{
  "abstract": "Billingsley's theorem states that the normalized logarithms of the prime factors of a uniform random integer in $[1,x]$ converge to the Poisson--Dirichlet law $PD(1)$, while weighting an integer $n$ by the generalized divisor function $d_θ(n)$ gives $PD(θ)$. We ask what remains of this picture when the integer is also rewarded according to its largest prime factor $P^+(n)$. For fixed $θ,β>0$ and $γ\\ge0$, we sample $N_x=n\\le x$ with probability proportional to $d_θ(n)\\exp\\{βH_γ(n)\\}$, where $H_γ(n)=(\\log P^+(n))^γ$ for $γ>0$, while $H_0(1)=0$ and $H_0(n)=1$ for $n\\ge2$. Two phase transitions occur. At $γ=0$ the $PD(θ)$ partition survives, whereas every fixed $γ>0$ forces one prime to carry asymptotically all logarithmic mass. The second transition, at $γ=1$, concerns the cofactor $R_x=N_x/P^+(N_x)$. With $a_x=βγ(\\log x)^{γ-1}$, its law is asymptotic in total variation to $Q_{a_x}(m)=d_θ(m)m^{-1-a_x}/ζ(1+a_x)^θ$. Thus, for $0<γ<1$, $a_x\\log R_x$ has a $Gamma(θ,1)$ limit, with shape $θ$ and rate one, and the normalized logarithmic prime factors of $R_x$ have an independent $PD(θ)$ limit; at $γ=1$, $R_x$ has a nondegenerate discrete limit; and for $γ>1$, $R_x=1$ with high probability. We also determine the joint limits involving $N_x/x$ and the normalizing constants, which include the classical Alladi--Erdős asymptotic as a special case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11735",
  "author_total": 1,
  "authors": [
    {
      "name": "Wen Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6avw14b3d8ccc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11735",
  "fulltext_url": null,
  "id": "6avw14b3d8ccc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11735",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212230373",
  "pdf_url": "https://arxiv.org/pdf/2609.11735",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6avw14b3d8ccc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11735",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11735"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Tilting Billingsley's model toward a giant prime: two phase transitions",
  "title_text": "Tilting Billingsley's model toward a giant prime: two phase transitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/6avw14b3d8ccc",
  "venue": "arXiv",
  "versions": []
}