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The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in ${\mathbb{R}}^n$ of intensity $e^{n ρ_n}$ with $ρ_n \to ρ$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $τ_d$ the degree threshold; $τ_p$ the percolation threshold; and $τ_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $ρ< τ_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $τ_d< ρ< τ_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $τ_p< ρ< τ_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $τ_d< ρ< τ_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $ρ> τ_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry.