### Abstract

Original language | English |
---|---|

Pages (from-to) | 273-305 |

Number of pages | 33 |

Journal | Probability Theory and Related Fields |

Volume | 156 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jun 2013 |

### Fingerprint

### Cite this

*Probability Theory and Related Fields*,

*156*(1-2), 273-305. https://doi.org/10.1007/s00440-012-0428-1

**Mobile geometric graphs : Detection, coverage and percolation.** / Peres, Yuval; Sinclair, Alistair; Sousi, Perla; Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 156, no. 1-2, pp. 273-305. https://doi.org/10.1007/s00440-012-0428-1

}

TY - JOUR

T1 - Mobile geometric graphs

T2 - Detection, coverage and percolation

AU - Peres, Yuval

AU - Sinclair, Alistair

AU - Sousi, Perla

AU - Stauffer, Alexandre

PY - 2013/6

Y1 - 2013/6

N2 - 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 Rd 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.

AB - 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 Rd 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.

UR - http://www.scopus.com/inward/record.url?scp=84878112315&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s00440-012-0428-1

U2 - 10.1007/s00440-012-0428-1

DO - 10.1007/s00440-012-0428-1

M3 - Article

VL - 156

SP - 273

EP - 305

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -