Poisson-Voronoi percolation in the hyperbolic plane with small intensities

Hansen, Benjamin T.; Müller, Tobias

doi:10.48550/arxiv.2111.04299

Cited by 2 publications

(3 citation statements)

References 38 publications

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“…which explains (5) in the case when Systole(S g ) → ∞. Note that recently percolation on hyperbolic Poisson-Voronoi tessellation with small intensity has been studied in [13]. Underneath the convergence of the above display as the intensity tends to 0 lies the fact that Vor λ (H) converges (in distribution for the Hausdorff topology on compact sets of H) towards a limiting object that we name the pointless Poisson-Voronoi tessellation of the hyperbolic disk (see Figure 1) and whose construction and properties will be studied in a forthcoming work.…”

Section: The Pointless Voronoi Tessellation Of Hmentioning

confidence: 79%

“…In that situation, the neighborhood of each point in S g looks like a piece of the hyperbolic plane and the Voronoi tessellation Vor λ (S g ) converges in distribution (in the local Hausdorff sense) towards the Voronoi tessellation Vor λ (H) of the hyperbolic plane with intensity λ, see Figure 1. This classical object has been studied in stochastic geometry [16,10] and in particular in relation to its percolation properties [3,13,14]. In particular, Isokawa [16] computed the mean characteristics of a typical 1 cell C λ in Vor λ (H) : this cell is an almost surely finite convex hyperbolic polygon satisfying…”

Section: The Pointless Voronoi Tessellation Of Hmentioning

confidence: 99%

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On Cheeger constants of hyperbolic surfaces

Budzinski¹,

Curien²,

Petri³

2022

Preprint

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It is a well-known result due to Bollobas that the maximal Cheeger constant of large d-regular graphs cannot be close to the Cheeger constant of the d-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by 2/π ≈ 0.63... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson-Voronoi tessellation of the surface with a vanishing intensity. Figure 1: From left to right: Poisson-Voronoi tessellations of the hyperbolic plane with decreasing intensity. Their limit (on the right) is the pointless Voronoi tessellation of the hyperbolic plane whose cells have been colored in black/white uniformly at random. This object has an average "linear" density equal to 2 × 2 π per unit of area.

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Section: The Pointless Voronoi Tessellation Of Hmentioning

confidence: 79%

Section: The Pointless Voronoi Tessellation Of Hmentioning

confidence: 99%

On Cheeger constants of hyperbolic surfaces

Budzinski¹,

Curien²,

Petri³

2022

Preprint

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“…The aim of this branch of stochastic geometry is to distinguish those properties of a random geometric system which are universal to some extent from the ones which are sensitive to the underlying geometry, especially to the curvature of the underlying space. We mention by way of example the studies [6,7,20] on random convex hulls, the papers [5,21,22,26,27,29,30,32] on random tessellations as well as the works [4,8,15,16,17,18,40] on geometric random graphs and networks. The present paper continues this line of research and naturally connects to the articles [26,32].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Intersections of Poisson $ k $-flats in constant curvature spaces

Betken¹,

Hug²,

Thäle³

2023

Preprint

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Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a ddimensional standard space of constant curvature κ ∈ {−1, 0, 1} are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d − m(d − k))-dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d − m(d − k) ≥ 0. For κ ∈ {−1, 0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k ≤ d + 1 a central limit theorem is shown for all possible values of m. However, while for κ = 0 asymptotic normality still holds if 2k > d + 1, we prove for κ = −1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m = 1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin-Stein method. We also show for general κ ∈ {−1, 0, 1} that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W . Along the way we derive a new integral-geometric formula of Blaschke-Petkantschin type in a standard space of constant curvature.

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Poisson-Voronoi percolation in the hyperbolic plane with small intensities

Cited by 2 publications

References 38 publications

On Cheeger constants of hyperbolic surfaces

On Cheeger constants of hyperbolic surfaces

Intersections of Poisson $ k $-flats in constant curvature spaces

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