{
  "abstract": "We establish a Thouless--Anderson--Palmer (TAP) variational formula for the free energy of spherical $2$-spin models whose interaction matrices have a compactly supported limiting spectral distribution. The formula holds at every inverse temperature and, in particular, extends the TAP representation previously obtained for the spherical Sherrington--Kirkpatrick model. Our main contribution is a new microcanonical proof designed to handle marginality. We partition the sphere according to the smallest collection of top eigendirections carrying a prescribed fraction of the norm. On every element of this partition, the remaining degrees of freedom are effectively at high temperature and their contribution can be computed through spherical-integral asymptotics. The resulting auxiliary variational problem separates the macroscopic low-temperature contribution from the high-temperature fluctuation term; its maximization recovers the usual TAP functional and Plefka condition. This decomposition also shows why neighborhoods of the TAP maximizers alone do not capture the Gibbs mass: all eigendirections associated with large eigenvalues contribute on the same exponential scale, a manifestation of the marginality of spherical $2$-spin states.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13670",
  "author_total": 3,
  "authors": [
    {
      "name": "David Belius"
    },
    {
      "name": "Simon Briend"
    },
    {
      "name": "Volodymyr Fomichov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35m7k4qcep2mj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13670",
  "fulltext_url": null,
  "id": "35m7k4qcep2mj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13670",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7221561896",
  "pdf_url": "https://arxiv.org/pdf/2610.13670",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35m7k4qcep2mj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:36Z",
      "native_id": "2610.13670",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13670"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "TAP variational formula for 2-spin models with general interaction matrix and marginality",
  "title_text": "TAP variational formula for 2-spin models with general interaction matrix and marginality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/35m7k4qcep2mj",
  "venue": "arXiv",
  "versions": []
}