{
  "abstract": "We investigate a compact two-term empirical approximation for the site-percolation threshold on (d)-dimensional hypercubic lattices:\n\n[\n\np_0(d)=\\frac{1}{2d-1}+\\frac{7(d-1)}{(2d-1)^3}.\n\n]\n\nThe 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.\n\nThe 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.\n\nAlthough 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.\n\nThe 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Jarosław Kaczorowski",
      "orcid": "0009-0000-7461-1179"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0m4jwmsban926.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23435460",
  "fulltext_url": null,
  "id": "0m4jwmsban926",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23435460",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23435460/files/Percolation_Hypercubic_Two_Term_Empirical_Formula_EN.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0m4jwmsban926.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:21:25Z",
      "native_id": "23435460",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23435460"
    }
  ],
  "subjects": [],
  "title": "A Two‐Term Empirical Approximation for Site‐Percolation Thresholds on Hypercubic Lattices",
  "title_text": "A Two‐Term Empirical Approximation for Site‐Percolation Thresholds on Hypercubic Lattices",
  "type": "preprint",
  "updated_at": "2026-10-31T12:21:25Z",
  "url": "https://research.draconitic.com/paper/0m4jwmsban926",
  "venue": "Zenodo",
  "versions": []
}