## Journal of Applied Probability

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

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

Venkat Anantharam and François Baccelli

#### Abstract

Consider a family of Boolean models, indexed by integers *n*≥1. The *n*th model features a Poisson point process in ℝ^{n} of intensity e^{{nρn}}, and balls of independent and identically distributed radii distributed like X̅_{n}√*n*. Assume that ρ_{n}→ρ as *n*→∞, and that X̅_{n} satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τ_{d}, the degree threshold, τ_{p}, the percolation probability threshold, and τ_{v}, the volume fraction threshold, such that, asymptotically as *n* tends to ∞, we have the following features. (i) For ρ<τ_{d}, almost every point is isolated, namely its ball intersects no other ball; (ii) for τ_{d}<ρ<τ_{p}, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τ_{p}<ρ<τ_{v}, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τ_{d}<ρ<τ_{v}, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τ_{v}, 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