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Secrecy coverage in two dimensions

Part of: Geometric probability and stochastic geometry Extremal combinatorics

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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References

[1]Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25. CrossRefGoogle Scholar
[2]Balister, P., Bollobás, B. and Sarkar, A. (2009). Percolation, connectivity, coverage and colouring of random geometric graphs. In Handbook of Large-Scale Random Networks, Springer, Berlin, pp. 117–142. Google Scholar
[3]Balister, P., Bollobás, B., Sarkar, A. and Walters, M. (2010). Sentry selection in wireless networks. Adv. Appl. Prob. 42, 1–25. CrossRefGoogle Scholar
[4]Gilbert, E. N. (1965). The probability of covering a sphere with N circular caps. Biometrika 52, 323–330. CrossRefGoogle Scholar
[5]Haenggi, M. (2008). The secrecy graph and some of its properties. In Proc. IEEE Internat. Symp. Information Theory, (Toronto, Canada), IEEE, New York, pp. 539–543. Google Scholar
[6]Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York. Google Scholar
[7]Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156, 83–118. CrossRefGoogle Scholar
[8]Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press. CrossRefGoogle Scholar
[9]Moran, P. A. P. and Fazekas de St Groth, S. (1962). Random circles on a sphere. Biometrika 49, 389–396. CrossRefGoogle Scholar
[10]Penrose, M. (2003). Random Geometric Graphs. Oxford University Press. CrossRefGoogle Scholar
[11]Sarkar, A. and Haenggi, M. (2013). Secrecy coverage. Internet Math. 9, 199–216. CrossRefGoogle Scholar