{
  "abstract": "We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with $-\\logλ_c$ locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit $tλ_c(t)\\toΔ\\log(1/(1-p_c(G)))$, where $Δ$ is the degree and $p_c(G)$ is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29699",
  "author_total": 2,
  "authors": [
    {
      "name": "Christian Mönch"
    },
    {
      "name": "Andreas Klippel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vc848385vfp4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29699",
  "fulltext_url": null,
  "id": "1vc848385vfp4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29699",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214370611",
  "pdf_url": "https://arxiv.org/pdf/2609.29699",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vc848385vfp4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.29699",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29699"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons",
  "title_text": "Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1vc848385vfp4",
  "venue": "arXiv",
  "versions": []
}