Space-time percolation and detection by mobile nodes

TL;DR

This paper analyzes a space-time percolation model where mobile nodes detect a moving target, establishing a phase transition in detection probability based on node density and using fractal percolation techniques.

Contribution

It introduces a novel space-time percolation framework with mobile nodes and proves a phase transition in detection capability depending on node density.

Findings

High node density ensures almost sure detection of the target.

Low node density allows the target to evade detection with positive probability.

Fractal percolation and multi-scale analysis are key tools in the proof.

Abstract

Consider the model where nodes are initially distributed as a Poisson point process with intensity $\lambda$ over $\mathbb{R}^d$ and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance $r$ of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of $\mathbb{R}^d$, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of $\lambda$ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for $\lambda$ since, for small enough $\lambda$, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.