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PreprintarXiv licencearXiv

Two-neighbor lattice percolation percolates in all dimensions

  • Jing Liu
  • Kainan Xiang
  • Lang Zou
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Abstract

It is trivial that $1$-dimensional $2$-neighbor lattice percolation percolates. Coupier, Henry, Jahnel and Köppl proved the $2$-neighbor lattice percolation percolates in dimension 2; and then for sufficiently large dimensions, Bäumler, Jahnel, Köppl, Lodewijks, Reeves and Tóbiás showed this is still true. In this paper, we verify that the $2$-neighbor lattice percolation percolates in all dimensions d≥ 3.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026twoneighbor,
  title = {{Two-neighbor lattice percolation percolates in all dimensions}},
  author = {Jing Liu and Kainan Xiang and Lang Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38200},
  eprint = {2610.38200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38200},
}
RIS
TY  - UNPB
TI  - Two-neighbor lattice percolation percolates in all dimensions
AU  - Jing Liu
AU  - Kainan Xiang
AU  - Lang Zou
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38200
UR  - https://arxiv.org/abs/2610.38200
AB  - It is trivial that $1$-dimensional $2$-neighbor lattice percolation percolates. Coupier, Henry, Jahnel and Köppl proved the $2$-neighbor lattice percolation percolates in dimension 2; and then for sufficiently large dimensions, Bäumler, Jahnel, Köppl, Lodewijks, Reeves and Tóbiás showed this is still true. In this paper, we verify that the $2$-neighbor lattice percolation percolates in all dimensions d≥ 3.
ER  -