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Mobile geometric graphs: detection, coverage and percolation
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  • Published: 11 May 2012

Mobile geometric graphs: detection, coverage and percolation

  • Yuval Peres1,
  • Alistair Sinclair2,
  • Perla Sousi3 &
  • …
  • Alexandre Stauffer4 

Probability Theory and Related Fields volume 156, pages 273–305 (2013)Cite this article

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  • 26 Citations

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Abstract

We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in \({\mathbb{R}^d}\) with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.

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Author information

Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Yuval Peres

  2. Computer Science Division, University of California, Berkeley, CA, 94720-1776, USA

    Alistair Sinclair

  3. University of Cambridge, Cambridge, UK

    Perla Sousi

  4. Computer Science Division, University of California, Berkeley, CA, USA

    Alexandre Stauffer

Authors
  1. Yuval Peres
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  2. Alistair Sinclair
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  3. Perla Sousi
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  4. Alexandre Stauffer
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Corresponding author

Correspondence to Perla Sousi.

Additional information

A. Sinclair research supported in part by NSF grant CCF-0635153 and by a UC Berkeley Chancellor’s Professorship.

A. Stauffer research supported by a Fulbright/CAPES scholarship and NSF grants CCF-0635153 and DMS-0528488. Part of this work was done while the author was doing a summer internship at Microsoft Research, Redmond, WA.

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Cite this article

Peres, Y., Sinclair, A., Sousi, P. et al. Mobile geometric graphs: detection, coverage and percolation. Probab. Theory Relat. Fields 156, 273–305 (2013). https://doi.org/10.1007/s00440-012-0428-1

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  • Received: 28 June 2011

  • Revised: 23 February 2012

  • Accepted: 27 March 2012

  • Published: 11 May 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0428-1

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Keywords

  • Poisson point process
  • Brownian motion
  • Coupling
  • Minkowski dimension

Mathematics Subject Classification

  • Primary 82C43
  • Secondary 60G55
  • 60D05
  • 60J65
  • 60K35
  • 82C21
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