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The Boolean model in the Shannon regime: three thresholds and related asymptotics. (English) Zbl 1356.60079

Summary: Consider a family of Boolean models, indexed by integers \(n\geq 1\). The \(n\)-th model features a Poisson point process in \(\mathbb{R}^{n}\) of intensity \(e^{n\rho_{n}}\), and balls of independent and identically distributed radii distributed like \(\bar{X}_{n}\sqrt{n}\). Assume that \(\rho_{n} \to \rho\) as \(n\to \infty\), and that \(\bar{X}_{n}\) satisfies a large deviations principle. We show that there then exist the three deterministic thresholds \(\tau_{d}\), the degree threshold, \(\tau_{p}\), the percolation probability threshold, and \(\tau_{v}\), the volume fraction threshold, such that, asymptotically as \(n\) tends to \(\infty\) we have the following features. (i) For \(\rho < \tau_{d}\), almost every point is isolated, namely its ball intersects no other ball; (ii) for \(\tau_{d}<\rho <\tau_{p}\), the mean number of balls intersected by a typical ball converges to \(\infty\) and nevertheless there is no percolation; (iii) for \(\tau_{p}<\rho <\tau_{v}\), the volume fraction is \(0\) and nevertheless percolation occurs; (iv) for \(\tau_{d}<\rho <\tau_{v}\), the mean number of balls intersected by a typical ball converges to \(\infty\) and nevertheless the volume fraction is \(0\); (v) for \(\rho >\tau_{v}\), 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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
60F10 Large deviations
94A15 Information theory (general)
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