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

Peter Gracar, Benedikt Jahnel, Lukas Lüchtrath, Anh Duc Vu

TL;DR

It is discovered that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates of the otherwise exponential decay rates of the dynamic network.

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 $α$-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 Lévy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system.

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

TL;DR

It is discovered that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates of the otherwise exponential decay rates of the dynamic network.

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 -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 Lévy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system.
Paper Structure (22 sections, 18 theorems, 152 equations, 3 figures)

This paper contains 22 sections, 18 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Detection times
  4. Coverage time
  5. Percolation time
  6. Discussion and outlook
  7. Proofs
  8. Proofs for detection times
  9. Prerequisite results on $\alpha$-stable processes
  10. Heat-kernel bounds, parabolic Harnack inequality and Hölder inequality
  11. Escape and hitting probabilities
  12. Proofs for coverage times
  13. Proof of percolation time
  14. Step 1: Construction and discretisation.
  15. Step 2: The discretised model is supercritical.
  16. ...and 7 more sections

Key Result

Theorem 2.1

Consider the dynamic Boolean model with $\alpha\in(0,2]$ and $d>\alpha$.

Figures (3)

  • Figure 1: A realisation of the dynamic Boolean model $\mathscr{G}$ based on the standard isotropic $\alpha$-stable process with $\alpha= 1.5$. Colour indicates time. As a Boolean model of a stationary Poisson point process, $\mathscr{G}_t$ has the same distribution as $\mathscr{G}_0$ for any $t\geq0$. Plotted is $(\mathscr{G}_t)_{t\in[0,10]}$. The starting configuration is depicted by red crosses.
  • Figure 2: The spatial multi-scale recursion. Note that the difference $V_{j-1}-V_j$ is proportional to the value $v_{j-1}$ by \ref{['eq:v_jm_j']} and \ref{['eq:v_geometric']}, allowing us to apply \ref{['thrm:mixing']}.
  • Figure 3: The temporal multi-scale recursion.

Theorems & Definitions (41)

  • Theorem 2.1: Detection time
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4: Coverage time
  • Theorem 2.5: Percolation time
  • Remark 2.6
  • proof : Proof of \ref{['lem:detection-time-is-void-probability']}
  • Lemma 3.1: Volume bounds
  • Lemma 3.2: Volume asymptotics for random radii
  • proof
  • ...and 31 more