The Annals of Probability

Nonoptimality of constant radii in high dimensional continuum percolation

Jean-Baptiste Gouéré and Régine Marchand

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Consider a Boolean model $\Sigma$ in $\mathbb{R}^{d}$. 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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 307-323.

Received: March 2013
Revised: September 2014
First available in Project Euclid: 2 February 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Percolation continuum percolation Boolean model


Gouéré, Jean-Baptiste; Marchand, Régine. Nonoptimality of constant radii in high dimensional continuum percolation. Ann. Probab. 44 (2016), no. 1, 307--323. doi:10.1214/14-AOP974.

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