Section: New Results

Information Theory: Boolean model in the Shannon Regime

In a paper accepted for publication in the Journal of Applied Probability, F. Baccelli and V. Anantharam consider a family of Boolean models, indexed by integers $n\ge 1$. The $n$-th model features a Poisson point process in ${ℝ}^n$ of intensity $e^{n{\rho }_n}$ and balls of independent and identically distributed radii distributed like ${\overline{X}}_n\sqrt{n}$. Assume that ${\rho }_n\to \rho$ as $n\to \infty$, and that ${\overline{X}}_n$ satisfies a large deviations principle. It is shown that there then exist three deterministic thresholds: ${\tau }_d$ the degree threshold; ${\tau }_p$ the percolation probability threshold; and ${\tau }_v$ the volume fraction threshold, such that asymptotically as $n$ tends to infinity, we have the following features. (i) For $\rho <{\tau }_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for ${\tau }_d<\rho <{\tau }_p$, the mean number of balls intersected by a typical ball converges to infinity and nevertheless there is no percolation; (iii) for ${\tau }_p<\rho <{\tau }_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for ${\tau }_d<\rho <{\tau }_v$, the mean number of balls intersected by a typical ball converges to infinity 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 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.