{
  "abstract": "We consider irreducible reversible Markov chains whose nonzero off-diagonal transition probabilities are bounded below by $ε>0$ and whose reversible measure satisfies $m_-\\le m(x)\\leκm_-$. Under the Bakry--Émery curvature condition $\\mathrm{CD}(K,\\infty)$, with $K\\in\\mathbb{R}$, we establish Wang's log-Harnack inequality $$ P_t(\\log f)(x)\\le\\log P_tf(y) +\\frac{C\\mathrm{dist}(x,y)^2}{\\int_0^t e^{2Ks}\\mathrm{d} s}, $$ for bounded positive $f$ bounded away from zero and all $x,y\\in V$, where $C$ is a constant dependent only on $\\varepsilon$. Building upon this inequality, we further prove volume doubling property: let $K_-:=\\max\\{-K,0\\}$ $$ \\mathrm{Vol}(B(o,2R))\\le C_0\\exp\\!\\bigl(C_1\\sqrt{K_-}\\,R\\bigr) \\mathrm{Vol}(B(o,R)),\\qquad R>0, $$ where $C_0,C_1$ depend only on $ε,κ$. If $d$ is the maximum combinatorial degree of the support graph, we further prove under $\\mathrm{CD}(0,\\infty)$ the sharp volume growth bound $\\mathrm{Vol}(B(o,R))\\le m_-CR^{\\lfloor d/2\\rfloor}$ for all $o$ and $R\\geq 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02010",
  "author_total": 3,
  "authors": [
    {
      "name": "Yanlong Ding"
    },
    {
      "name": "Shiping Liu"
    },
    {
      "name": "Chiyu Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wr88qpj8868m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02010",
  "fulltext_url": null,
  "id": "0wr88qpj8868m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02010",
  "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.CO"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02010",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wr88qpj8868m.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02010",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02010"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature",
  "title_text": "Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/0wr88qpj8868m",
  "venue": "arXiv",
  "versions": []
}