{
  "abstract": "We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size $n$, vertices are added sequentially, with the $i$th vertex starting a new branch with a prescribed probability $f(n,i)$; otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity $tdt$ in the construction of the Brownian continuum random tree by one of intensity $f(t)dt$. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When $f(n,i)=(i/n)^β$, with $β>0$, after rescaling the graph distance by $n^{-β/(β+1)}$, the random tree converges to the compact aggregation tree with $f(t)=t^β$. This recovers convergence to the Brownian continuum random tree when $β=1$ (Aldous, 1991), as well as scaling limits of choice spanning trees for integer $β$ (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with $f(t)=\\log^γ(1+t)$ for every $γ>1$. Finally, we identify necessary conditions for compactness. Consequently, the threshold $γ>1$ in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13066",
  "author_total": 2,
  "authors": [
    {
      "name": "Serte Donderwinkel"
    },
    {
      "name": "Jeroen van Haastert"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/546kkndkrdcrj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13066",
  "fulltext_url": null,
  "id": "546kkndkrdcrj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13066",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212956052",
  "pdf_url": "https://arxiv.org/pdf/2609.13066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/546kkndkrdcrj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convergence and compactness of discrete aggregation trees",
  "title_text": "Convergence and compactness of discrete aggregation trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/546kkndkrdcrj",
  "venue": "arXiv",
  "versions": []
}