{
  "abstract": "Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\\sqrt{{\\log(β+1)}/β}\\exp(Ct)+\\exp(-ct)$ in terms of the temperature parameter $β^{-1}\\to 0$ and inference time $t\\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\\to\\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\\logβ$, which correspond to the early-phase behavior most relevant for practical finite-depth models, the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $β$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.10931",
  "author_total": 4,
  "authors": [
    {
      "name": "Albert Alcalde"
    },
    {
      "name": "Leon Bungert"
    },
    {
      "name": "Konstantin Riedl"
    },
    {
      "name": "Tim Roith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9me477nbpj3ae.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.10931",
  "fulltext_url": null,
  "id": "9me477nbpj3ae",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.10931",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7160930630",
  "pdf_url": "https://arxiv.org/pdf/2605.10931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9me477nbpj3ae.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:19Z",
      "native_id": "2605.10931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.10931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime",
  "title_text": "Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/9me477nbpj3ae",
  "venue": "arXiv",
  "versions": []
}