Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

Hilhorst, H. J.; Calka, Pierre

doi:10.1007/s10955-008-9577-0

Cited by 16 publications

(21 citation statements)

References 33 publications

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“…We will select β 1 as this special degree of freedom and express the remaining β m and γ m as where ρ 0 ≡ ρ n . This ratio (2.16) is the same as the one encountered in the Voronoi and Crofton cell problems [6,8], except for an interchange of β m and γ m . It follows that one may express the ρ m exclusively in terms of the angles by solving them from (2.16) together with (2.6).…”

Section: New Angular Variablesmentioning

confidence: 88%

“…Then, because of the δx m , the second term will become a sum of independent zero-average random variables and hence its contribution will be of relative order n −δ/2 with respect to that coming from the first term. We may therefore neglect the δx m u m term in (3.7) when our purpose is to study the variation of r m on a scale that increases as a power of n. Taking now second order differences we find the recursion r m−1 − 2r m + r m+1 = 2πn 8) with the right hand member defined by…”

Section: Second Order Recursionmentioning

confidence: 99%

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Sylvester’s question and the random acceleration process

2008

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Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(φ) of the polar angle φ, satisfies the equation of the random acceleration process (RAP), d2 r/dφ 2 = f (φ), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p n = −2n log n + n log(2π 2 e 2 ) − c 0 n 1/5 + . . ., of which the first two terms agree with a rigorous result due to Bárány. The nonanalyticity in n of the third term is a new result. The value 1 5 of the exponent follows from recent work on the RAP due to Györgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width ∼ n −4/5 along the edge of the disk. The distance δ n of closest approach to the edge is exponentially distributed with average (2n) −1 .

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Section: New Angular Variablesmentioning

confidence: 88%

Section: Second Order Recursionmentioning

confidence: 99%

Sylvester’s question and the random acceleration process

2008

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“…We now proceed to characterize f (A v ), for which we define the following terms, before stating our main result in Theorem 1. Although the result presented in Theorem 1 exists in the literature on random polygon tessellations (see [4]), it has, to the best of our knowledge, never been applied to a localization setting previously. Since its proof is quite straightforward, we include it for completeness.…”

Section: Asymptotic Blind-spot Probability Letmentioning

confidence: 99%

Asymptotic Blind-Spot Analysis of Localization Networks Under Correlated Blocking Using a Poisson Line Process

Aditya

Dhillon

Molisch

et al. 2017

IEEE Wireless Commun. Lett.

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Abstract-In a localization network, the line-of-sight between anchors (transceivers) and targets may be blocked due to the presence of obstacles in the environment. Due to the non-zero size of the obstacles, the blocking is typically correlated across both anchor and target locations, with the extent of correlation increasing with obstacle size. If a target does not have line-ofsight to a minimum number of anchors, then its position cannot be estimated unambiguously and is, therefore, said to be in a blind-spot. However, the analysis of the blind-spot probability of a given target is challenging due to the inherent randomness in the obstacle locations and sizes. In this letter, we develop a new framework to analyze the worst-case impact of correlated blocking on the blind-spot probability of a typical target; in particular, we model the obstacles by a Poisson line process and the anchor locations by a Poisson point process. For this setup, we define the notion of the asymptotic blind-spot probability of the typical target and derive a closed-form expression for it as a function of the area distribution of a typical Poisson-Voronoi cell. As an upper bound for the more realistic case when obstacles have finite dimensions, the asymptotic blind-spot probability is useful as a design tool to ensure that the blind-spot probability of a typical target does not exceed a desired threshold, ǫ.

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“…Again, this probability distribution is unknown and one might hope to find its asymptotic large-n behavior by the methods of Refs. [3,6,8]. However, we do not know how to solve that problem.…”

Section: The Edgedness Of a Face Between 3d Cellsmentioning

confidence: 99%

Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation

Hilhorst¹

2016

J. Stat. Mech.

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We consider the 3D Poisson-Voronoi tessellation. We investigate the joint probability distribution π n (L) for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of its adjacent cells to be equal to 2L. We derive an exact expression for this quantity, valid in the limit n → ∞ with n 1/6 L fixed. The leading order correction term is determined. Good agreement with earlier Monte Carlo data is obtained. The cell face is surrounded by a three-dimensional excluded domain that is the union of n balls; it is pumpkin-shaped and analogous to the flower of the 2D Voronoi cell. For n → ∞ this domain tends towards a torus of equal major and minor radii. The radii scale as n 1/3 , in agreement with earlier heuristic work. We achieve a detailed understanding of several other statistical properties of the n-edged cell face.

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Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

Cited by 16 publications

References 33 publications

Sylvester’s question and the random acceleration process

Sylvester’s question and the random acceleration process

Asymptotic Blind-Spot Analysis of Localization Networks Under Correlated Blocking Using a Poisson Line Process

Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation

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