The Boolean model in the Shannon regime: three thresholds and related asymptotics

Abstract

Consider a family of Boolean models, indexed by integers n ≥ 1, where the n-th model features a Poisson point process in R n of intensity e nρn with ρ n → ρ as n → ∞, and balls of independent and identically distributed radii distributed likeX n √ n, withX n satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: τ d the degree threshold; τ p the percolation threshold; and τ v the volume fraction threshold; such that asymptotically as n tends to infinity, in a sense made precise in the paper: (i) for ρ < τ d , almost every point is isolated, namely its ball intersects no other ball; (ii) for τ d < ρ < τ p , almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for τ p < ρ < τ v , the volume fraction is 0 and nevertheless percolation occurs; (iv) for τ d < ρ < τ v , almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for ρ > τ v , the whole space 1 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.

Reach of repulsion for determinantal point processes in high dimensions

“…Poisson-Boolean models in the Shannon regime were studied in [1], and the degree threshold results can be extended to Laguerre-Gaussian DPPs using Proposition 4.1.…”

“…(1.1) This justifies the interest in considering this regime where the intensities grow exponentially with the dimension and distances grow with the square root of the dimension. This regime also naturally arises in information theory, and following [1] we call it the Shannon regime. In this paper we study the effect of repulsion in this regime and quantify the range and strength at which DPPs asymptotically exhibit repulsion between points.…”

“…An application of these results is presented in Section 5. We show that some threshold results of [1] for Poisson-Boolean models can be extended to generalized Laguerre-Gaussian DPP Boolean models in the Shannon regime using the rates of convergence computed for these DPPs. Finally, concluding remarks and open questions are stated in Section 6.…”

High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas

“…While the scaling of the direct (occupied space) percolation threshold is under analytical control, lim d→∞ Φ c ∼ 2 −d [18,36] (see also Refs. 17 and 19), little is formally known about the void (vacant space) percolation threshold, Φ p , other than that Φ c provides a lower bound for it,…”

“…Despite the superficial similarities between the two phenomena, our understanding of them differs markedly. The asymptotic, highdimensional scaling of the direct percolation threshold has been physically argued [17], rigorously proven [18], and numerically assessed up to d = 11 [19]. By contrast, reports of percolation thresholds for d > 3 are limited [20], and the asymptotic high-dimensional scaling of that threshold remains an open question.…”