Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces

Abstract

Ideal Poisson--Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson--Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson--Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson--Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson--Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg--Poisson--Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.

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BibTeX RIS

BibTeX

@misc{davalo2026ideal,
  title = {{Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces}},
  author = {Colin Davalo and J. Maxwell Riestenberg and Anna Wienhard},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17045},
  eprint = {2610.17045},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17045},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces
AU  - Colin Davalo
AU  - J. Maxwell Riestenberg
AU  - Anna Wienhard
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17045
UR  - https://arxiv.org/abs/2610.17045
AB  - Ideal Poisson--Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson--Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson--Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson--Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson--Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg--Poisson--Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.
ER  -

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