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

Part of: Stochastic processes Communication, information Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  09 December 2016

Venkat Anantharam  and
François Baccelli
Venkat Anantharam*
Affiliation:
University of California, Berkeley
François Baccelli*
Affiliation:
The University of Texas at Austin
*
* Postal address: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA. Email address: ananth@eecs.berkeley.edu
** Postal address: Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1202, USA. Email address: baccelli@math.utexas.edu
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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 X̅n√n. 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.

Keywords

Point processBoolean modelhigh-dimensional stochastic geometryinformation theorylarge deviations theory

MSC classification

Primary: 60G55: Point processes 94A15: Information theory, general
Secondary: 60D05: Geometric probability and stochastic geometry 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1001 - 1018
Copyright
Copyright © Applied Probability Trust 2016 

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