Speaker:
Alexandre Stauffer
Title:
Phase transitions in the detection of a target by mobile nodes
Abstract:
We consider a Poisson point process of intensity \lambda in R^d. We denote
the points as nodes and let each node move as an independent Brownian
motion. Consider a target particle that is initially placed at the origin
at time 0 and can move according to any continuous function. We say that
the target is detected at time t if there exists at least one node within
distance 1 of the target at time t. We establish the existence of a
critical intensity \lambda_c so that
if \lambda>\lambda_c, the target will eventually be detected almost surely.
In the proof we use coupling to establish local mixing of the nodes
and multi-scale analysis to see this problem as a fractal percolation process.
We will discuss some extensions and open problems.