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Secrecy coverage in two dimensions

Part of: Extremal combinatorics Geometric probability and stochastic geometry

Published online by Cambridge University Press:  24 March 2016

Amites Sarkar
Amites Sarkar*
Affiliation:
Western Washington University
*
* Postal address: Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA. Email address: amites.sarkar@wwu.edu
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Abstract

Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2y ≤ pλ(n) ≤ e-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.

Keywords

Poisson processcoverage

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05D40: Probabilistic methods
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 1 - 12
Copyright
Copyright © Applied Probability Trust 2016 

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