The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

TL;DR

This paper analyzes high-dimensional Boolean models, identifying three critical thresholds that determine connectivity, percolation, and coverage, revealing complex phase transitions in asymptotic regimes relevant to stochastic geometry and information theory.

Contribution

It introduces a precise asymptotic analysis of Boolean models in high dimensions, establishing three thresholds and their implications for connectivity and coverage.

Findings

Identifies three deterministic thresholds: degree, percolation, and volume fraction.

Describes phase transitions in connectivity and coverage as dimension grows.

Provides asymptotic regimes with distinct geometric and percolation properties.

Abstract

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.