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  "slug": "dyck-transpositions-n-log-n",
  "title": "Random transpositions on Dyck paths mix in order n log n",
  "shortTitle": "Shuffling a constrained path",
  "url": "https://evidencepress.org/releases/dyck-transpositions-n-log-n/",
  "oneLine": "A two-cut entropy argument gives a proof candidate for n log n mixing of the original Dyck-path random-transposition rejection walk.",
  "abstract": "For Dyck words of length 2n, choose two coordinates independently and uniformly, swap their letters if the result remains Dyck, and otherwise hold. This unrefereed analytic proof candidate gives worst-start total-variation mixing time Theta(n log n) in that original proposal clock. A two-cut matroid conditional-entropy bound, uniform height-channel classical log-Sobolev estimate and discrete hypercontractive composition lead to entropy factorization. A constant-factor retained-fibre comparison returns to the original chain. A midpoint drift argument gives the lower bound (n/2) log n - O(n). Finite exact audits corroborate identities, not the all-size theorem. No cutoff, sharp upper constant, sampler optimality, external validation or priority clearance is claimed.",
  "datePublished": "2026-09-06",
  "dateModified": "2026-09-06",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22556925",
  "doiUrl": "https://doi.org/10.5281/zenodo.22556925",
  "conceptDoi": "10.5281/zenodo.22556924",
  "pdfUrl": "https://github.com/ipitchford/dyck-transpositions-n-log-n/releases/download/v0.1.0-candidate/dyck-transpositions-n-log-n-0.1.0-candidate.pdf",
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  "authors": [
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    "detail": "Unrefereed analytic proof candidate for the original length-2n ordered independent-coordinate Dyck rejection walk. No cutoff, sharp upper constant, numerical burn-in guarantee, computational-runtime bound, sampler optimality, external peer review, formal verification or priority clearance."
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    },
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      "note": "Standard-library Python replay and source supplied; no hermetic environment or cross-platform PDF byte-identity claim."
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  "problem": {
    "name": "AIM-PROBABILITY-0039: Random transpositions on Dyck paths",
    "url": "https://www.unsolvedmath.com"
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  "keywords": [
    "Dyck paths",
    "mixing time",
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    "Analytic proof candidate for Theta(n log n) worst-start mixing in the original ordered-coordinate rejection clock.",
    "An absolute one-step entropy contraction of order 1/n.",
    "Lower bound (n/2) log n - O(n); no sharp upper constant or cutoff claim.",
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  "openProblems": [
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    "Derive an explicit upper constant and investigate sharp asymptotics or cutoff as new work.",
    "Assess extensions without transferring the result to different Catalan kernels by analogy alone."
  ],
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  "relatedWorks": [
    {
      "citation": "Cohen, Tetali and Yeliussizov (2015). Lattice Path Matroids: Negative Correlation and Fast Mixing.",
      "url": "https://arxiv.org/abs/1505.06710",
      "doi": "10.48550/arXiv.1505.06710"
    },
    {
      "citation": "Cryan, Guo and Mousa (2021). Modified log-Sobolev inequalities for strongly log-concave distributions. Deletion entropy input.",
      "url": "https://arxiv.org/abs/1903.06081v3",
      "doi": "10.48550/arXiv.1903.06081"
    },
    {
      "citation": "Barthe and Roberto (2003). Sobolev inequalities for probability measures on the real line. Hardy framework.",
      "url": "https://doi.org/10.4064/sm159-3-9",
      "doi": "10.4064/sm159-3-9"
    },
    {
      "citation": "Carlen and Cordero-Erausquin (2009). Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities.",
      "url": "https://doi.org/10.1007/s00039-009-0001-y",
      "doi": "10.1007/s00039-009-0001-y"
    },
    {
      "citation": "Anari et al. (2021). Log-concave polynomials IV. General down-up mixing for a different kernel.",
      "url": "https://arxiv.org/abs/2004.07220v2",
      "doi": "10.48550/arXiv.2004.07220"
    },
    {
      "citation": "Feng and Paguyo (2026), version 2. Analysis of a twisted Bose-Einstein Markov chain with applications to sampling Catalan structures. Different sampler.",
      "url": "https://arxiv.org/abs/2605.16244v2",
      "doi": "10.48550/arXiv.2605.16244"
    }
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