{
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  "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 e^(−i/n). We prove that for every fixed K, uniformly in 2 ≤ i ≤ K n^(3/2), E N_i = exp(−i/n − i²/(2n³))(1 + O(n^(−1/2) log n)), and that E N_i e^(i/n) → 0 when i/n^(3/2) → ∞. Consequently E N_i ∼ e^(−i/n) holds if and only if i = o(n^(3/2)). This range contains fixed i, the scale i ∼ xn 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 ∼ y n^(3/2) the ratio E N_i/e^(−i/n) tends to e^(−y²/2), so the conjecture read literally for all i is false. The proof is elementary. It combines the construction of a uniform Eulerian circuit from a uniform spanning tree (the BEST theorem, in the form used by Kandel, Matias, Unger and Winkler) with a hazard representation of the first excursion and a convexity bound. Exact computations for n ≤ 6 confirm the hazard representation and the closed forms for E N_2 and E N_3, and Monte Carlo simulations up to n = 6400 are consistent with the asymptotic results. This is an unrefereed note.",
  "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",
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  "date_modified": "2026-09-30",
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  "canonical_url": "https://eulersolve.org/papers/amr-096-0015/",
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  "doi": "10.5281/zenodo.23051216",
  "zenodo_record_url": "https://zenodo.org/records/23051216",
  "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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  "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.",
  "concept_doi": "10.5281/zenodo.23049689",
  "concept_url": "https://doi.org/10.5281/zenodo.23049689",
  "revision_published_at": "2026-09-30T04:22:18.175641+00:00",
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  "revision_note": "A numerical remark in Section 7, which the proofs do not use, is corrected; the text now states that all circuits were enumerated only for n ≤ 5, with the n = 6 values taken from the exact formula of Remark 2.5; related work is cited, bibliographic data are corrected, and the verification record is updated. The theorems are unchanged.",
  "review_disclosure": "Internal AI-assisted checks only; unrefereed preprint, no independent human peer review claimed."
}