Title

Motivated by mobile wireless networks we consider a random graph over R^2, where nodes perform independent Brownian motions and edges are kept between pairs of nodes within distance r of each other. Combining ideas from stochastic geometry, coupling and multi-scale analysis, we obtain precise asymptotics for detection (the time until a given target is within distance r to some node of the graph) and percolation (the time until a given node belongs to the infinite connected component of the graph). (This is a joint work with Yuval Peres, Alistair Sinclair and Perla Sousi.)