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December 2016 The Boolean model in the Shannon regime: three thresholds and related asymptotics
Venkat Anantharam, François Baccelli
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J. Appl. Probab. 53(4): 1001-1018 (December 2016).

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.

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Venkat Anantharam. François Baccelli. "The Boolean model in the Shannon regime: three thresholds and related asymptotics." J. Appl. Probab. 53 (4) 1001 - 1018, December 2016.

Information

Published: December 2016
First available in Project Euclid: 7 December 2016

zbMATH: 1356.60079
MathSciNet: MR3581237

Subjects:
Primary: 60G55 , 94A15
Secondary: 60D05 , 60F10

Keywords: Boolean model , high-dimensional stochastic geometry , information theory , large deviations theory , point process

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 4 • December 2016
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