Continuum percolation in high dimensions

TL;DR

This paper investigates the behavior of the critical covered volume in high-dimensional Boolean models, showing that the minimality conjecture for Dirac measures does not hold in high dimensions.

Contribution

It proves that in high dimensions, the critical covered volume is not minimized by Dirac measures, contrasting previous heuristic expectations.

Findings

Critical covered volume does not attain its minimum at Dirac measures in high dimensions.

Geometrical dependencies persist in high dimensions, unlike in constant radii models.

Asymptotic analysis reveals differences from low-dimensional behavior.

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 at least in high dimension. To establish this result we study the asymptotic behaviour, as $d$ tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension.