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Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces

  • Kohki Sakamoto
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Abstract

We prove that the harmonic measure of simple random walk on the Poisson--Voronoi tessellation in ℍd is almost surely singular with respect to Lebesgue measure on the boundary. This answers a question of Benjamini, Paquette, and Pfeffer in the hyperbolic plane and establishes the analogous result in higher dimensions. Our proof is based on ergodic theory of rerooting relations and their boundary extensions.

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Cite

BibTeXRIS
BibTeX
@misc{sakamoto2026singularity,
  title = {{Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces}},
  author = {Kohki Sakamoto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40067},
  eprint = {2610.40067},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40067},
}
RIS
TY  - UNPB
TI  - Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
AU  - Kohki Sakamoto
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40067
UR  - https://arxiv.org/abs/2610.40067
AB  - We prove that the harmonic measure of simple random walk on the Poisson--Voronoi tessellation in ℍ^d is almost surely singular with respect to Lebesgue measure on the boundary. This answers a question of Benjamini, Paquette, and Pfeffer in the hyperbolic plane and establishes the analogous result in higher dimensions. Our proof is based on ergodic theory of rerooting relations and their boundary extensions.
ER  -