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 .
The -th model features a Poisson point process in
of intensity
and balls of independent and identically distributed
radii distributed like . Assume that
as , and
that satisfies a large deviations principle.
It is shown that there then exist three deterministic thresholds:
the degree threshold; the percolation probability threshold;
and the volume fraction threshold, such that
asymptotically as tends to infinity,
we have the following features.
(i) For , almost every point is isolated, namely its ball
intersects no other ball;
(ii) for ,
the mean number of balls intersected by a typical ball
converges to infinity and
nevertheless there is no percolation;
(iii) for ,
the volume fraction is 0 and nevertheless percolation occurs;
(iv) for ,
the mean number of balls intersected by a typical ball
converges to infinity and
nevertheless the volume fraction is 0;
(v) for , 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.