### Secrecy coverage in two dimensions

Amites Sarkar

#### 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-8π2ypλ(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.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 1-12.

Dates
First available in Project Euclid: 8 March 2016

https://projecteuclid.org/euclid.aap/1457466152

Mathematical Reviews number (MathSciNet)
MR3473564

Zentralblatt MATH identifier
1338.60028

Subjects
Secondary: 05D40: Probabilistic methods

Keywords
Poisson process coverage

#### Citation

Sarkar, Amites. Secrecy coverage in two dimensions. Adv. in Appl. Probab. 48 (2016), no. 1, 1--12. https://projecteuclid.org/euclid.aap/1457466152

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