Coverage of Random Discs Driven by a Poisson Point Process

Lan, Guo-Lie; Ma, Zhiming; Sun, Su-Yong

doi:10.1007/978-1-4614-1966-2_4

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…The case k = 1 of the second equality of (2.7) can be found in [6]. It provides a stronger asymptotic result than the one in [18]. A similar statement to the case k = 1 of (2.6) can be found in [16].…”

Section: Convergence In Distributionmentioning

confidence: 52%

“…This is a very natural type of question with a long history; see for example [3,7,10,11,12,15,20]. Potential applications include wireless communications [3,18], ballistics [12], genomics [2], statistics [8], immunology [20], and topological data analysis [4,17].…”

Section: Introductionmentioning

confidence: 99%

“…We use R t,k and RZt,k mainly as stepping stones towards deriving results for R n,k and Rn,k respectively, but they are also of interest in their own right. Indeed, some of the literature [12,11,3,18] is concerned more with R t than with R n . We mention some notation used throughout.…”

Section: Introductionmentioning

confidence: 99%

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Random Euclidean coverage from within

Penrose¹

2021

Preprint

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Let X 1 , X 2 , . . . be i.i.d. random uniform points in a bounded domain A ⊂ R d with smooth boundary. Define the coverage threshold R n to be the smallest r such that A is covered by the balls of radius r centred on X 1 , . . . , X n . We obtain the limiting distribution of R n and also a strong law of large numbers for R n in the large-n limit. For example, if d = 3 and A has volume 1 and perimeter |∂A| then P[nπR 3 n −log n−2 log(log n) ≤ x] converges to exp(−2 −4 π 5/3 |∂A|e −2x/3 ), and (nπR 3 n )/(log n) → 1 almost surely. We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

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Section: Convergence In Distributionmentioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

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Random Euclidean coverage from within

Penrose¹

2021

Preprint

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“…To recast (23) into an equation for the generating function we multiply ( 23) by (L + 1)x L and sum over all L ⩾ 1. After a bit of algebra we arrive at a differential equation…”

Section: Full Counting Statisticsmentioning

confidence: 99%

“…The least dense sphere coverings initially attracted interest thanks to their beauty and intriguing connections with number theory [2][3][4][12][13][14][15][16]. There are also intriguing applications to genomics [17], ballistics [18], topological data analysis [19,20], stochastic optimization [21], immunology [22], wireless communications [23], ground states of interacting particles [24], and other subjects.…”

Section: Introductionmentioning

confidence: 99%

Random sequential covering

Krapivsky¹

2023

J. Stat. Mech.

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In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts increasing the coverage are accepted. A finite system eventually gets congested, and we study the statistics of congested configurations. For the covering of an interval by dimers, we determine the average number of deposited dimers, compute all higher cumulants, and establish the probabilities of reaching minimally and maximally congested configurations. We also investigate random covering by segments with ℓ sites and sticks. Covering an infinite substrate continues indefinitely, and we analyze the dynamics of random sequential covering of Z and R d .

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Random Euclidean coverage from within

Penrose

2023

Probab. Theory Relat. Fields

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Let $$X_1,X_2, \ldots $$ X 1 , X 2 , … be independent random uniform points in a bounded domain $$A \subset \mathbb {R}^d$$ A ⊂ R d with smooth boundary. Define the coverage threshold$$R_n$$ R n to be the smallest r such that A is covered by the balls of radius r centred on $$X_1,\ldots ,X_n$$ X 1 , … , X n . We obtain the limiting distribution of $$R_n$$ R n and also a strong law of large numbers for $$R_n$$ R n in the large-n limit. For example, if A has volume 1 and perimeter $$|\partial A|$$ | ∂ A | , if $$d=3$$ d = 3 then $$\mathbb {P}[n\pi R_n^3 - \log n - 2 \log (\log n) \le x]$$ P [ n π R n 3 - log n - 2 log ( log n ) ≤ x ] converges to $$\exp (-2^{-4}\pi ^{5/3} |\partial A| e^{-2 x/3})$$ exp ( - 2 - 4 π 5 / 3 | ∂ A | e - 2 x / 3 ) and $$(n \pi R_n^3)/(\log n) \rightarrow 1$$ ( n π R n 3 ) / ( log n ) → 1 almost surely, and if $$d=2$$ d = 2 then $$\mathbb {P}[n \pi R_n^2 - \log n - \log (\log n) \le x]$$ P [ n π R n 2 - log n - log ( log n ) ≤ x ] converges to $$\exp (- e^{-x}- |\partial A|\pi ^{-1/2} e^{-x/2})$$ exp ( - e - x - | ∂ A | π - 1 / 2 e - x / 2 ) . We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

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Coverage of Random Discs Driven by a Poisson Point Process

Cited by 3 publications

References 7 publications

Random Euclidean coverage from within

Random Euclidean coverage from within

Random sequential covering

Random Euclidean coverage from within

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