Detection, coverage and percolation in dynamic Boolean models with random radii based on $\alpha$-stable processes

TL;DR

This paper analyzes the tail behavior of detection, coverage, and discovery times in a dynamic continuum network with nodes moving via $ ext{α}$-stable processes and equipped with random detection radii, revealing exponential decay rates influenced by stability and radii.

Contribution

It extends previous results on mobile networks by incorporating $ ext{α}$-stable processes and random radii, providing new tail estimates and analyzing their effects on network connectivity.

Findings

Tail decay rates are exponential, with constants affected by stability index and radii.

Heat-kernel bounds and multiscale analysis are used to control correlations.

Stability index and radii influence network detection and coverage times.

Abstract

We consider a dynamic network in continuum time and space in which nodes, with initial locations given by a Poisson point process, move according to i.i.d. isotropic $\alpha$-stable processes. Each node is additionally equipped with an i.i.d. detection radius. Inspired by corresponding results by Peres et. al. on mobile networks based on Brownian sausages with fixed width, we investigate the tail behaviour of three stopping times: The detection time of the first discovery of a designated node, the first coverage of an entire set, and the first discovery of a node by the infinite connected component of the system. Broadly speaking, we discover that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates. The proofs rest on heat-kernel bounds for the underlying L\'{e}vy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system.