The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Venkat Anantharam 1 François Baccelli 2, 3
2 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
Abstract : Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in ${\mathbb{R}}^n$ of intensity $e^{n \rho_n}$ with $\rho_n \to \rho$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $\tau_d$ the degree threshold; $\tau_p$ the percolation threshold; and $\tau_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\rho < \tau_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $\tau_d< \rho< \tau_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $\tau_p< \rho< \tau_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $\tau_d< \rho< \tau_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\rho > \tau_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry.
Liste complète des métadonnées

https://hal.inria.fr/hal-01259177
Contributeur : Alain Monteil <>
Soumis le : mercredi 20 janvier 2016 - 09:35:28
Dernière modification le : jeudi 11 janvier 2018 - 06:28:02

Identifiants

  • HAL Id : hal-01259177, version 1
  • ARXIV : 1408.1338

Collections

Citation

Venkat Anantharam, François Baccelli. The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics. Journal of Applied Probability, Applied Probability Trust, 2016, 53 (4), pp.1001 - 1018. 〈https://projecteuclid.org/euclid.jap/1481132832〉. 〈hal-01259177〉

Partager

Métriques

Consultations de la notice

151