Skip to content
DraconiticResearch
Search Sources API
197,693 studies31 sourcesUpdated 43 min ago
PreprintCC BY 4.0arXiv

Lengths and incidences in Poisson hypersphere and spherical splitting tessellations

  • Daniel Hug
  • Christoph Thäle
Open at sourcePDFDOICite

Abstract

We investigate distributional properties of two random tessellation models on the d-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For d=2, we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.

Related

  1. Radial reduction, spherical splitting and root counts for two-sided quaternionic polynomials
    R. Laniewski
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. A combinatorial approach to point-flat incidences over finite fields
    Ben Lund, Tao Zhang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. Discrepancy of geometric incidences
    Azem Adibelli, István Tomon
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsGeometry & topologyarXiv
  4. Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
    Kohki Sakamoto
    arXivPreprintarXiv licenceFull textDynamical systemsProbabilityarXiv
  5. Hyperplane Incidences and Distance Sets in Higher Dimensions
    Marianna Csornyei, D. M. Stull
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{hug2026lengths,
  title = {{Lengths and incidences in Poisson hypersphere and spherical splitting tessellations}},
  author = {Daniel Hug and Christoph Thäle},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07725},
  eprint = {2609.07725},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07725},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lengths and incidences in Poisson hypersphere and spherical splitting tessellations
AU  - Daniel Hug
AU  - Christoph Thäle
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07725
UR  - https://arxiv.org/abs/2609.07725
AB  - We investigate distributional properties of two random tessellation models on the d-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For d=2, we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.
ER  -