Non-optimality of constant radii in high dimensional continuum percolation

TL;DR

This paper proves that in high dimensions, the critical covered volume in continuum percolation is not minimized by constant radii, challenging previous heuristic assumptions.

Contribution

The authors demonstrate that constant radii do not minimize the critical covered volume in high-dimensional continuum percolation, providing a rigorous mathematical result.

Findings

Constant radii are not optimal in high dimensions.

Critical covered volume is not minimized by Dirac measures.

High-dimensional behavior differs from low-dimensional heuristics.

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 $\nu$. 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 $\nu$ is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.