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

Consider a family of Boolean models, indexed by integers n ≥ 1, where the n-th model features a Poisson point process in R of intensity en with ρn → ρ as n → ∞, and balls of independent and identically distributed radii distributed like X̄n √ n, with X̄n satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: τd the degree threshold; τp the percolation threshold; and τv the volume fraction threshold; such that asymptotically as n tends to infinity, in a sense made precise in the paper: (i) for ρ < τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd < ρ < τp, almost every ball intersects an infinite number of balls 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, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for ρ > τv, the whole space