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

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^{nρ_n}, 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.