{
  "schema_version": 1,
  "problem_number": "OWR-12872-018",
  "title": "On the Albert–Bousquet-Mélou Conjecture for Corners in Quadrant Loops: The Case of Nonnegative Corner Weight",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a real number a let Q_n(a) be the sum of a^c(w) over the walks w of length 2n with steps N, E, S, W that start and end at the origin and stay in the quadrant {x ≥ 0, y ≥ 0}, where c(w) is the number of factors NW and ES of w, and let L_n(a) be the same sum without the quadrant condition. Albert and Bousquet-Mélou conjectured, and Bousquet-Mélou asked in the problem session of Oberwolfach Report 12/2014, that for every a ≥ −1 the series Σ_n Q_n(a)u^n has the same radius of convergence as Σ_n L_n(a)u^n, whose radius they determined. They proved this for a = 1 and a = −1, and for a ≥ 0 they proved that the radius for quadrant loops is at least (2 + √(2+2a))^(−2). We prove the conjecture for every a ≥ 0: the limits of Q_n(a)^(1/n) and of L_n(a)^(1/n) exist and both are equal to (2 + √(2+2a))², and Q_n(a) ≤ (2 + √(2+2a))^(2n) for all n. The proof is elementary and self-contained. It combines a formula for corner-weighted walks under an exponential tilt, a large-deviation bound for the loops that go far to the left of or below the origin, a padding without corners that moves the other loops into the quadrant, and Fekete's lemma; all these tools are standard. The question remains open for −1 < a < 0, and this range contains the values in [−1/3, 0) on which the conjectural growth rate of permutations sortable by two stacks in parallel depends. We show where the argument needs a ≥ 0 and report numerical evidence, not a proof, for the open range. No proof of the case a ≥ 0 was found in the sources we searched; for a ≥ 1 the result seems to follow, after a short argument, from known theorems on weighted walks in cones. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "keywords": [
    "quadrant loops",
    "quarter-plane walks",
    "corners",
    "NW and ES factors",
    "exponential growth rate",
    "radius of convergence",
    "permutations sortable by two stacks in parallel",
    "Albert–Bousquet-Mélou conjecture",
    "exponential tilting",
    "Fekete's lemma",
    "case a ≥ 0 (scoped result)",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-12872-018",
    "math.CO",
    "math.PR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12872-018/",
  "pdf_url": "https://eulersolve.org/papers/owr-12872-018/paper.pdf?v=e3fd08bd8b0a",
  "doi": "10.5281/zenodo.23489441",
  "zenodo_record_url": "https://zenodo.org/records/23489441",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scoped result: the conjecture is proved for a >= 0 only (with the known case a = -1); the open range between -1 and 0 includes the values in [-1/3, 0) needed for two-stack sorting.",
  "files": {
    "paper.pdf": {
      "sha256": "e3fd08bd8b0a7972a123cac3390572bb8cdd32a11cafb7e391f84c6901390f5e"
    },
    "source.zip": {
      "sha256": "3931dfe70dbea5684518f39c82df5a8b572010a50b90e201df7866c3744742c4"
    },
    "verification_report.md": {
      "sha256": "9d50ce47f30f49b2ef30a0d71d89a4105a23f3eb6b542586f91781338ceffc8a"
    }
  },
  "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."
}