2014/08/06 by Anantharam, Venkat, Baccelli, François
#60D05 #94A15 #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Information Theory (cs.IT) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1408.1338
Consider a family of Boolean models, indexed by integers n ≥ 1, where the n-th model features a Poisson point process in ℝn of intensity en ρn with ρn → ρ as n → ∞, and balls of independent and identically distributed radii distributed like Xn √(n), with Xn 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.