{
  "abstract": "We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let $J:\\mathbb Z^d\\to[0,\\infty)$ be symmetric, satisfy $J(0)=0$, and $\\|J\\|_1:=\\sum_{x\\in\\mathbb Z^d}J(x)<\\infty$. For a nonempty finite set $Λ\\subset\\mathbb Z^d$, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration $σ\\in\\{-1,+1\\}^Λ$ has probability proportional to \\[ \\exp\\!\\left\\{ β\\sum_{x,y\\inΛ} J(x-y)σ_xσ_y \\right\\}. \\] Let $S_Λ:=\\sum_{x\\inΛ}σ_x$ be the total spin and $β_{\\mathrm c}$ denote the critical inverse temperature. For every fixed $0\\leqβ<β_{\\mathrm c}$, we prove that \\[ d_{\\mathrm K}\\!\\left( \\frac{S_Λ}{\\sqrt{\\operatorname{Var}(S_Λ)}},G \\right) \\leq C_{β,J}|Λ|^{-1/2}, \\] where $d_{\\mathrm K}$ is the Kolmogorov distance, $G\\sim \\mathcal{N}(0,1)$, and $C_{β,J}$ depends only on $β$ and $J$. For the fully periodized model on $(\\mathbb Z/L\\mathbb Z)^d$, the corresponding rate is $L^{-d/2}$, uniformly in $L$. For the classical nearest-neighbor model, the Kolmogorov bound also holds for $-β_{\\mathrm c}<β<β_{\\mathrm c}$, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22835",
  "author_total": 4,
  "authors": [
    {
      "name": "Song-Hao Liu"
    },
    {
      "name": "Qi-Man Shao"
    },
    {
      "name": "Yang Xie"
    },
    {
      "name": "Jing-Yu Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/97gv6824cpmbm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22835",
  "fulltext_url": null,
  "id": "97gv6824cpmbm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22835",
  "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": "W7222943327",
  "pdf_url": "https://arxiv.org/pdf/2610.22835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/97gv6824cpmbm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Normal approximation for total magnetization in subcritical long-range Ising models",
  "title_text": "Normal approximation for total magnetization in subcritical long-range Ising models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/97gv6824cpmbm",
  "venue": "arXiv",
  "versions": []
}