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New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Bibliography
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Bibliography


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 n1, where the n-th model features a Poisson point process in n of intensity enρn with ρnρ as n, and balls of independent and identically distributed radii distributed like X¯nn, with X¯n satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: τd the degree threshold; τp the percolation threshold; and τv the volume fraction threshold; such that asymptotically as n tends to infinity, in a sense made precise in the paper: (i) for ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for ρ>τv, 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.