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Poisson-Voronoi percolation in the hyperbolic plane with small intensities

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Benjamin T. Hansen, Tobias Müller

Probability · Submitted 8 Nov 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm.

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arXiv ID2111.04299 (version 3)
Pith ID57IKTQA5, Pith's permanent record of this paper and its review
CitationBenjamin T. Hansen, Tobias M\"uller. "Poisson-Voronoi percolation in the hyperbolic plane with small intensities." arXiv:2111.04299 (2021). https://pith.science/paper/57IKTQA5
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,327 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.02 · states quantitative result 0.96 · compares with prior results 0.27 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. background 0.50 0.38 0.03 unclear 0.54
s1 We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm. result 1.00 0.95 0.95 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • references and citing sentences
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  • P abstract
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm.

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4 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 3 papers
  • This classical object has been studied in stochastic geometry [ref] and in particular in relation to its percolation properties [ref] .

    On Cheeger constants of hyperbolic surfaces · The pointless Voronoi tessellation of $ℍ$ · 1.00

  • We mention by way of example the studies [ref] on random convex hulls, the papers [ref] on random tessellations as well as the works [ref] on geometric random graphs and networks.

    Intersections of Poisson $ k $-flats in constant curvature spaces · Introduction and statement of the results · 1.00

  • The recent contributions [ref] on Poisson–Voronoi percolation in the hyperbolic plane explore the behavior of the critical probability for an infinite cluster as the intensity of the underlying Poisson process approaches infinity or zero.

    Boolean models in hyperbolic space · Introduction · 0.96

What it cites · 1 sentence without a confident label

Cites as background 2 papers
  • Works on continuum percolation models over Poisson processes in the hyperbolic plane include [ref] .

    Hyperbolic Radial Spanning Tree · Introduction and statement of main result · 1.00

  • Aspects of hyperbolic Poisson-Voronoi tessellations besides percolation that have been studied include the (expected) combinatorial structure of their cells, random walks on them and anchored expansions – see for example [ref] .

    Beta-star polytopes and hyperbolic stochastic geometry · Introduction and statement of main result · 1.00

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