{
  "abstract": "We study the high-temperature derivation of $Φ^4_d$ measures from grand-canonical Bose gases on two- and three-dimensional tori. Building on the upper-symbol variational approach of the author's earlier work, we use relative entropy to give a simple, unified proof of relative free-energy convergence and Hilbert--Schmidt convergence of all fixed-order rescaled reduced density matrices to their Hartree counterparts. For interaction ranges $\\varepsilon=λ^η$, where $λ$ is the inverse temperature, these results hold for every fixed $0<η<1/2$ in two dimensions and $0<η<1/26$ in three. In two dimensions, we improve the result of Jougla and Rougerie (2026) and the local $Φ^4_2$ derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local $Φ^4_2$ derivation valid for every fixed exponent $0<η<1/2$, allowing polynomial exponents arbitrarily close to the diluteness threshold from below. In three dimensions, we use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert--Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26518",
  "author_total": 1,
  "authors": [
    {
      "name": "Hao Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/97ck71avv0nbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26518",
  "fulltext_url": null,
  "id": "97ck71avv0nbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26518",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
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  "openalex": "W7214110021",
  "pdf_url": "https://arxiv.org/pdf/2609.26518",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/97ck71avv0nbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.26518",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26518"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An upper-symbol approach to $Φ^4_d$ limits of Bose gases",
  "title_text": "An upper-symbol approach to Φ^4_d limits of Bose gases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/97ck71avv0nbf",
  "venue": "arXiv",
  "versions": []
}