{
  "abstract": "We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing and a fully polynomial-time approximation scheme (\\textbf{FPTAS}) whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. We further extend the framework to proper colorings of sparse Erdős-Rényi random graphs using recursion on permissive blocks. For every fixed $η\\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\\ge(2+η)d$, we obtain an \\textbf{FPTAS} for counting proper $q$-colorings of $G\\sim\\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12352",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhidan Li"
    },
    {
      "name": "Kuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gjfnk18171hv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12352",
  "fulltext_url": null,
  "id": "2gjfnk18171hv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12352",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7212646440",
  "pdf_url": "https://arxiv.org/pdf/2609.12352",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gjfnk18171hv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.12352",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12352"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Spatial Mixing and Deterministic Approximate Counting of Multi-spin Systems beyond Bounded Degree Graphs",
  "title_text": "Spatial Mixing and Deterministic Approximate Counting of Multi-spin Systems beyond Bounded Degree Graphs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/2gjfnk18171hv",
  "venue": "arXiv",
  "versions": []
}