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October 2015 Space–time percolation and detection by mobile nodes
Alexandre Stauffer
Ann. Appl. Probab. 25(5): 2416-2461 (October 2015). DOI: 10.1214/14-AAP1052

Abstract

Consider the model where nodes are initially distributed as a Poisson point process with intensity $\lambda$ over $\mathbb{R}^{d}$ and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance $r$ of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of $\mathbb{R}^{d}$, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of $\lambda$ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for $\lambda$ since, for small enough $\lambda$, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.

Citation

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Alexandre Stauffer. "Space–time percolation and detection by mobile nodes." Ann. Appl. Probab. 25 (5) 2416 - 2461, October 2015. https://doi.org/10.1214/14-AAP1052

Information

Received: 1 March 2012; Revised: 1 July 2014; Published: October 2015
First available in Project Euclid: 30 July 2015

zbMATH: 1331.82046
MathSciNet: MR3375880
Digital Object Identifier: 10.1214/14-AAP1052

Subjects:
Primary: 82C43
Secondary: 60G55, 60J65, 60K35, 82C21

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 5 • October 2015
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