  Consider a family of Boolean models, indexed by integers $n \ge 1$, where the
$n$-th model features a Poisson point process in ${\mathbb{R}}^n$ 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 $\bar X_n
\sqrt{n}$, with $\bar X_n$ 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.
