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Existence of the thermodynamic limit for the Hopfield model — arXiv:2610.41938

Source

arXiv (en)

Original

https://arxiv.org/abs/2610.41938

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CC0-1.0

Archived

2026-11-03T08:17:43.958Z

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Existence of the thermodynamic limit for the Hopfield model — arXiv:2610.41938 — arXiv (en), https://arxiv.org/abs/2610.41938 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7e82faa4e151 (Metraipolis archive dc_7e82faa4e151, sha256 ff38f5573019efd2078a78583c1afcdee600a9fc3961cb09adac569767c9e0f1)

@misc{dc7e82faa4e151,
  title = {Existence of the thermodynamic limit for the Hopfield model — arXiv:2610.41938},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41938},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7e82faa4e151; sha256 ff38f5573019efd2078a78583c1afcdee600a9fc3961cb09adac569767c9e0f1},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41938. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove that the quenched pressure of the Hopfield model with $K\sim\alpha N$ i.i.d. symmetric Rademacher patterns has a thermodynamic limit at every load $\alpha\in(0,\infty)$, finite inverse temperature and uniform external field, with no replica-symmetry, order-parameter concentration or non-condensation hypothesis required. The limit is the supremum over $r$ of limiting pressures of mixed models with $r$ binary patterns in a Gaussian bulk. Via the Hubbard--Stratonovich representation we extract the finitely many patterns whose hidden coordinates exceed $\delta\sqrt N$; a collective interpolation replaces the other binary patterns by Gaussian ones with error $O(\delta^2)$ uniform in the volume, and convex concentration on the discrete cube controls the disorder-dependent choice of retained patterns. For the mixed models we prove a Gaussian hidden-spin truncation bound uniform in the volume and in an arbitrary Ising potential, and convergence for Gaussian bulks with finitely many site-field types, reducing to results of Ho and of Chen, Ho and Issa on non-convex vector spin glasses with compact spins. For Gaussian patterns we give an alternative proof of the existence result announced by Camilli, Contucci and Mingione, extended to a uniform field and an extra binary pattern. In both cases the pressure has exponentially small deviation probabilities and converges a.s.; internal energy and entropy converge wherever the limit is differentiable. An extra binary Mattis pattern admits a scalar variational reduction. At zero field the Rademacher pressure dominates the Gaussian one with an extra binary pattern, which dominates the pure Gaussian one. No explicit formula is given. The proofs were generated through extensive interaction with the AI models Claude (Opus~5.5) and ChatGPT (Sol~6.1), based on the problem formulation and guidance of the authors.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Disordered Systems and Neural Networks (cond-mat.dis-nn) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2610.41938 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T08:04:53.769Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).