[Paper Review] 4 NON-OPTIMALITY OF CONSTANT RADII IN HIGH DIMENSIONAL CONTINUUM PERCOLATION

Jean-Baptiste Gouéré, Régine Marchand|arXiv (Cornell University)|Sep 25, 2014

Stochastic processes and statistical mechanics12 references4 citations

TL;DR

This paper proves that in high-dimensional continuum percolation, constant radii do not minimize the critical covered volume—contrary to long-standing conjectures. Using a two-type Boolean model with alternating radii 1 and ρ, the authors show that for sufficiently high dimensions, random or mixed radii yield lower critical covered volumes than deterministic radii, demonstrating the non-optimality of constant radii via stochastic domination and asymptotic analysis of the critical intensity.

ABSTRACT

Consider a Boolean model $\\Sigma$ in $\\R^d$. The centers are given by a homogeneous Poisson point process with intensity $\\lambda$ and the radii of distinct balls are i.i.d.\\ with common distribution $\ u$. The critical covered volume is the proportion of space covered by $\\Sigma$ when the intensity $\\lambda$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $\ u$ is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.

Motivation & Objective

• To challenge the widely held conjecture that constant radii minimize the critical covered volume in high-dimensional continuum percolation.
• To investigate whether random or mixed radii distributions can yield lower critical covered volumes than deterministic radii.
• To rigorously establish the non-optimality of constant radii in high dimensions using stochastic domination and asymptotic analysis.
• To analyze the behavior of the normalized critical intensity and its dependence on the radius distribution in high-dimensional spaces.

Proposed method

• Construct a two-type Boolean model in ℝ^d with balls of radius 1 and ρ, where centers are given by a Poisson point process with intensity λ.
• Define a site process on a hierarchical grid structure (W_{a,n}) to track paths of alternating radii across levels.
• Use oriented percolation on a lattice to stochastically dominate the path existence in the two-type Boolean model.
• Establish stochastic domination by Bernoulli oriented percolation with parameter p > p_c, ensuring positive probability of infinite open paths.
• Analyze the asymptotic behavior of the critical intensity λ_d^c(μ_d) as d → ∞, using bounds derived from the two-type model.
• Apply scaling invariance of the critical covered volume to compare different radius distributions independently of scale.

Experimental results

Research questions

• RQ1 Is the critical covered volume minimized when all radii are constant (i.e., Dirac measure) in high-dimensional continuum percolation?
• RQ2 Can a two-type Boolean model with alternating radii 1 and ρ achieve a lower critical covered volume than a single-radius model in high dimensions?
• RQ3 What is the asymptotic behavior of the normalized critical intensity as dimension d → ∞ for mixed-radius distributions?
• RQ4 Does the critical covered volume remain bounded away from zero for mixed-radius measures, and can it be strictly smaller than for constant radii?
• RQ5 Is the critical intensity λ_d^c(ν) minimized when ν is a Dirac measure, or can other distributions yield lower values in high dimensions?

Key findings

• The critical covered volume is not minimized by constant radii in high dimensions, contradicting the conjecture that Dirac-distributed radii are optimal.
• For any ρ > 0, there exists a dimension d_0 such that for all d > d_0, the critical covered volume for a two-type model with radii 1 and ρ is strictly less than that for a constant-radius model.
• The normalized critical intensity ̃λ_d^c(μ_d) for a two-type measure μ_d with radii 1 and ρ satisfies lim_{d→∞} (1/d) ln(λ_d^c(μ_d)) = ln(κ^c_ρ), where κ^c_ρ is the critical threshold for the two-type model.
• The critical covered volume for the two-type model can be made arbitrarily close to 1 in high dimensions, while remaining bounded away from 1 for the constant-radius case.
• The critical covered volume is strictly smaller for mixed radii than for constant radii when d is sufficiently large, proving non-optimality of constant radii.