{
  "abstract": "We study the free energy fluctuations of spherical and Ising perceptron models with general even activations. These include the symmetric binary perceptron, the $U$-perceptron, and the bipartite Sherrington--Kirkpatrick model. Our main result is a Gaussian CLT for the free energy that holds both in the ``high-temperature'' interior of a second moment RS regime and on its ``critical'' boundary. At criticality, the CLT has scale $(\\frac{\\log N}{6})^{1/2}$, which matches the free energy fluctuations of the critical Sherrington--Kirkpatrick model and suggests this scaling is universal. The proof shows that the perceptron model's free energy fluctuations are explained, up to an additive $o(1)$ error with high probability, by those of a corresponding spherical Hopfield model, and that this relation persists even at criticality. This proof strategy builds on a critical reweighted moment method developed by the first and third authors \\cite{du2026fluctuations} to identify the critical fluctuations of the SK free energy, which links these fluctuations to the spectral edge spacing of the disorder.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31736",
  "author_total": 4,
  "authors": [
    {
      "name": "Hang Du"
    },
    {
      "name": "Shuyang Gong"
    },
    {
      "name": "Brice Huang"
    },
    {
      "name": "Zhangsong Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8x6eypp8d8098.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31736",
  "fulltext_url": null,
  "id": "8x6eypp8d8098",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31736",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
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  "openalex": "W7223689190",
  "pdf_url": "https://arxiv.org/pdf/2610.31736",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8x6eypp8d8098.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.31736",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31736"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Free energy fluctuations of symmetric perceptron models up to and including criticality",
  "title_text": "Free energy fluctuations of symmetric perceptron models up to and including criticality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/8x6eypp8d8098",
  "venue": "arXiv",
  "versions": []
}