{
  "abstract": "We study the number of infinite clusters in the $2dp$-neighbor graph on $\\mathbb Z^d$, $d\\ge2$, in which every vertex independently selects $\\lfloor 2dp\\rfloor$ of its nearest neighbors uniformly at random, and one more with probability $2dp-\\lfloor 2dp\\rfloor$. The model has negatively correlated edges and, for $1\\le2dp\\le2d-1$, neither finite energy nor insertion or deletion tolerance. We prove that the undirected graph has at most one infinite cluster for every $p\\in [0,1]$, and that in the planar case the same holds for forward and strong forward clusters of the directed graph when $p\\in[1/2,1]$. We further exhibit a second phase transition: for $p\\ge(d+1)/(2d)$ the open cluster of the origin is almost surely the whole lattice, and for $p>(d+1)/(2d)$ any two vertices of a box are connected inside the box of twice the size with probability exponentially close to one. In particular, the percolation function is continuous but not analytic on the supercritical interval $(p_c,1]$. Finally, we prove that the full-coverage phase transition at $p_f = (d+1)/2d$ is of order $2^d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06800",
  "author_total": 4,
  "authors": [
    {
      "name": "David Coupier"
    },
    {
      "name": "Benoît Henry"
    },
    {
      "name": "Benedikt Jahnel"
    },
    {
      "name": "Jonas Köppl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a2nxe9s3wmkrd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06800",
  "fulltext_url": null,
  "id": "a2nxe9s3wmkrd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06800",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a2nxe9s3wmkrd.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Uniqueness and full coverage in $k$-neighbor graphs",
  "title_text": "Uniqueness and full coverage in k-neighbor graphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/a2nxe9s3wmkrd",
  "venue": "arXiv",
  "versions": []
}