{
  "abstract": "We study causal self-attention dynamics -- a toy model for decoder Transformers -- which we interpret as a non-exchangeable interacting particle system. Adapting cumulant expansions to the triangular causal dependency structure of the model, and appealing to non-hierarchical methods to estimate correlations using Glauber calculus, we prove a quantitative mean-field limit result and a next-order characterization of correlations. For iid uniformly distributed tokens, the limiting correlation equation can be solved in closed form and we obtain a rigorous explanation of the empirically observed \\emph{lost-in-the-middle} phenomenon: the token retrieval profile, as a function of the source position in the prompt, is $\\mathsf{U}$-shaped, with primacy, recency, and a unique interior minimum under an explicit smallness condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.09213",
  "author_total": 3,
  "authors": [
    {
      "name": "Mitia Duerinckx"
    },
    {
      "name": "Borjan Geshkovski"
    },
    {
      "name": "Stefano Rossi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b405ebe49v4de.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.09213",
  "fulltext_url": null,
  "id": "b405ebe49v4de",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.09213",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7160891242",
  "pdf_url": "https://arxiv.org/pdf/2605.09213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b405ebe49v4de.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2605.09213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.09213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Kinetic theory for Transformers and the lost-in-the-middle phenomenon",
  "title_text": "Kinetic theory for Transformers and the lost-in-the-middle phenomenon",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/b405ebe49v4de",
  "venue": "arXiv",
  "versions": []
}