Continuum percolation in high dimensions

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.
Type de document :
Pré-publication, Document de travail
26 pages, 3 figures. 2011
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https://hal.archives-ouvertes.fr/hal-00617809
Contributeur : Régine Marchand <>
Soumis le : mercredi 20 mars 2013 - 10:15:42
Dernière modification le : lundi 22 février 2016 - 13:48:16
Document(s) archivé(s) le : dimanche 2 avril 2017 - 15:28:21

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Boolean-high-dimension-3.pdf
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  • HAL Id : hal-00617809, version 2
  • ARXIV : 1108.6133

Citation

Jean-Baptiste Gouéré, Regine Marchand. Continuum percolation in high dimensions. 26 pages, 3 figures. 2011. 〈hal-00617809v2〉

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