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.