{
  "abstract": "We study the random-cluster model on locally treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p \\in (0,1)$ and a clustering weight $q \\in [1,\\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, at parameter $p$ below the tree uniqueness threshold $p_{\\mathsf u}(q,d)$, it is expected that sampling is easy and local Markov chains mix rapidly on all graphs with degree bounded by $d$. On typical graphs, the same is predicted at $p>p_{\\mathsf s}(q,d)$, where $p_{\\mathsf s}(q,d)$ is a second uniqueness transition point on the $d$-regular wired tree. Our result confirms this prediction for random regular graphs, showing that the random-cluster Glauber dynamics mix in time $O(n^{1+o(1)})$ on a random $d$-regular graph for all fixed $q>2$ and $p>p_{\\mathsf s}$. For comparison, the best prior sampling results for these models that held for all $p>p_{\\mathsf s}$ required $q\\gtrsim \\log d$ \\cite{BGPZ26}. The key difficulty is that simply utilizing the contraction of cavity messages on $d$-regular trees does not suffice to cover the entire regime $q>2$. To address this issue, motivated by \\cite{LS26}, we develop a careful bootstrap argument to show the shrinkage of disagreement under grand coupling. The core ideas of the proof were generated using GPT-6 and the Pharos system; the proof was then refined and simplified by the author.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05794",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhangsong Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pq1evnk6rr8w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05794",
  "fulltext_url": null,
  "id": "5pq1evnk6rr8w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05794",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pq1evnk6rr8w.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:56Z",
      "native_id": "2611.05794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Near linear mixing of random-cluster Glauber dynamics on random regular graphs",
  "title_text": "Near linear mixing of random-cluster Glauber dynamics on random regular graphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/5pq1evnk6rr8w",
  "venue": "arXiv",
  "versions": []
}