Journal of Applied Probability

The Boolean model in the Shannon regime: three thresholds and related asymptotics

Venkat Anantharam and François Baccelli

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Abstract

Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅nn. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.

Article information

Source
J. Appl. Probab., Volume 53, Number 4 (2016), 1001-1018.

Dates
First available in Project Euclid: 7 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1481132832

Mathematical Reviews number (MathSciNet)
MR3581237

Zentralblatt MATH identifier
1356.60079

Subjects
Primary: 60G55: Point processes 94A15: Information theory, general [See also 62B10, 81P94]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F10: Large deviations

Keywords
Point process Boolean model high-dimensional stochastic geometry information theory large deviations theory

Citation

Anantharam, Venkat; Baccelli, François. The Boolean model in the Shannon regime: three thresholds and related asymptotics. J. Appl. Probab. 53 (2016), no. 4, 1001--1018. https://projecteuclid.org/euclid.jap/1481132832


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