{
  "abstract": "Motivated by the continuous-time critical beta-splitting tree, we investigate chronological trees recording the successive block splits and their times in homogeneous fragmentation processes restricted to $n$ labels. The separation time of a subset is the first time its labels cease to lie in a common block. For each fixed integer $q\\geq2$, we study extremal separation times of $q$-element subsets via their associated point process. Under mild conditions, as $n\\to\\infty$, the last such time has a randomly shifted Gumbel limit after deterministic centering. Above a threshold $θ_*$, the coefficients of both the leading $\\log n$ term and the $\\log\\log n$ correction in the centering become independent of $q$, as does the limiting law up to deterministic translation. The extremal process converges to a randomly shifted Poisson point process, without and with decorations, for $q \\le θ_*$ and $q>θ_*$ respectively. Finally, in the same centered time window, the number of size-$q$ blocks (clades) converges to a pure-death process for $q\\leqθ_*$, and to a process whose trajectories are non-monotone with positive probability for $q>θ_*$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14325",
  "author_total": 1,
  "authors": [
    {
      "name": "Heng Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ek531fkxvtwt0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14325",
  "fulltext_url": null,
  "id": "ek531fkxvtwt0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14325",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213268872",
  "pdf_url": "https://arxiv.org/pdf/2609.14325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ek531fkxvtwt0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.14325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition",
  "title_text": "Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/ek531fkxvtwt0",
  "venue": "arXiv",
  "versions": []
}