{
  "abstract": "Let $(Z_n)_{n\\geq 0}$ be a supercritical branching process in a random environment in which every individual produces at least one offspring, and let $W$ be the almost sure limit of the normalized $Z_n$. Write $p_1$ for the (random) probability that an individual has exactly one child and $m_0$ for the mean offspring number in a given environment. Assume that $p_1$ is positive almost surely and bounded away from $1$, that $m_0$ is bounded away from $1$ and from infinity together with a mild moment condition, and that $\\log m_0$ is non-lattice. Let $a_0>0$ be the unique solution of the equation $\\mathbb{E}[p_1 m_0^{a_0}]=1$. Under these assumptions, we determine the precise left tail asymptotics of the annealed law of $W$: the distribution function $F(x)=\\mathbb{P}(W\\leq x)$ satisfies $F(x)\\sim D x^{a_0}$ as $x\\downarrow 0$ with a constant $D\\in(0,\\infty)$ admitting an explicit representation, and the annealed density $w$ of $W$, which we show exists and can be chosen continuous on $(0,\\infty)$, satisfies $w(x)\\sim G x^{a_0-1}$ as $x\\downarrow 0$ with $G=a_0 D>0$. These results extend the classical power-law left tail behavior of the martingale limit of a Galton--Watson process in the literature to the random environment setting, where the exponent $a_0$ is determined by the averaged environmental law rather than by a fixed offspring distribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25848",
  "author_total": 2,
  "authors": [
    {
      "name": "Panyi Chen"
    },
    {
      "name": "Mei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vet7tye3gwmg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25848",
  "fulltext_url": null,
  "id": "2vet7tye3gwmg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25848",
  "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": "W7223026273",
  "pdf_url": "https://arxiv.org/pdf/2610.25848",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vet7tye3gwmg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25848",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25848"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Left tail asymptotics for the density of supercritical branching processes in random environment",
  "title_text": "Left tail asymptotics for the density of supercritical branching processes in random environment",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2vet7tye3gwmg",
  "venue": "arXiv",
  "versions": []
}