Article
Secrecy coverage in two dimensions
Advances in Applied Probability
(2016)
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 process,
- Coverage
Disciplines
Publication Date
January 3, 2016
DOI
10.1017/apr.2015.3
Publisher Statement
COPYRIGHT: © Applied Probability Trust 2016
Citation Information
Amites Sarkar. "Secrecy coverage in two dimensions" Advances in Applied Probability Vol. 48 Iss. 1 (2016) p. 1 - 12 Available at: http://works.bepress.com/amites_sarkar/21/