{
  "schema_version": 1,
  "problem_number": "OWR-14298808-012",
  "title": "The Critical Window for High-Dimensional Limits of Hyperbolic Poisson k-Plane Processes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For 2k > d + 1, the centred and normalized total k-volume of a stationary Poisson process of k-planes in hyperbolic space H^d, observed in a growing ball, converges to a non-Gaussian infinitely divisible law Z_{d,k}. Bühler and Hug asked how the standardized law Z*_{d,k} = Z_{d,k}/(Var Z_{d,k})^{1/2} behaves as d → ∞. Bühler, Hug and Thäle proved that, when k/d → 1/2, Z*_{d,k} tends to the standard Gaussian law if d^{−1}(2k − d − 1)^{d/k} stays asymptotically below eπ and to 0 if it stays above eπ, and they left the critical rate open. We complete the picture. Put m = 2k − d − 1, m_c = (2πe(k − 1))^{1/2} and y = (m − m_c)/m_c^{1/2}. Along every sequence of admissible pairs with d → ∞, the Lévy distance between Z*_{d,k} and the centred Gaussian law with variance Φ(−y/√2) tends to 0, where Φ is the standard normal distribution function. Hence Z*_{d,k} converges in distribution if and only if y converges in [−∞, ∞], every limit law is a centred, possibly degenerate, Gaussian N(0, τ²) with τ² ∈ [0, 1] (where N(0, 0) = δ₀), and every τ² ∈ [0, 1] occurs. At the critical rate k = d/2 + ½(eπd)^{1/2} + O(1) the limit is N(0, 1/2); a shift by c·d^{1/4} gives N(0, Φ(−√2·c/(eπ)^{1/4})). The proof combines an exact Beta representation of the Kolmogorov measure, a reduction lemma for Kolmogorov measures that split between 0 and ∞, an anti-concentration bound, and a central limit theorem for the logarithm of a Beta variable. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "keywords": [
    "hyperbolic stochastic geometry",
    "Poisson k-plane process",
    "infinitely divisible distribution",
    "high-dimensional limit",
    "Gaussian limit",
    "Beta distribution",
    "Oberwolfach Reports",
    "OWR-14298808-012",
    "math.PR",
    "math.MG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14298808-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-14298808-012/paper.pdf?v=3bc81f6385ec",
  "doi": "10.5281/zenodo.23041944",
  "zenodo_record_url": "https://zenodo.org/records/23041944",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Completes the Bühler–Hug–Thäle theorem (ECP 31 (2026)), whose Gaussian/degenerate dichotomy away from the critical scale is credited; the new content is the critical window. Rates of convergence are not studied. Unrefereed.",
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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."
}