{
  "abstract": "For independent bond percolation on $\\mathbb{Z}^d$ with parameter $p$, let $p_c^b(d)$ be the critical probability. We prove that for each $d \\geq 9$, there is $ε_d>0$ such that for each $p \\in (p_c^b(d), p_c^b(d)+ε_d)$, the complement of the infinite open cluster stochastically dominates a supercritical site percolation on $\\mathbb{Z}^d$. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of $p_c^b(d)$ and $p_c^s(d)$ (the site critical probability) suggest that a similar stochastic domination result holds for all $d \\geq 4$. For the Ising model on $\\mathbb{Z}^d$ with inverse temperature $β$, let $β_c(d)$ be the critical inverse temperature. We prove that for each $d \\geq 8$, there is $ε_d>0$ such that for each $β\\in [0,β_c(d)+ε_d)$, the $+$ spins under the minus phase stochastically dominate a supercritical site percolation on $\\mathbb{Z}^d$. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of $β_c(d)$ and $p_c^s(d)$ suggest that a similar stochastic domination result holds for all $d \\geq 5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29806",
  "author_total": 2,
  "authors": [
    {
      "name": "Jianping Jiang"
    },
    {
      "name": "Sike Lang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8044z209bpzq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29806",
  "fulltext_url": null,
  "id": "d8044z209bpzq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29806",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214383342",
  "pdf_url": "https://arxiv.org/pdf/2609.29806",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8044z209bpzq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.29806",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29806"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Coexistence of infinite clusters for percolation and Ising model on $\\mathbb{Z}^d$",
  "title_text": "Coexistence of infinite clusters for percolation and Ising model on ℤ^d",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/d8044z209bpzq",
  "venue": "arXiv",
  "versions": []
}