{
  "abstract": "The random-cluster model with cluster parameter \\(0<q<1\\) is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees $(d_n)$, even for edge pairs at distance \\(o(d_n/\\log d_n)\\). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed \\(p\\in(0,1)\\), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance \\(o(d_n/\\log d_n)\\), uniformly in \\(q\\in[0,1)\\) and in the choice of the two edges. In particular, the result includes the \\(q=0\\) endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11578",
  "author_total": 2,
  "authors": [
    {
      "name": "Pengfei Tang"
    },
    {
      "name": "Zibo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cn4b77cjqqake.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11578",
  "fulltext_url": null,
  "id": "cn4b77cjqqake",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11578",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212406438",
  "pdf_url": "https://arxiv.org/pdf/2609.11578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cn4b77cjqqake.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2609.11578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Negative correlation for the random-cluster model below one on high-degree regular graphs",
  "title_text": "Negative correlation for the random-cluster model below one on high-degree regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cn4b77cjqqake",
  "venue": "arXiv",
  "versions": []
}