{
  "schema_version": 1,
  "problem_number": "AMR-096-0015",
  "title": "Excursion Lengths in a Uniform Eulerian Circuit of the Complete Graph: Aldous's Conjecture Holds if and only if i = o(n^(3/2))",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let N_i be the number of excursions of length i from a fixed vertex in a uniformly random Eulerian circuit of the complete graph K_n in which every edge is replaced by two opposite arcs. Aldous and Yu (Open Problems in Mathematics, 2014) stated as the natural conjecture that E N_i is asymptotic to exp(-i/n). We prove that for every fixed K, uniformly in 2 <= i <= K n^(3/2), E N_i = exp(-i/n - i^2/(2n^3))(1 + O(n^(-1/2) log n)), and that E N_i exp(i/n) tends to zero when i/n^(3/2) tends to infinity. Consequently E N_i is asymptotic to exp(-i/n) if and only if i = o(n^(3/2)). This range contains fixed i, the scale i proportional to n and essentially all excursions, and the length of a uniformly chosen excursion, divided by n, converges in distribution to the standard exponential law. At the scale i proportional to y n^(3/2) the ratio E N_i/exp(-i/n) tends to exp(-y^2/2), so the conjecture read literally for all i is false. The proof combines the uniform-spanning-tree construction of a uniform Eulerian circuit (the BEST theorem, in the form used by Kandel, Matias, Unger and Winkler), a hazard representation of the first excursion, and a convexity bound. Exact computations for n <= 6 and Monte Carlo simulations up to n = 6400 agree with the results.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "AMR-096-0015",
    "random Eulerian circuit",
    "uniform spanning tree",
    "BEST theorem",
    "excursion lengths",
    "complete graph",
    "exponential limit law",
    "Aldous-Yu conjecture",
    "math.PR",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-096-0015/",
  "pdf_url": "https://eulersolve.org/papers/amr-096-0015/paper.pdf?v=4480f96317ac",
  "doi": "10.5281/zenodo.23049690",
  "zenodo_record_url": "https://zenodo.org/records/23049690",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: This result answers example (a), the complete bidirected graph, of Aldous and Yu's random Eulerian circuits note, corresponding to corpus record AMR-096-0015. It proves the conjectured excursion-count asymptotics in the sharp range i = o(n^(3/2)) and disproves the literal all-i formulation. The torus conjecture, the Hamming-cube example, and the broader random-Eulerian-circuit programme are not solved here. Monte Carlo computations are evidence only and are not used as proof. No absolute priority claim is made. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
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      "sha256": "4480f96317ac81bff78d343759616f84cce69ef75e8e57602ec1f1fbda93da99"
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    "source.zip": {
      "sha256": "0ee2cd4afa31572b4cbdf419416b5ed88ae0140128d81bc64fe260b364d49921"
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    "verification_report.md": {
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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."
}