Details

Autor(en) / Beteiligte

• Peres, Yuval
• Sinclair, Alistair
• Sousi, Perla
• Stauffer, Alexandre

Titel

Mobile geometric graphs: detection, coverage and percolation

Ist Teil von

• Probability theory and related fields, 2013-06, Vol.156 (1-2), p.273-305

Ort / Verlag

Berlin/Heidelberg: Springer-Verlag

Erscheinungsjahr

2013

Quelle

Business Source Ultimate

Beschreibungen/Notizen

• 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 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.