{
  "abstract": "We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load $α$ and temperature $T$, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto $N$ Ising spins coupled to $M=αN$ Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies $T_g=1+\\sqrtα$ as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below $T_g$ we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32618",
  "author_total": 3,
  "authors": [
    {
      "name": "Francesco Camilli"
    },
    {
      "name": "Pierluigi Contucci"
    },
    {
      "name": "Emanuele Mingione"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc5hwtq8zjg9p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32618",
  "fulltext_url": null,
  "id": "cc5hwtq8zjg9p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32618",
  "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",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7223686957",
  "pdf_url": "https://arxiv.org/pdf/2610.32618",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc5hwtq8zjg9p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.32618",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32618"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The solution of the Hopfield model with Gaussian patterns",
  "title_text": "The solution of the Hopfield model with Gaussian patterns",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/cc5hwtq8zjg9p",
  "venue": "arXiv",
  "versions": []
}