{
  "abstract": "In this work, we prove exponential decay for the spin correlation of the spin $O(\\mathsf{N})$ model for $\\mathsf{N}\\geq3$ at every inverse temperature $ β>0 $. In particular, our work confirms Polyakov's prediction in 1975 on the Heisenberg model, i.e., the case of $\\mathsf{N}=3$. To this end, we investigate a generalized $O(\\mathsf{N})$ model where $ \\mathsf{N} $ is a positive \\emph{real number}, as follows: we sample a loop soup on the metric graph (say of a box) with intensity $ \\tfrac{\\mathsf{N}}{2} $ subject to the conditioning that the local time at each lattice vertex is $β$, and then for each loop we independently assign it to copies $1\\leq \\mathsf{i}\\leq \\lceil\\mathsf{N}\\rceil$ such that it is assigned to copy $\\mathsf{i}$ with probability $ \\tfrac{1}{\\mathsf{N}} $ for $ 1\\leq \\mathsf{i}\\leq \\lfloor\\mathsf{N}\\rfloor $. Then, by Lupu's isomorphism theorem, we can construct a spin field by assigning a random sign to each loop soup cluster in each copy independently, such that the field coincides with the spin $O(\\mathsf{N})$ model when $\\mathsf{N}$ is an integer. We then establish exponential decay for every real-valued $\\mathsf{N}>2$. Our result reveals an underlying mechanism for exponential decay, which emerges as soon as $\\mathsf{N}>2$ (i.e., as soon as beyond the range of the XY model) and does not necessarily rely on the rotation symmetry in the sphere $ \\mathbb{S}^{2} $.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04908",
  "author_total": 3,
  "authors": [
    {
      "name": "Jian Ding"
    },
    {
      "name": "Fenglin Huang"
    },
    {
      "name": "Peng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3xyc64y5ncjgz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04908",
  "fulltext_url": null,
  "id": "3xyc64y5ncjgz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04908",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3xyc64y5ncjgz.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.04908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Exponential decay for the two-dimensional Heisenberg model",
  "title_text": "Exponential decay for the two-dimensional Heisenberg model",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/3xyc64y5ncjgz",
  "venue": "arXiv",
  "versions": []
}