{
  "abstract": "We revisit the statistical mechanics of Transformer attention via modern Hopfield models. We present a free-energy formalism to characterize parameter regimes and critical thresholds for robust attention inference. The resulting free-energy landscape incorporates randomness in query states and attention weights. Attention then emerges as the equation of motion of the corresponding statistical-mechanical system (i.e., modern Hopfield model) through the standard free-energy minimization principle. This allows us to identify first-order phase transitions in attention between glassy and robust-inference phases. We further extend to attentions with stochastic queries and keys. Together, our results provide a rigorous statistical-mechanics framework for characterizing phase structure and inference reliability across a family of Transformer attention mechanisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17347",
  "author_total": 4,
  "authors": [
    {
      "name": "Jerry Yao-Chieh Hu"
    },
    {
      "name": "Yu-An Chen"
    },
    {
      "name": "Maojiang Su"
    },
    {
      "name": "Han Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4s7p6qr28cnj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17347",
  "fulltext_url": null,
  "id": "b4s7p6qr28cnj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17347",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7221788516",
  "pdf_url": "https://arxiv.org/pdf/2610.17347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4s7p6qr28cnj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.17347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "On the Statistical Mechanics of Attention",
  "title_text": "On the Statistical Mechanics of Attention",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/b4s7p6qr28cnj",
  "venue": "arXiv",
  "versions": []
}