[Paper Review] The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Venkat Anantharam, François Baccelli|arXiv (Cornell University)|Aug 6, 2014

Point processes and geometric inequalities17 references4 citations

TL;DR

This paper analyzes the asymptotic behavior of a high-dimensional Boolean model in the Shannon regime, where Poisson point processes with intensity $ e^{n ho_n} $ and ball radii scaled as $ \bar{X}_n\sqrt{n} $ are studied. It identifies three distinct thresholds—degree, percolation, and volume fraction—using large deviations principles, showing phase transitions in connectivity, coverage, and infinite cluster formation as $ \rho $ crosses these critical values.

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 $e^{n \ ho_n}$ with $\ ho_n \ o \ ho$ as $n \ o \\infty$, 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: $\ au_d$ the degree threshold; $\ au_p$ the percolation threshold; and $\ au_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\ ho < \ au_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $\ au_d< \ ho< \ au_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $\ au_p< \ ho< \ au_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $\ au_d< \ ho< \ au_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\ ho > \ au_v$, 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.

Motivation & Objective

• To characterize the asymptotic behavior of a Boolean model in high-dimensional space ($ \mathbb{R}^n $) as $ n \to \infty $, under a scaling regime where intensity and radius scale with dimension.
• To identify and rigorously define three critical thresholds: the degree threshold $ \tau_d $, percolation threshold $ \tau_p $, and volume fraction threshold $ \tau_v $, based on the large deviations behavior of the radius distribution.
• To derive explicit representations of these thresholds as solutions to optimization problems involving the rate function of the underlying large deviations principle.
• To establish precise asymptotics for convergence rates near each threshold, particularly in the neighborhood of phase transitions.
• To extend prior deterministic results to a stochastic setting with random radii satisfying a large deviations principle, and to recover known results (e.g., for Gaussian radii) as special cases.

Proposed method

• Formalize the Boolean model in $ \mathbb{R}^n $ with a Poisson point process of intensity $ e^{n\rho_n} $, where $ \rho_n \to \rho $, and balls of radius $ \bar{X}_n\sqrt{n} $, with $ \bar{X}_n $ i.i.d. and satisfying a large deviations principle (LDP) with good, convex rate function $ I(\cdot) $.
• Define the three thresholds via asymptotic probabilistic behavior: $ \tau_d $ via mean number of intersecting grains in the Palm distribution, $ \tau_p $ via percolation probability, and $ \tau_v $ via coverage probability of the origin.
• Represent each threshold as the solution to a variational optimization problem involving the rate function $ I(\cdot) $, derived from Laplace's principle and large deviations asymptotics.
• Use the Gärtner-Ellis theorem to ensure the validity of the LDP and to derive exponential order statistics for the radius distribution.
• Apply Slivnyak’s theorem to analyze the Palm distribution, which allows the study of the typical grain’s interaction with the rest of the process.
• Derive explicit formulas for the thresholds in the case of Gaussian-distributed radii, with $ \bar{X}_n \sim \mathcal{N}(0,\sigma^2) $, and verify the ordering $ \tau_d < \tau_p < \tau_v $.

Experimental results

Research questions

• RQ1 What are the three critical thresholds—degree, percolation, and volume fraction—in the high-dimensional Boolean model under the Shannon regime?
• RQ2 How do these thresholds depend on the large deviations rate function of the radius distribution?
• RQ3 What is the asymptotic behavior of the system when the intensity parameter $ \rho $ lies in the intervals between these thresholds?
• RQ4 How do the convergence rates near the thresholds behave, and can they be quantified using large deviations theory?
• RQ5 What happens in the special case of Gaussian-distributed radii, and how do the thresholds compare to known results in information theory?

Key findings

• Three distinct thresholds exist: $ \tau_d < \tau_p < \tau_v $, with $ \tau_d $ marking the transition from finite to infinite expected number of intersecting grains, $ \tau_p $ from no percolation to percolation, and $ \tau_v $ from zero to full coverage.
• For Gaussian radii with variance $ \sigma^2 $, the volume fraction threshold is $ \tau_v = -\frac{1}{2}\log(2\pi e\sigma^2) - \frac{1}{2}(\log 4 - 1) $, corresponding to $ R_v = \sigma\sqrt{2} $.
• The degree threshold is $ \tau_d = -\frac{1}{2}\log(2\pi e\sigma^2) - \frac{1}{2}(\log(27/2) - 1) $, corresponding to $ R_d = \sigma\sqrt{3/2} $.
• The percolation threshold satisfies $ \tau_p = -\frac{1}{2}\log(2\pi e\sigma^2) - \frac{1}{2}(\log(c^2(1+c)^2) - c^2 + 1) $, where $ c \approx 1.24698 $ is the unique positive root of $ c^3 + c^2 - 2c - 1 = 0 $.
• In the deterministic case where $ \bar{X}_n = R^*_n \to R^* $, the percolation threshold coincides with the degree threshold, and both are $ \log 2 $ below the volume fraction threshold.
• Near each threshold, the paper derives precise asymptotics for convergence rates, showing exponential decay or growth in probabilities depending on the regime, validated via Laplace’s principle and large deviations.