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## Section: New Results

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

In  we consider a family of Boolean models, indexed by integers $n\ge 1$, where the $n$-th model features a Poisson point process in ${ℝ}^{n}$ of intensity ${e}^{n{\rho }_{n}}$ with ${\rho }_{n}\to \rho$ as $n\to \infty$, and balls of independent and identically distributed radii distributed like ${\overline{X}}_{n}\sqrt{n}$, with ${\overline{X}}_{n}$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: ${\tau }_{d}$ the degree threshold; ${\tau }_{p}$ the percolation threshold; and ${\tau }_{v}$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\rho <{\tau }_{d}$, almost every point is isolated, namely its ball intersects no other ball; (ii) for ${\tau }_{d}<\rho <{\tau }_{p}$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for ${\tau }_{p}<\rho <{\tau }_{v}$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for ${\tau }_{d}<\rho <{\tau }_{v}$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\rho >{\tau }_{v}$, 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.