Section: New Results

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

In [4] 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.