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 , where the -th model features a Poisson point process in of intensity with as , and balls of independent and identically distributed radii distributed like , with satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: the degree threshold; the percolation threshold; and the volume fraction threshold; such that asymptotically as tends to infinity, in a sense made precise in the paper: (i) for , almost every point is isolated, namely its ball intersects no other ball; (ii) for , almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for , the volume fraction is 0 and nevertheless percolation occurs; (iv) for , almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for , 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.