{
  "abstract": "We study heterogeneous bootstrap percolation on uniform simple graphs with prescribed degrees of mean $d\\to\\infty$ and deviations at most $C\\sqrt d$. Thresholds are independently one with probability $(1-c/d)/d$ and two otherwise; seeds form an independent uniform set of prescribed size $a$. Let $T_a$ be the terminal number of active vertices and $b$ be one minus the mean number of threshold-one descendants of a reached vertex. Let $K\\sim d^2$ be the quadratic response coefficient. If $b>0$ and $nb^3/K\\to\\infty$, we prove, uniformly over seed counts $a$, $ \\mathbb{P}(T_a\\ge n/2)=Φ(\\frac{a-nα_{\\mathrm{det}}}{w})+o(1)$ with $w=\\sqrt{nb/K}$, where $α_{\\mathrm{det}}$ is the exact empirical response-barrier center and $Φ$ is the standard normal distribution function. Put $a_0=(n/K)^{1/3}$. If $d^4/n\\to0$, $a/a_0\\to x>0$ and $λ=ba_0\\to\\ell\\in\\mathbb{R}$, then $\\mathbb{P}(T_a\\ge n/2)\\to p(x,\\ell)$, the one-sided Brownian parabolic survival probability. A sequential absolute-error law joins these regimes. A graphical counterexample shows that replacing the exact center by its leading term can cause limiting probability error $1/2$. The proof balances adaptive early-exploration transfer against late concentration at a tunable physical horizon, despite exponentially rare simplicity. We also prove quantitative and mesoscopic-seed refinements in explicitly narrower regimes. The only external graph-enumeration input is a stated reduction of McKay--Wormald's estimate; the probabilistic estimates for the cascade argument are proved in the text and appendices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38912",
  "author_total": 1,
  "authors": [
    {
      "name": "Aleksandr Rodionov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7n4q98tsxhgy8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38912",
  "fulltext_url": null,
  "id": "7n4q98tsxhgy8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38912",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7n4q98tsxhgy8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Finite-size ignition laws near a vanishing branching deficit on near-regular simple graphs",
  "title_text": "Finite-size ignition laws near a vanishing branching deficit on near-regular simple graphs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7n4q98tsxhgy8",
  "venue": "arXiv",
  "versions": []
}