubic and non-compact neighborhoods in three dimensions
AU - Zhe Zhang
AU - Yuxuan Fan
AU - Yang Yang
AU - Yusong Tu
AU - Zhipeng Xun
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15441
UR - https://arxiv.org/abs/2610.15441
AB - Using MC simulations based on an efficient single-cluster growth algorithm, we investigate two types of three-dimensional extended-range percolation models, namely cubic and complex (non-compact) neighborhoods. We obtain high-precision percolation thresholds p_c for approximately 100 distinct structures and further explore the relationship between these thresholds and the topology of the lattices, which includes the total coordination number z, the number of sites z_i and these sites' distance to the central site r_i of the given neighborhood in the ith coordination zone. Both bond and site percolation are analyzed for cubic neighborhoods on the SC lattice. For bond percolation, the asymptotic value of zp_c tends to Bethe-lattice behavior (zp_c∼1), and finite-z correction is found to be consistent with zp_c−1∼ a_1z^−x, where x=(d−1)/d. For site percolation, the value of zp_c varies non-monotonically, exhibiting an initial increase followed by a decrease with the coordination number z. Linear fitting leads to zp_c∼2.5987(1) for large z, predicting a value of continuum percolation threshold of aligned cubes η_c=0.324838(12) using zp_c=2^dη_c. We then examine site percolation for non-compact neighborhoods extending to the fifth coordination zone on SC, FCC, and BCC lattices. Through a comparison of the three indices of z, ξ=∑_iz_ir_i^2/i, and ζ=∑_iz_ir_i, we find that ζ is effective in removing the degeneracy. The results obtained show power-law dependence p_c∝(ζ/K)^−γ with γ_SC^ζ=0.703(12), γ_FCC^ζ=0.728(10), and γ_BCC^ζ=0.699(12) for SC, FCC, and BCC lattices, and p_c∝ζ^−γ with γ^ζ=0.719(9) independent of the underlying lattice. The number K indicates lattice connectivity.
ER -