{
  "schema_version": 1,
  "problem_number": "AMR-099-0065",
  "title": "On a Question of Benjamini on First-Passage Percolation in Hierarchical Graphs: The Recursive Constant Exceeds the Branching Random Walk Constant",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Question 9.1 of his survey \"Euclidean vs. Graph Metric\" (2013), Benjamini considers first-passage percolation on the hierarchical graphs G_n obtained by replacing, recursively, every edge by one edge followed by two parallel edges. With independent positive edge weights of law ξ, the source–sink distance satisfies D_n =d D_{n−1} + min(D′_{n−1}, D″_{n−1}). The question asks to compute γ_rec = lim 2^{−n} E D_n in terms of ξ, and in particular to show, if true, that γ_rec > γ_brw, the speed of the minimal position of the binary branching random walk with step law ξ, which is a lower bound for γ_rec. We give a partial answer. We prove γ_rec > γ_brw for every non-degenerate law ξ that is bounded below and has finite mean, hence for every non-degenerate law of positive weights with finite mean. The proof is a standard first-moment (union) bound taken one level up the hierarchy, combined with the inequality e^{−t min(x,y)} < e^{−tx} + e^{−ty}; at level zero the first-moment bound is γ_brw, by the classical theorem of Hammersley, Kingman and Biggins. We also show that γ_rec is the increasing limit of explicit first-moment bounds and the decreasing limit of 2^{−k} E D_k, that D_n/2^n → γ_rec in L¹, as anticipated in the source, and that 2^{−k} E D_k − γ_rec ≤ ((1+√2)/2) ‖ξ − ess inf ξ‖₂ 2^{−k/2}. Certified computer-assisted computations give γ_rec ∈ [0.268785599, 0.268808710] for weights uniform on [0,1], where γ_brw = 0.18482…, and γ_rec ∈ [0.379282228, 0.379412906] for exponential weights of mean one, where γ_brw = 0.23196…; in particular, the remark in the UnsolvedMath collection that the two constants coincide in the exponential case is incorrect. A closed formula for γ_rec in terms of ξ, the \"compute\" part of the question, remains open. The method is standard; no answer to the question was found in the literature accessible to us, and no priority is claimed. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.PR"
  ],
  "keywords": [
    "first-passage percolation",
    "hierarchical graphs",
    "series-parallel graphs",
    "recursive distributional equation",
    "branching random walk",
    "minimal displacement",
    "time constant",
    "first-moment method",
    "computer-assisted proof",
    "interval arithmetic",
    "partial answer",
    "Benjamini, Euclidean vs. Graph Metric",
    "UnsolvedMath",
    "AMR-099-0065",
    "math.PR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-099-0065/",
  "pdf_url": "https://eulersolve.org/papers/amr-099-0065/paper.pdf?v=466648b02a82",
  "doi": "10.5281/zenodo.23369628",
  "zenodo_record_url": "https://zenodo.org/records/23369628",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Question 9.1 of I. Benjamini's survey Euclidean vs. Graph Metric (2013): for first-passage percolation on the hierarchical graphs G_n, γ_rec > γ_brw is proved for every non-degenerate law of positive weights with finite mean; γ_rec is characterised as a limit of explicit first-moment bounds with a rate, and certified enclosures are given for uniform, exponential and two-point weights. A closed formula for γ_rec (the 'compute' part of the question) remains open. The method is standard; novelty is not certified.",
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    "source.zip": {
      "sha256": "61b30ee0431781f07ac6c46da52a828c71bec83de8b718296f08cdaf22db08ee"
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    "verification_report.md": {
      "sha256": "ace2a101f542de834a61a9c2c24bdea2e51f340685322e2f592ff2f0d5077f70"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}