## Advances in Applied Probability

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

### Secrecy coverage in two dimensions

#### Abstract

Working in the infinite plane **R**^{2}, 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 *A*_{λ}, 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 *p*_{λ}(*n*) that this disc is covered by *A*_{λ}? We prove that if λ^{3}*n*log*n* = *y* then, for sufficiently large *n*, e^{-8π2y} ≤ *p*_{λ}(*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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR3473564

**Zentralblatt MATH identifier**

1338.60028

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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