{
  "abstract": "We consider weakly reinforced Pólya urns on a countable graph of uniformly bounded degree. The vertices have positive bounded firing rates, which need not be bounded away from zero. For every reinforcement exponent $α\\in[0,1)$ and every deterministic choice of finite positive initial weights, the normalized edge weights converge almost surely, coordinate-wise, to a deterministic vector independent of the initial weights. We identify this vector as the unique non-vanishing equilibrium of the limiting equation. This proves the homogenization conjecture of Couzinié and Hirsch, including its formulation for non-uniform firing rates in the MATRIX open-problems report. The main step is a Liouville theorem for the limiting equation, established by a logarithmic-norm comparison of complete solutions using the row-sum balance at each vertex.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11780",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuo Qin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn79cxbmhy10n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11780",
  "fulltext_url": null,
  "id": "bn79cxbmhy10n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11780",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7221172110",
  "pdf_url": "https://arxiv.org/pdf/2610.11780",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn79cxbmhy10n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11780",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11780"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Homogenization of weakly reinforced Pólya urns on countable networks",
  "title_text": "Homogenization of weakly reinforced Pólya urns on countable networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/bn79cxbmhy10n",
  "venue": "arXiv",
  "versions": []
}